Split (1)

train · 275k rows

id stringlengths 6 15	solution stringlengths 8 36.1k	answer stringlengths 0 543 ⌀	metadata dict	problem stringlengths 15 7.84k	tags listlengths 1 56	idea stringlengths 29 420	problem_embedding listlengths 2.56k 2.56k
ours_352	Let $ I $ be the incenter. By Pascal's Theorem on hexagon $ PBYXZC $, $ I $ lies on $ MN $. Consider $ \triangle BIZ $, which is isosceles. We have $ \angle MIB = \angle MBI = \angle PBY $, and this is equal to $ \angle ICB $. Hence $ MI $ is tangent to the circumcircle of triangle $ BIC $, as is $ NI $. So $ MN $ is the tangent. Since $ X $ is the desired circumcenter, $ I $ is the foot of the altitude from $ X $ to $ MN $. Now we can compute $ IX = BX = BC \cdot \frac{\sin \frac{1}{2} \angle A}{\sin A} = \frac{15}{\cos \frac{1}{2} \angle A} $. Standard computations give that $ \cos \frac{1}{2} \angle A = \frac{3}{\sqrt{13}} $, hence $ BX^2 = 325 $. $\boxed{325}$	325	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns-2.md" }	Let $ \triangle ABC $ be a triangle with $ AB = 26 $, $ AC = 28 $, $ BC = 30 $. Let $ X, Y, Z $ be the midpoints of arcs $ BC $, $ CA $, $ AB $ (not containing the opposite vertices) respectively on the circumcircle of $ \triangle ABC $. Let $ P $ be the midpoint of arc $ BC $ containing point $ A $. Suppose lines $ BP $ and $ XZ $ meet at $ M $, while lines $ CP $ and $ XY $ meet at $ N $. Find the square of the distance from $ X $ to $ MN $.	[ "/Mathematics/Geometry/InversiveGeometry/CirclePower", "/Mathematics/Geometry/PlaneGeometry/Arcs", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/Trigonometry/Angles", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions" ]	Apply Pascal's theorem to the hexagon PBYXZC to show the incenter I lies on MN, making MN a tangent to the BIC circle so the distance from X equals the radius.	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ours_353	Let $ y^{-1} = 1/y $ denote the inverse of $ y \pmod{p} $. We work in $\mathbb{Z}/p\mathbb{Z}$, or $\mathbb{F}_p$. The functional equation is equivalent to $ f(x) f(y) = f(xy) + f(x/y) $, since $ y^{p-2} = y^{-1} $ by Fermat's Little Theorem. Let $ P(x, y) $ be the assertion $ f(x) f(y) = f(xy) + f(x/y) $. I claim that all functions satisfying this equation are of the form $ f(x) = x^s + x^{-s} $ for integers $ s $. Firstly, the assertion $ P(1, 1) $ gives that $ f(1)^2 = 2f(1) $, so either $ f(1) = 0 $ or $ f(1) = 2 $. If $ f(1) = 0 $, then the assertion $ P(x, 1) $ gives $ 0 = 2f(x) \Rightarrow f(x) = 0 $ for all $ x $. Since we only care about the product of the nonzero values, we can ignore this case. So $ f(1) = 2 $. Lemma: $ 3 $ is a primitive root $(\bmod p)$. It suffices to show that $ 3^{2^{15}} \equiv -1 \pmod{p} $. This follows from \[ 3^{2^{15}} \equiv \left(\frac{3}{p}\right) = \left(\frac{p}{3}\right) = -1 \] by the Quadratic Reciprocity Law. Now, let $ k $ be a positive integer, and look at the assertion $ P(3^k, 3) $. This gives $ P(3^k) P(3) = P(3^{k+1}) + P(3^{k-1}) $. Let $ f(3) = g $, and find an $\omega$ such that $ f(3) = \omega + \omega^{-1} $. We can solve, using the quadratic formula, that $\omega = \frac{g + \sqrt{g^2 - 4}}{2}$. If $ g^2 - 4 $ is a perfect square $(\bmod p)$, then $\omega$ is in $\mathbb{F}_p$. We have another case, though, where $ g^2 - 4 $ is not a perfect square $(\bmod p)$. Let $ c = g^2 - 4 $. In this case, consider all numbers of the form $ a + b\sqrt{c} $, where $ 0 \leq a, b \leq p-1 $. Since $ c $ is not a square, all these numbers are distinct. Also, $(a + b\sqrt{c})(d + e\sqrt{c}) = ad + bec + (ae + bd)\sqrt{c}$, so these numbers are closed under multiplication. Since they are obviously also closed under addition, these numbers form a field of order $ p^2 $, so we'll just call this set of numbers $\mathbb{F}_{p^2}$. We'll work in this field from now on. Using $ f(1) = 2, f(3) = \omega + \omega^{-1} $, we can use induction and the previous assertion to show that $ f(3^k) = \omega^k + \omega^{-k} $. Choosing $ k = p-1 $, we get that $ 2 = f(1) = f(3^{p-1}) = \omega^{p-1} + \omega^{-(p-1)} $. Solving gives that $\omega^{p-1} = 1$, so $\omega$ satisfies the polynomial $ x^{p-1} - 1 $. Since the polynomial can have at most $ p-1 $ roots in $\mathbb{F}_{p^2}$, and the integers $ 1, 2, \ldots, p-1 $ are roots, $\omega$ must lie in $\mathbb{F}_p$, contradicting our assumption that $ g^2 - 4 $ is not a perfect square. So $ f(3) = \omega + \omega^{-1} $ for some $\omega \in \mathbb{F}_p$. Since $ 3 $ is a primitive root, we can find a $ k $ such that $ 3^k = \omega $, so $ f(3) = 3^k + 3^{-k} $ for some $ k $. Therefore, $ f(81) = 3^{4k} + 3^{-4k} $. These values are distinct for $ k = 0, 1, \ldots, \frac{p-1}{8} $, excluding the value $\frac{p-1}{16}$, so it suffices to compute \[ \prod_{0 \leq i \leq \frac{p-1}{8}, i \neq \frac{p-1}{16}} \left(3^{4i} + 3^{-4i}\right) = -2 \cdot \prod_{0 \leq i < \frac{p-1}{8}, i \neq \frac{p-1}{16}} \frac{\left(3^{8i} + 1\right)}{3^{4i}} \] To evaluate this sum, consider the polynomial \[ \prod_{0 \leq i < \frac{p-1}{8}, i \neq \frac{p-1}{16}} \frac{\left(3^{8i} - x\right)}{3^{4i}} \] For $ 0 \leq i < \frac{p-1}{8}, 3^{8i} $ are roots of this polynomial except when $ 3^{8i} = -1 $, this numerator simplifies to $\frac{1 - x^{\frac{p-1}{8}}}{x + 1}$, since it satisfies the same properties, and has the same number of roots. The denominator simplifies to $ 3^{\frac{(p-1)(p-5)}{8}} \equiv -1 \pmod{p} $. So to find our desired product, we must find the value of $\frac{1 - x^{\frac{p-1}{8}}}{x + 1}$ at $ x = -1 $. Applying L'Hopital's rule, this easily evaluates to $-\frac{p-1}{8}$. Therefore, our final product is \[ -2 \cdot \prod_{0 \leq i < \frac{p-1}{8}, i \neq \frac{p-1}{16}} \frac{\left(3^{8i} + 1\right)}{3^{4i}} = -2 \cdot \frac{p-1}{8} \cdot -1 = 16384 \] Thus, the remainder when $ N $ is divided by $ p $ is $\boxed{16384}$.	16384	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns-2.md" }	Let $ p = 2^{16} + 1 $ be a prime, and let $ S $ be the set of positive integers not divisible by $ p $. Let $ f: S \rightarrow \{0, 1, 2, \ldots, p-1\} $ be a function satisfying \[ f(x) f(y) \equiv f(xy) + f\left(xy^{p-2}\right) \pmod{p} \quad \text{and} \quad f(x+p) = f(x) \] for all $ x, y \in S $. Let $ N $ be the product of all possible nonzero values of $ f(81) $. Find the remainder when $ N $ is divided by $ p $.	[ "/Mathematics/Algebra/FieldTheory/ExtensionField", "/Mathematics/Algebra/FieldTheory/Field", "/Mathematics/Algebra/FieldTheory/FieldCharacteristic", "/Mathematics/Algebra/FieldTheory/FieldOrder", "/Mathematics/Algebra/FieldTheory/FiniteExtension", "/Mathematics/Algebra/FieldTheory/FiniteField", "/Mathematics/Algebra/FieldTheory/PrimeField", "/Mathematics/Algebra/FieldTheory/PrimeSubfield", "/Mathematics/Algebra/FieldTheory/PrimitiveElement", "/Mathematics/Algebra/NumberTheory/Congruences", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/PrimeNumbers", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/FermatsLittleTheorem", "/Mathematics/NumberTheory/Congruences/FunctionalCongruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/ModularInverse", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/PrimitiveRoot", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticReciprocityTheorem" ]	Show the functional equation forces f(x) to equal x^s + x^{-s} by using the inverse exponent via Fermat and a primitive root of p.	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ours_354	First, note that $(a, b) = (1, -1)$ is a solution. Let $ z = a - bi $. Adding the first equation to $-bi$ times the second equation yields $ z^3 + z^2 + z = -1 + 5i $. Factoring out $ z - (1+i) $ since $(1, -1)$ was a solution, we obtain $ z^2 + (2+i)z + (2+3i) = 0 $. Using Vieta's formulas, if the remaining solutions are $(a_1, a_2)$ and $(b_1, b_2)$, then we have: \[ \begin{aligned} a_1 + a_2 &= -2, \\ b_1 + b_2 &= 1, \\ a_1 a_2 - b_2 b_1 &= 2, \\ a_1 b_2 + a_2 b_1 &= -3. \end{aligned} \] From here, it is easy to derive $ a_1 b_1 + a_2 b_2 = (-2)(1) - (-3) = 1 $. Thus, $ C = 1 + 1(-1) = 0 $. Furthermore, $ A + B = a_1 a_2(1) + b_1 b_2(-1) = 2 $. Therefore, the value of $ A + B + C $ is $\boxed{2}$.	2	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns-2.md" }	Let $ S $ be the set of all pairs $(a, b)$ of real numbers satisfying $1+a+a^{2}+a^{3}=b^{2}(1+3a)$ and $1+2a+3a^{2}=b^{2}-\frac{5}{b}$. Find $A+B+C$, where \[ A=\prod_{(a, b) \in S} a, \quad B=\prod_{(a, b) \in S} b, \quad \text{and} \quad C=\sum_{(a, b) \in S} ab. \]	[ "/Mathematics/Algebra/AlgebraicEquations/CubicEquation", "/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations", "/Mathematics/Algebra/Polynomials/CubicEquation", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/PolynomialFactorization", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial", "/Mathematics/Algebra/Polynomials/VietasFormulas", "/Mathematics/Algebra/Products/Product" ]	Convert the system into a single polynomial by setting z = a - bi and combining the equations.	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ours_355	Let $ M_A $ be the midpoint of the minor arc $ BC $. Let $ a = BC, b = CA, c = AB $ and observe that $ 2a = b + c $. By Ptolemy's Theorem, we obtain \[ BC \cdot AM_A = (AB + AC) \cdot IM_A \] where we have used the fact that $ IM_A = IB = IC $. It follows that $ IM_A = \frac{1}{2} AM_A $, so $ I $ is the midpoint of $ \overline{AM_A} $. Now let the tangents to the circumcircle of $ \triangle ABC $ at $ M $ meet line $ AI $ at $ X $. Evidently, $ X $ lies on the circumcircle of $ \triangle IMO $, since $ \angle OMX = \angle OIX = 90^\circ $. So $ XD $ is also a tangent to the circumcircle of $ \triangle ABC $. Next, notice that $ \overline{DE} \parallel \overline{AM_A} $, as both lines are perpendicular to line $ OI $. So it follows that line $ EM $ bisects line $ AM_A $, meaning that $ E, I, M $ are collinear. That means $ E $ is the tangency point of the $ A $-mixtilinear incircle with the circumcircle of $ \triangle ABC $. Now let the $ A $-excircle be tangent to $ \overline{BC} $ at $ K $. We find that $ \angle CAK = \angle BAE $. So if we extend ray $ AK $ to meet the circumcircle at $ F $, we get that $ BE = CF $, $ BF = CE $. Let $ s = \frac{1}{2}(a + b + c) $. We have $ \frac{BK}{CK} = \frac{s-c}{s-b} $, from which it follows that $ \frac{CF}{BF} = \frac{c(s-b)}{b(s-c)} $. Hence $ 1000 \frac{BE}{CE} = 1000 \cdot \frac{CF}{BF} $, so the requested ratio is \[ 1000 \cdot \frac{c(s-b)}{b(s-c)} = 1000 \cdot \frac{7 \cdot 3}{9 \cdot 5} = 466 + \frac{2}{3} \] and the answer is $ \boxed{467} $.	467	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns-2.md" }	Let $ \triangle ABC $ be a triangle with circumcenter $ O $, incenter $ I $, and circumcircle $ \Gamma $. It is known that $ AB = 7 $, $ BC = 8 $, $ CA = 9 $. Let $ M $ denote the midpoint of the major arc $ \widehat{BAC} $ of $ \Gamma $, and let $ D $ denote the intersection of $ \Gamma $ with the circumcircle of $ \triangle IMO $ (other than $ M $). Let $ E $ denote the reflection of $ D $ over line $ IO $. Find the integer closest to $ 1000 \cdot \frac{BE}{CE} $.	[ "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Arcs", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Prove I is the midpoint of AM_A to identify E as the A‑mixtilinear incircle touchpoint, then use excircle tangency ratios to relate BE and CE.	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ours_356	The first step is to notice that $ 1 + p H_{n} \equiv \binom{p+n}{n} \pmod{p^2} $. This is true because \[ \binom{p+n}{n} = \frac{p+n}{n} \cdot \frac{p+n-1}{n-1} \cdots \frac{p+1}{1} = \left(1 + \frac{p}{n}\right)\left(1 + \frac{p}{n-1}\right) \cdots \left(1 + \frac{p}{1}\right) \equiv 1 + p H_{n} \pmod{p^2}. \] Let \[ S = \sum_{n=1}^{p-1} H_{n} \cdot 4^{n} \cdot \binom{2p-2n}{p-n}. \] Then \[ \begin{aligned} pS &= \sum_{n=1}^{p-1} p H_{n} \cdot 4^{n} \cdot \binom{2p-2n}{p-n} \\ &= \sum_{n=1}^{p-1} \left(1 + p H_{n}\right) \cdot 4^{n} \cdot \binom{2p-2n}{p-n} - \sum_{n=1}^{p-1} 4^{n} \cdot \binom{2p-2n}{p-n} \\ &\equiv \sum_{n=1}^{p-1} \binom{p+n}{n} \cdot 4^{n} \cdot \binom{2p-2n}{p-n} - \sum_{n=1}^{p-1} 4^{n} \cdot \binom{2p-2n}{p-n} \pmod{p^2}. \end{aligned} \] Using Vandermonde's Convolution Identity and Wolstenholme's Theorem, we simplify further: \[ \begin{aligned} pS &\equiv \sum_{n=0}^{p} \binom{p+n}{n} \cdot 4^{n} \cdot \binom{2p-2n}{p-n} - \sum_{n=0}^{p} 4^{n} \cdot \binom{2p-2n}{p-n} - 4^p \pmod{p^2}. \end{aligned} \] Now, using properties of binomial coefficients and simplifications: \[ \begin{aligned} -4^p \sum_{n=0}^{p} \binom{-p-1}{n} \cdot \binom{-1/2}{p-n} + 4^p \sum_{n=0}^{p} (-1)^n \cdot \binom{-1/2}{p-n} - 4^p \equiv -4^p \binom{\frac{-2p-3}{2}}{p} + 4^p \binom{-3/2}{p} - 4^p \pmod{p^2}. \end{aligned} \] After further simplification and using L'Hopital's Rule: \[ \frac{4^p - 4}{p} \equiv -2^{15} \pmod{x+1}. \] Thus, the final answer is: \[ \frac{4^p - 4}{p} - 8 \equiv 2^{15} - 7 = 32761 \pmod{p}. \] Therefore, the remainder is $\boxed{32761}$.	32761	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns-2.md" }	Let $ p = 2^{16} + 1 $ be an odd prime. Define $ H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} $. Compute the remainder when \[ (p-1)!\sum_{n=1}^{p-1} H_{n} \cdot 4^{n} \cdot\binom{2p-2n}{p-n} \] is divided by $ p $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/DivisionProblems/SubsetSumProblem", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/Congruences/ResidueClass", "/Mathematics/NumberTheory/Congruences/WilsonsTheorem", "/Mathematics/NumberTheory/Congruences/WolstenholmesTheorem", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/MiscellaneousPrimes", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties" ]	Replace p·H_n modulo p^2 by the binomial coefficient binom{p+n}{n} and then evaluate the resulting sum using binomial identities.	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ours_357	Solution. If the initial length and width are $a$ and $b$, respectively, then the change in area is given by: \[ x = (a-1)(b+1) - ab = ab - b + a - 1 - ab = -b + a - 1 \] To find the smallest positive value of $x$, we set $x = -b + a - 1 = 1$. Solving for $a$, we get: \[ a = b + 2 \] Choosing $b = 1$, we find $a = 3$, which gives: \[ x = -1 + 3 - 1 = 1 \] Thus, the smallest possible positive value of $x$ is $\boxed{1}$.	1	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$?	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicExpression", "/Mathematics/Algebra/AlgebraicEquations/LinearEquation", "/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations", "/Mathematics/Algebra/GeneralAlgebra/Algebra", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals", "/Mathematics/Geometry/PlaneGeometry/Rectangles" ]	Simplify the area change to a linear expression a - b - 1 and set it equal to the smallest positive integer.	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ours_358	Let $s_{n}=a_{n}+b_{n}+c_{n}$. We observe that $s_{n}$ is also an arithmetic progression. From $s_{1}=0$ and $s_{2}=1$, we get that $s_{n}=n-1$. Therefore, $s_{2014}=2013$. $\boxed{2013}$	2013	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Suppose $\left(a_{n}\right),\left(b_{n}\right),\left(c_{n}\right)$ are arithmetic progressions. Given that $a_{1}+b_{1}+c_{1}=0$ and $a_{2}+b_{2}+c_{2}=1$, compute $a_{2014}+b_{2014}+c_{2014}$.	[ "/Mathematics/Algebra/GeneralAlgebra/Algebra", "/Mathematics/Algebra/Sums/Sum" ]	Recognize that the termwise sum of arithmetic progressions forms another arithmetic progression.	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ours_359	For any possible red point $ F $, $\angle BFC = \frac{\pi}{2}$, so $ F $ lies on the circle with diameter $ BC $. Conversely, any point on this circle can be formed by some projection onto a line through $ B $, so the set of red points is simply this circle. The diameter of the circle is $\sqrt{(20 - 18)^2 + (14 - 0)^2} = \sqrt{2^2 + 14^2}$, so its area is \[ \frac{1}{4}(2^2 + 14^2) \pi = 50 \pi \] Thus, $\lfloor \mathcal{A} \rfloor = \lfloor 50 \pi \rfloor = 157$. $\boxed{157}$	157	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $ B = (20, 14) $ and $ C = (18, 0) $ be two points in the plane. For every line $\ell$ passing through $ B $, we color red the foot of the perpendicular from $ C $ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal{A}$).	[ "/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry", "/Mathematics/Geometry/CoordinateGeometry/Cartesian", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates", "/Mathematics/Geometry/Distance/Locus", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/Points/Locus", "/Mathematics/Geometry/Points/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/Points/PointDistances" ]	Recognize that all feet of perpendiculars from C to lines through B form the circle having BC as diameter (right‑angle locus).	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ours_360	When we have $3$ emons, they form an equilateral triangle. When the physicist adds a fourth emon, they form a rhombus with angles $60^{\circ}$ and $120^{\circ}$. For adding the fifth emon, we have $2$ cases: using the long diagonal of the rhombus, or using a side of the rhombus. In the first case (using the long diagonal), the ten lengths are: $1, 1, 1, 1, 1, 1, \sqrt{3}, \sqrt{3}, \sqrt{3}, 2$, so $P^{2}=108$. In the second case (using a side of the rhombus), the lengths are $1, 1, 1, 1, 1, 1, 1, \sqrt{3}, \sqrt{3}, 2$, so $P^{2}=36$. Therefore, the greatest possible value of $P^{2}$ is $\boxed{108}$.	108	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom{5}{2}=10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^{2}$.	[ "/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/GeometricConstruction/EuclideanTools", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Place the fifth emon at the far end of the rhombus's long diagonal to maximize the set of distances.	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ours_361	Consider the difference in the mass between the two particles. At each step, it becomes cut in half, so in order for it to remain an integer at each step, we want the initial difference to be divisible by the greatest power of $2$ possible. This greatest possible power of $2$ dividing the difference is clearly $512=2^{9}$, so the maximal number of times he can use his machine is $9$, which can be obtained if we start with omons of mass $1$ and $513$ and arbitrarily destroy an omon at each step. $\boxed{9}$	9	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac{1}{2}(a+b)$. The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$. What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?	[ "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/NonnegativeInteger", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Observe that each use halves the mass difference, so the number of steps equals the exponent of 2 in the initial difference.	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ours_362	Suppose the triangle has angles $10a$, $10b$, and $10c$ in degrees, where $a < b < c < 9$ and $a + b + c = 18$. We consider several cases: - If $c = 8$, then $a + b = 10$ and $a < b \leq 7$. This gives possibilities $(a, b) = (3, 7)$ and $(a, b) = (4, 6)$, neither of which work. - If $c = 7$, then $a + b = 11$ and $a < b \leq 6$. The only possibility here is $(a, b) = (5, 6)$, which also fails. - If $c \leq 6$, then $a, b < 6$, so $a + b + c < 18$. Hence no solutions can occur with $c \leq 6$. Hence, Tina unfortunately cannot draw any triangles and the answer is $\boxed{0}$.	0	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^{\circ}$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^{\circ}$ or $60^{\circ}$, and the triangle should definitely not be isosceles. How many different triangles can Tina draw? (Similar triangles are considered the same.)	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Translate the angle conditions into an integer triple a<b<c with a+b+c=18 and each ≤8, then use the small sum to limit possible values of c and check the few cases.	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ours_363	Let $ g(x, y, z) = f(x, y, z) - z^{y^{x}} $. By symmetry, we have \[ g(1,2,3) + g(1,3,2) + g(2,1,3) + g(2,3,1) + g(3,1,2) + g(3,2,1) = 0. \] Thus, \[ \begin{aligned} & f(1,2,3) + f(1,3,2) + f(2,1,3) + f(2,3,1) + f(3,1,2) + f(3,2,1) \\ = & \, g(1,2,3) + g(1,3,2) + g(2,1,3) + g(2,3,1) + g(3,1,2) + g(3,2,1) \\ & + \left(1^{2^{3}} + 1^{3^{2}} + 2^{1^{3}} + 2^{3^{1}} + 3^{1^{2}} + 3^{2^{1}}\right) = 24. \end{aligned} \] $\boxed{24}$	24	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Define the function $ f(x, y, z) $ by \[ f(x, y, z) = x^{y^{z}} - x^{z^{y}} + y^{z^{x}} - y^{x^{z}} + z^{x^{y}}. \] Evaluate $ f(1,2,3) + f(1,3,2) + f(2,1,3) + f(2,3,1) + f(3,1,2) + f(3,2,1) $.	[ "/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]	Use symmetry by defining an auxiliary function that cancels the cyclic terms when summed over all permutations.	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ours_364	Compute \[ \frac{\binom{9}{2} \cdot \binom{10}{2}^{2}}{\binom{900}{2}} = \frac{36 \cdot 45^{2}}{450 \cdot 899} = \frac{162}{899} \] The numerator represents the number of ways to select a pair of hundreds digits, tens digits, and unit digits in a pair of clearly bigger numbers. The denominator represents the total number of pairs $ a > b $ of three-digit numbers. Hence the answer is $ 162 + 899 = 1061 $. $\boxed{1061}$	1061	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $ a $ and $ b $ be randomly selected three-digit integers and suppose $ a > b $. We say that $ a $ is clearly bigger than $ b $ if each digit of $ a $ is larger than the corresponding digit of $ b $. If the probability that $ a $ is clearly bigger than $ b $ is $\frac{m}{n}$, where $ m $ and $ n $ are relatively prime integers, compute $ m+n $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace" ]	Count digit‑wise selections using combinations of two distinct digits for each place, then divide by total ordered three‑digit pairs.	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ours_365	We seek $\min(\nu_2(N), \nu_5(N))$, where $\nu_p(n)$ denotes the exponent of $ p $ in the prime factorization of $ n $. Consider the expression: \[ \frac{N}{2014!} = 1 + 2015 + 2015 \cdot 2016 + 2015 \cdot 2016 \cdot 2017 + \cdots \] This shows that $\frac{N}{2014!}$ is an integer not divisible by 5. Therefore, $\nu_5(N) = \nu_5(2014!)$. Using Legendre's formula, $\nu_5(2014!)$ is calculated as: \[ \nu_5(2014!) = \left\lfloor \frac{2014}{5} \right\rfloor + \left\lfloor \frac{2014}{25} \right\rfloor + \left\lfloor \frac{2014}{125} \right\rfloor + \left\lfloor \frac{2014}{625} \right\rfloor \] Calculating each term: \[ \left\lfloor \frac{2014}{5} \right\rfloor = 402, \quad \left\lfloor \frac{2014}{25} \right\rfloor = 80, \quad \left\lfloor \frac{2014}{125} \right\rfloor = 16, \quad \left\lfloor \frac{2014}{625} \right\rfloor = 3 \] Adding these, we get: \[ \nu_5(2014!) = 402 + 80 + 16 + 3 = 501 \] Since $\nu_2(2014!) > 501$, it follows that $\nu_2(N) > 501$ as well. Therefore, the number of zeros at the end of $ N $ is $\boxed{501}$.	501	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $ N = 2014! + 2015! + 2016! + \cdots + 9999! $. How many zeros are at the end of the decimal representation of $ N $?	[ "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Factor out 2014! and note the remaining sum is not divisible by 5, so the 5‑adic valuation of N equals that of 2014! and can be found with Legendre's formula.	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ours_366	The resulting quotient is $$ N=0103 \ldots 97 $$ (where we have added a leading $0$). If we consider $N^{\prime}$, which is $N$ with $99$ appended to the right, then we obtain a 100-digit number for which the average of the odd-indexed digits is $4.5$ and the average of the even-indexed digits is $5$. So, the sum of the digits of $N^{\prime}$ is $9.5 \cdot 50=475$. Hence the final answer is $475-18=457$. $\boxed{457}$	457	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Find the sum of the decimal digits of $$ \left\lfloor\frac{51525354555657 \ldots 979899}{50}\right\rfloor $$ Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.	[ "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Integers/Z-Star", "/Mathematics/NumberTheory/Numbers/LargeNumbers", "/Mathematics/NumberTheory/Numbers/NumbersRepresentation", "/Mathematics/NumberTheory/Rounding/FloorFunction", "/Mathematics/NumberTheory/Rounding/IntegerPart", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Observe that dividing the concatenated number by 50 produces a predictable alternating digit pattern whose average can be used to compute the total digit sum.	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ours_367	Let $ M $ be the midpoint of $ EF $ and let the semicircle be tangent to $\overline{AB}$ and $\overline{AC}$ at $ X $ and $ Y $, respectively. It is easy to see that $ BM = 13 $, so $ BX = 5 $. Similarly, $ CM = 15 $, so $ CY = 9 $. Now let $ AX = AY = k $. Thus, $ AB = k + 5 $ and $ AC = k + 9 $, and then $ AM^2 = k^2 + 12^2 $. Finally, $ BC = 28 $. Apply Stewart's Theorem to obtain \[ (k^2 + 12^2 + 13 \cdot 15) \cdot 28 = (k+5)^2 \cdot 15 + (k+9)^2 \cdot 13 \] Solving, we find that $ k = 21 $. So the perimeter $ AB + AC + BC = 2k + 14 + 28 = 84 $. $\boxed{84}$	84	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Given a triangle $ \triangle ABC $, consider the semicircle with diameter $\overline{EF}$ on $\overline{BC}$ tangent to $\overline{AB}$ and $\overline{AC}$. If $ BE = 1 $, $ EF = 24 $, and $ FC = 3 $, find the perimeter of $\triangle ABC$.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Apply Stewart's Theorem to triangle ABC with cevian AM, using the tangent lengths from B and C to the semicircle to find BM and CM.	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ours_368	We have the equations: \[ ab + ac = 5, \quad bc + ba = 10, \quad ca + cb = 13. \] Adding these, we get: \[ ab + bc + ca = \frac{5 + 10 + 13}{2} = 14. \] From the individual equations, we find: \[ ab = 1, \quad bc = 9, \quad ca = 4. \] Thus, the product $abc$ is: \[ abc = \sqrt{1 \cdot 4 \cdot 9} = 6. \] Now, we calculate $a+b+c$ using: \[ a+b+c = \frac{6}{1} + \frac{6}{4} + \frac{6}{9} = \frac{49}{6}. \] Therefore, the answer is $49 + 6 = 55$. $\boxed{55}$	55	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $a, b, c$ be positive real numbers for which \[ \frac{5}{a}=b+c, \quad \frac{10}{b}=c+a, \quad \text{and} \quad \frac{13}{c}=a+b. \] If $a+b+c=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations", "/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations" ]	Convert the given reciprocal equations into equations for the pairwise products ab, bc, ca and solve for those products.	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ours_369	By linearity of expectation, it suffices to sum the expected value of the scores for each question. We see that the sum is $$ \begin{aligned} \sum_{n=1}^{30} n \cdot\left(\frac{1}{n^{2}}+\frac{1}{n+1}+\frac{1022}{n^{2}(n+1)}\right) & =\sum_{n=1}^{30}\left(\frac{1}{n}+1-\frac{1}{n+1}+1022\left(\frac{1}{n}-\frac{1}{n+1}\right)\right) \\ & =30+1023 \sum_{n=1}^{30}\left(\frac{1}{n}-\frac{1}{n+1}\right) \\ & =30+1023\left(\frac{1}{1}-\frac{1}{31}\right) \\ & =30+1023-33 \\ & =1020 . \end{aligned} $$ $\boxed{1020}$	1020	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Two ducks, Wat and Q, are taking a math test with $1022$ other ducklings. The test has $30$ questions, and the $n$th question is worth $n$ points. The ducks work independently on the test. Wat gets the $n$th problem correct with probability $\frac{1}{n^{2}}$ while Q gets the $n$th problem correct with probability $\frac{1}{n+1}$. Unfortunately, the remaining ducklings each answer all $30$ questions incorrectly. Just before turning in their test, the ducks and ducklings decide to share answers! On any question which Wat and Q have the same answer, the ducklings change their answers to agree with them. After this process, what is the expected value of the sum of all $1024$ scores?	[ "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilityAxioms" ]	Apply linearity of expectation to each question and use the probability that Wat and Q give the same answer to get a telescoping sum.	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ours_370	Solution. Observe that the numbers in the problem are $111111111^2$ and $111111^2$ (with nine and six ones). So we seek $\operatorname{gcd}(111111111, 111111)^2$. By the Euclidean Algorithm, this is $\operatorname{gcd}(111000000, 111111)^2 = \operatorname{gcd}(111, 111111)^2 = 12321$. $\boxed{12321}$	12321	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	What is the greatest common factor of $12345678987654321$ and $12345654321$?	[ "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Rewrite each number as a square of a repunit (111…1) and find the gcd of the repunit bases.	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ours_371	Since $\phi^{2}=\phi+1$, we can perform the transformation: $100_{\phi} \rightarrow 011_{\phi}$. In $\underbrace{(100 \ldots 100)_{\phi}}$, for each of the twenty $100$'s, we can choose whether to change it to $011$. This gives us $2^{20}$ equivalent numbers. Consider the first two $100$'s in the sequence of twenty. If we apply the transformation to the first $100$, it becomes $011$. Regardless of what we do to the second $100$, neither $1$ in the $011$ will have two zeroes after it, so we cannot apply the transformation again. Thus, the total number of base-$\phi$ numbers is $2^{20} = 1048576$. $\boxed{1048576}$	1048576	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $\phi=\frac{1+\sqrt{5}}{2}$. A base-$\phi$ number $\left(a_{n} a_{n-1} \ldots a_{1} a_{0}\right)_{\phi}$, where $0 \leq a_{n}, a_{n-1}, \ldots, a_{0} \leq 1$ are integers, is defined by \[ \left(a_{n} a_{n-1} \ldots a_{1} a_{0}\right)_{\phi}=a_{n} \cdot \phi^{n}+a_{n-1} \cdot \phi^{n-1}+\ldots+a_{1} \cdot \phi^{1}+a_{0} \] Compute the number of base-$\phi$ numbers $\left(b_{j} b_{j-1} \ldots b_{1} b_{0}\right)_{\phi}$ which satisfy $b_{j} \neq 0$ and \[ \left(b_{j} b_{j-1} \ldots b_{1} b_{0}\right)_{\phi}=\underbrace{(100 \ldots 100)_{\phi}}_{\text{Twenty } 100\text{'s}} \]	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/BinarySequences", "/Mathematics/NumberTheory/IrrationalNumbers/IrrationalNumber", "/Mathematics/NumberTheory/IrrationalNumbers/Surd", "/Mathematics/NumberTheory/Sequences/BeattySequence", "/Mathematics/NumberTheory/Sequences/Sequence", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Apply the relation φ² = φ + 1 to replace each block ‘100’ with ‘011’, giving an independent binary choice for each of the twenty blocks.	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ours_372	The 3D Pythagorean Theorem (also called De Gua's Theorem) states that the sum of the squares of the areas of the faces of a right-angled tetrahedron adjacent to the right angle is equal to the square of the area of the face opposite to it; in other words, \[ [O A B]^{2} + [O B C]^{2} + [O C A]^{2} = [A B C]^{2} \] Let $ u = [O C A] $ and $ v = [A B C] $. Then $(v-u)(v+u) = v^{2} - u^{2} = 596 = 4 \cdot 149$, and since the parities of $v-u$ and $v+u$ must be the same, $v+u = 2 \cdot 149$ and $v-u = 2$. Thus the only integer solution $(u, v)$ is $(150, 148)$, and so our answer is $150 \cdot 148 = 22200$. $\boxed{22200}$	22200	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $ O A B C $ be a tetrahedron such that $\angle A O B = \angle B O C = \angle C O A = 90^{\circ}$ and its faces have integral surface areas. If $[O A B] = 20$ and $[O B C] = 14$, find the sum of all possible values of $[O C A][A B C]$. (Here $[\triangle]$ denotes the area of $\triangle$.)	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry", "/Mathematics/Geometry/SolidGeometry/Polyhedra" ]	Apply De Gua’s theorem to turn the area condition into a difference‑of‑squares equation.	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ours_373	Solution 1. Let $ M $ be the midpoint of side $ BC $ and let $ G' $ be the reflection of $ G $ over $ A $. By considering the homothety at $ A $ with ratio $ 2 $, we see that $ G' $ lies on the circumcircle of $ \triangle ABC $. By Power of a Point, we have \[ AM \cdot MG' = BM \cdot MC = 5^2 = 25 \] But $ MG' = MG = \frac{1}{3} AM $ (since $ G $ is the centroid), so we derive that $ AM = 5\sqrt{3} $. Now let us reflect $ A $ over $ M $ to $ A' $, yielding parallelogram $ ABA'C $. It is a well-known fact about parallelograms that we have \[ BC^2 + AA'^2 = 2(AB^2 + AC^2) \] We know $ AA'^2 = 4 \cdot AM^2 $, so the answer to the question is $\frac{1}{2} BC^2 + 2 AM^2 = 50 + 150 = 200$. Solution 2. As before, we see that $ G' $ lies on the circumcircle of $ \triangle ABC $. The barycentric coordinates of $ G' $ are thus $ G' = (2: 2: -1) $. Letting $ a = BC, b = CA, c = AB $ and using the circumcircle formula, we detect \[ -a^2(2)(2) - b^2(-1)(2) - c^2(2)(-1) = 0 \] Hence, $ b^2 + c^2 = 2a^2 = 200 $. Solution 3. The concyclicity of $ AEGF $ implies \[ \angle AEG + \angle AFG = 180^\circ \] so \[ \angle AEB + \angle AFC = 180^\circ \] implying that there is a point $ X $ on $ BC $ such that $ \angle AXC = \angle AFC $ and $ \angle AXB = \angle AEB $. So $ AEXB, AFXC $ are concyclic, implying \[ \begin{aligned} & BC \cdot BX = BF \cdot BA = \frac{1}{2} BA^2 \\ & BC \cdot CX = CE \cdot CA = \frac{1}{2} CA^2 \end{aligned} \] so upon adding these together, we recover \[ BC^2 = BC(BX + CX) = \frac{1}{2}(AB^2 + AC^2) \] Thus our answer is $ 2BC^2 = 200 $. $\boxed{200}$	200	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $ \triangle ABC $ be a triangle with area $ 5 $ and $ BC = 10 $. Let $ E $ and $ F $ be the midpoints of sides $ AC $ and $ AB $ respectively, and let $ BE $ and $ CF $ intersect at $ G $. Suppose that quadrilateral $ AEGF $ can be inscribed in a circle. Determine the value of $ AB^2 + AC^2 $.	[ "/Mathematics/Geometry/Centroids/GeometricCentroid", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/CircularTriangles", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Reflect the centroid across A (equivalently apply a 2‑scale homothety at A) to obtain a point on the circumcircle, then use power of a point with the midpoint of BC to relate AM and the side lengths.	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ours_374	Define $c=\lfloor\alpha\rfloor$. Switch the order of summation: \[ \sum_{a=1}^{1000} \sum_{b=1}^{a}\left\lfloor\frac{b+c}{a}\right\rfloor \] One can check that in fact (using the Hermite identity) we have \[ \sum_{b=1}^{a}\left\lfloor\frac{b+c}{a}\right\rfloor=1+c \] for every $a$. Hence \[ S=1000 \cdot \frac{c+1}{c+\epsilon}=1000+1000 \cdot \frac{1-\epsilon}{c+\epsilon} \] where $\epsilon=\alpha-c$. Hence $S \geq 1200$ is just $5-5 \epsilon \geq c+\epsilon$ or $\epsilon \leq \frac{5-c}{6}$. Over $c=0,1, \ldots, 4$ the intervals for which $S \geq 1200$ sum to $\frac{5}{2}$. Hence $1000 p=1000 \cdot \frac{5}{2} \cdot \frac{1}{500}=5$. $\boxed{5}$	5	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S=\frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000}\left\lfloor\frac{m+\alpha}{n}\right\rfloor . \] Let $p$ denote the probability that $S \geq 1200$. Compute $1000 p$.	[ "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace" ]	Apply the Hermite identity to evaluate the inner floor sum after swapping the summation order.	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ours_375	By construction, $C$ is the midpoint of arc $AD$ not containing $B$. Hence, it follows that $IC=IA=ID$. Moreover, it is easy to see that $\angle IAE=\angle IEA$. One can use Stewart's theorem to compute $AI^2=\frac{5^3 \cdot 3 + 2^2 \cdot 5 - 2 \cdot 5 \cdot 7}{7}=\frac{25}{7}$. By Heron's formula, the area of $\triangle ABC$ is $\frac{15}{4} \sqrt{3}$, so the height from $A$ to $BC$ has length $\frac{15}{14} \sqrt{3}$. Then \[ \cos^2 \angle IAE = \frac{\left(\frac{15}{14} \sqrt{3}\right)^2}{\frac{25}{7}} = \frac{27}{28} \] Thus the answer is $27+28=55$. $\boxed{55}$	55	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos^2 \angle AEI=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, determine $m+n$.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Use that C is the midpoint of arc AD (not containing B) to deduce IC = IA = ID, giving ∠IAE = ∠IEA and simplifying the cosine calculation.	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ours_376	We use vectors/complex numbers. Let $ a_j^{(i)} $ be the number at the point $ A_j^{(i)} $. Since the centroid is just the average of the vertices, \[ a_j^{(i)} = \frac{1}{3} \left( a_j^{(i-1)} + a_{j+3^{7-i}}^{(i-1)} + a_{j+2 \cdot 3^{7-i}}^{(i-1)} \right) \] I claim now that \[ a_j^{(i)} = \frac{1}{3^i} \sum_{k=0}^{3^i-1} a_{j+3^{7-i} \cdot k}^{(0)} \] This readily follows from induction and the above statement. So \[ a_{2014}^{(7)} = \frac{1}{2187} \sum_{k=0}^{2186} a_{2014+3^{0} \cdot k}^{(0)} = -\frac{1}{2187} a_{2014}^{(0)} \] Therefore, \[ \frac{\left\|a_{2014}^{(7)}\right\|}{\left\|a_{2014}^{(0)}\right\|} = \frac{1}{2187} \Longrightarrow 1 + 2187 = 2188 \] Thus, the answer is $\boxed{2188}$.	2188	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall14Solns.md" }	Let $ n = 2188 = 3^7 + 1 $ and let $ A_0^{(0)}, A_1^{(0)}, \ldots, A_{n-1}^{(0)} $ be the vertices of a regular $ n $-gon (in that order) with center $ O $. For $ i = 1, 2, \ldots, 7 $ and $ j = 0, 1, \ldots, n-1 $, let $ A_j^{(i)} $ denote the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)} \] Here the subscripts are taken modulo $ n $. If \[ \frac{\left\|O A_{2014}^{(7)}\right\|}{\left\|O A_{2014}^{(0)}\right\|} = \frac{p}{q} \] for relatively prime positive integers $ p $ and $ q $, find $ p+q $.	[ "/Mathematics/Geometry/Centroids/GeometricCentroid", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Polygons", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Iteratively taking centroids averages over equally spaced vertices, so after 7 steps the point equals the uniform average of all n vertices, scaling by 1/3^7.	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ours_377	Solution. The main observation is that $$ \binom{8}{2}+\binom{9}{2}=8^{2} \quad \text{and} \quad \binom{15}{2}+\binom{16}{2}=15^{2} $$ so that the desired sum is $\sqrt{8^{2}+15^{2}}=17$, a well-known Pythagorean triple. $\boxed{17}$	17	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Evaluate $$ \sqrt{\binom{8}{2}+\binom{9}{2}+\binom{15}{2}+\binom{16}{2}} $$	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Pair consecutive binomial coefficients to form perfect squares using the formula for n choose 2.	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ours_379	These integers must take on the form $\overline{22n}$ or $\overline{2n2}$, where $n \neq 2$. Since there are $9$ choices for $n$ in each case, the answer is $2 \cdot 9 = 18$. $\boxed{18}$	18	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	How many integers between $123$ and $321$ inclusive have exactly two digits that are 2?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Fix the positions of the two 2's in the three‑digit number and count the allowed choices for the remaining digit.	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ours_380	Let $O_{A}$ be the center of $\omega_{A}$ and $O_{B}$ the center of $\omega_{B}$. Notice that $O_{A}$ and $O_{B}$ must lie on the same side of line $AB$, since the assumptions of the problem imply the circles are not tangent at $O$. By symmetry, we have that $\overline{OC} \perp \overline{AB}$; so $O_{A}$ and $O_{B}$ are the midpoints of $AC$ and $BC$. In particular, $\triangle AOO_{A}$ and $\triangle BOO_{B}$ are equilateral; thus, we deduce $\triangle ABC$ is equilateral too. Since $OC = \sqrt{3}$, we find $AB = BC = CA = 2$, so the perimeter is $6$. $\boxed{6}$	6	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $\omega$ be a circle with diameter $AB$ and center $O$. We draw a circle $\omega_{A}$ through $O$ and $A$, and another circle $\omega_{B}$ through $O$ and $B$; the circles $\omega_{A}$ and $\omega_{B}$ intersect at a point $C$ distinct from $O$. Assume that all three circles $\omega, \omega_{A}, \omega_{B}$ are congruent. If $CO = \sqrt{3}$, what is the perimeter of $\triangle ABC$?	[ "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/CircularTriangles", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Use symmetry of the three congruent circles to see that O_A and O_B are midpoints of AC and BC, making triangle ABC equilateral.	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ours_381	By the Pythagorean Theorem in $n$-dimensional space, the maximal length of the diagonal of the hypercube is given by \[ \sqrt{\underbrace{(1-0)^{2}+(1-0)^{2}+\cdots+(1-0)^{2}}_{n \text{ times}}} = \sqrt{n} \] This represents the distance from $(0, \ldots, 0)$ to $(1, \ldots, 1)$ in $\mathbb{R}^{n}$. We require $\sqrt{n} \geq 25.6$, which implies $n \geq 655.36$. The smallest integer $n$ that satisfies this condition is $n = 656$. $\boxed{656}$	656	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Merlin wants to buy a magical box, which happens to be an $n$-dimensional hypercube with side length $1 \text{ cm}$. The box needs to be large enough to fit his wand, which is $25.6 \text{ cm}$ long. What is the minimal possible value of $n$?	[ "/Mathematics/Geometry/Distance", "/Mathematics/Geometry/MultidimensionalGeometry/n-DimensionalGeometry" ]	Use the n‑dimensional space diagonal length √n of a unit hypercube and compare it to the wand length.	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ours_382	It's clear that $ A_{0} $ is achieved when the fence forms a circle, $ A_{1} $ when it forms a half-circle, and $ A_{2} $ when it forms a quarter circle. After some computations, we find that $ A_{1} = 2 A_{0} $, $ A_{2} = 2 A_{1} $, and thus $ n = 4 $. Therefore, the answer is $\boxed{4000}$.	4000	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Farmer John has a flexible fence of length $ L $ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he can enclose a maximum possible area of $ A_{0} $, and when he uses one of the walls or both walls, he gets a maximum area of $ A_{1} $ and $ A_{2} $ respectively. If $ n=\frac{A_{1}}{A_{0}}+\frac{A_{2}}{A_{1}} $, find $ \lfloor 1000 n\rfloor $.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/PlaneGeometry/Arcs", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/PlaneCurves" ]	Recognize that the maximal area with 0, 1, or 2 walls is achieved by a full circle, a semicircle, and a quarter‑circle respectively.	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ours_383	We claim that $a_{n}=n^{2}$ for every integer $n$. The proof is by induction on $n$ with the base cases $n=0$ and $n=1$ given. For the inductive step, we observe that \[ 2(i-1)^{2}-(i-2)^{2}+2=i^{2} \] as desired. Therefore, $a_{1000}=1000^{2}=1000000$. $\boxed{1000000}$	1000000	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Define sequence $\{a_{n}\}$ as follows: $a_{0}=0$, $a_{1}=1$, and $a_{i}=2a_{i-1}-a_{i-2}+2$ for all $i \geq 2$. Determine the value of $a_{1000}$.	[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/DiscreteMathematics/RecurrenceEquations/LinearRecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecursiveSequence" ]	Guess that the sequence follows a quadratic form and verify it satisfies the recurrence	[ -0.000244140625, -0.01134490966796875, 0.053619384765625, 0.01103973388671875, -0.0007748603820800781, 0.01270294189453125, 0.0270233154296875, -0.02484130859375, 0.002838134765625, -0.00005263090133666992, 0.0169219970703125, -0.082275390625, -0.0019054412841796875, 0.023681640625, 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ours_384	All numbers are integers at all points, so we will tacitly take modulo $2$ everywhere. We claim that after $k$ operations, the numbers on the board are \[ \underbrace{011011011 \ldots 011}_{\frac{2^{k}-1}{2} \text{ blocks }} 01 \] when $k$ is even and \[ \underbrace{011011011 \ldots 011}_{\frac{2^{k}+1}{3} \text{ blocks }} \] when $k$ is odd. (Note that in total, there are $2^{k}+1$ numbers written.) The proof of this observation is a direct induction on $k \geq 0$. Applying this to $k=10$, we see the number of odd numbers is $\frac{2^{10}-1}{3} \cdot 2+1=683$. $\boxed{683}$	683	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a, b$ on the board. How many odd numbers will be on the board after $10$ such operations?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/BinarySequences", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/OddNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Parity/OddNumber", "/Mathematics/NumberTheory/Sequences/OddSequence", "/Mathematics/NumberTheory/Sequences/SternsDiatomicSeries", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Track numbers modulo 2 and prove by induction that the parity pattern consists of repeating 011 blocks.	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ours_385	The main observation is that $ s_{s_{1}}, s_{s_{2}}, s_{s_{3}} $ must be in arithmetic progression since $ s_{1}, s_{2}, $ and $ s_{3} $ are. From this, we have that $ x+2, x^{2}+18, $ and $ 2x^{2}+18 $ are in arithmetic progression. Therefore, we set up the equation: \[ 2(x^{2}+18) = (2x^{2}+18) + (x+2) \] Simplifying this equation gives: \[ 2x^{2} + 36 = 2x^{2} + 18 + x + 2 \] \[ 36 = 18 + x + 2 \] \[ 36 = 20 + x \] \[ x = 16 \] Thus, the value of $ x $ is $\boxed{16}$.	16	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $ s_{1}, s_{2}, \ldots $ be an arithmetic progression of positive integers. Suppose that \[ s_{s_{1}} = x+2, \quad s_{s_{2}} = x^{2}+18, \quad \text{and} \quad s_{s_{3}} = 2x^{2}+18 \] Determine the value of $ x $.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/LinearEquation" ]	Use that s_{s1}, s_{s2}, s_{s3} must be in arithmetic progression because s1, s2, s3 are.	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ours_386	By observing the base-2 expansion of the integer, we see that the function is equivalent to removing the frontmost nonzero digit (which is $ 1 $ ) and adding a $ 1 $ at the end. Thus, $ f^n(n) = 2^{s(n)} - 1 $, where $ s(n) $ is the sum of binary digits of $ n $. Since $ 2015 = 11111011111_2 $ has $ s(2015) = 10 $, it suffices to find the number of positive integers with at most $ 16 $ binary digits, exactly $ 10 $ of which are $ 1 $. This is $\binom{16}{10} = 8008$. $\boxed{8008}$	8008	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	For any positive integer $ n $, define a function $ f $ by \[ f(n) = 2n + 1 - 2^{\left\lfloor \log_2 n \right\rfloor + 1} \] Let $ f^m $ denote the function $ f $ applied $ m $ times. Determine the number of integers $ n $ between $ 1 $ and $ 65535 $ inclusive such that $ f^n(n) = f^{2015}(2015) $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/BinarySequences", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Interpret f as moving the leading binary 1 to the least‑significant position, so fⁿ(n) depends only on the number of 1‑bits of n.	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ours_387	The slope of line $CD$ is $\frac{17}{6}$. This can be determined using coordinate geometry or vector methods. Therefore, the absolute value of the slope is $\frac{17}{6}$, where $m = 17$ and $n = 6$. Thus, $100m + n = 100 \times 17 + 6 = 1706$. $\boxed{1706}$	1706	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	A trapezoid $ABCD$ lies on the $xy$-plane. The slopes of lines $BC$ and $AD$ are both $\frac{1}{3}$, and the slope of line $AB$ is $-\frac{2}{3}$. Given that $AB = CD$ and $BC < AD$, the absolute value of the slope of line $CD$ can be expressed as $\frac{m}{n}$, where $m, n$ are two relatively prime positive integers. Find $100m + n$.	[ "/Mathematics/Algebra/VectorAlgebra/Vector", "/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry", "/Mathematics/Geometry/CoordinateGeometry/Cartesian", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates", "/Mathematics/Geometry/CoordinateGeometry/CartesianGeometry", "/Mathematics/Geometry/CoordinateGeometry/CartesianPlane", "/Mathematics/Geometry/CoordinateGeometry/CoordinatePlane", "/Mathematics/Geometry/CoordinateGeometry/CoordinateSystem", "/Mathematics/Geometry/CoordinateGeometry/Coordinates", "/Mathematics/Geometry/PlaneGeometry/Polygons", "/Mathematics/Geometry/PlaneGeometry/Quadrangles", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals" ]	Set the length of AB equal to CD using the distance formula with the given slopes to solve for the unknown slope of CD.	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ours_388	To find the sum of the coefficients of $Q(x)$, we need to evaluate $Q(1)$. The roots of $Q(x)$ are $bc-a^2$, $ca-b^2$, and $ab-c^2$. By Vieta's formulas for the polynomial $P(x)$, we have: - $a + b + c = 10$, - $ab + bc + ca = 1$, - $abc = 2015$. The polynomial $Q(x)$ is monic, so it can be expressed as: \[ Q(x) = (x - (bc-a^2))(x - (ca-b^2))(x - (ab-c^2)) \] The sum of the coefficients of $Q(x)$ is $Q(1)$, which is: \[ Q(1) = (1 - (bc-a^2))(1 - (ca-b^2))(1 - (ab-c^2)) \] Substituting the expressions for the roots, we have: \[ Q(1) = (1 - (bc-a^2))(1 - (ca-b^2))(1 - (ab-c^2)) \] Using the identity $1 = ab + bc + ca$, we can simplify: \[ Q(1) = (a(a+b+c))(b(a+b+c))(c(a+b+c)) = abc(a+b+c)^3 = 2015 \times 10^3 = 2015000 \] Thus, the sum of the coefficients of $Q(x)$ is $\boxed{2015000}$.	2015000	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $a, b, c$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$. The cubic polynomial $Q(x)$ is monic and has distinct roots $bc-a^2, ca-b^2, ab-c^2$. What is the sum of the coefficients of $Q$?	[ "/Mathematics/Algebra/GeneralAlgebra/Algebra", "/Mathematics/Algebra/Polynomials/CubicEquation", "/Mathematics/Algebra/Polynomials/CubicPolynomial", "/Mathematics/Algebra/Polynomials/MonicPolynomial", "/Mathematics/Algebra/Polynomials/VietasFormulas", "/Mathematics/Algebra/Sums/Sum" ]	Replace 1 by ab+bc+ca (using Vieta) so each factor 1-(bc‑a^2) becomes a(a+b+c), then multiply.	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ours_390	Consider the prefix products, i.e. $p_{i}=a_{1} a_{2} \ldots a_{i}$ with $p_{0}=1$. Note that $(x, y)$ is good if and only if $p_{x-1} \equiv p_{y} \pmod{101}$. Let there be $s_{i}$ prefix products that evaluate to $i \pmod{101}$. Then $\sum_{i=1}^{100} s_{i}=2016$. So our answer is \[ \sum_{i=1}^{100}\binom{s_{i}}{2} \geq 84\binom{20}{2}+16\binom{21}{2}=19320 \] by convexity. $\boxed{19320}$	19320	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $a_{1}, a_{2}, \ldots, a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence good if the product of the integers in it leaves a remainder of $1$ when divided by 101. In other words, it is a pair of integers $(x, y)$ such that $1 \leq x \leq y \leq 2015$ and \[ a_{x} a_{x+1} \ldots a_{y-1} a_{y} \equiv 1 \pmod{101} \] Find the minimum possible number of good subsequences across all possible $\left(a_{i}\right)$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Congruences/CommonResidue", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/Congruences/ResidueClass", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Map each good subsequence to a pair of equal prefix‑product residues modulo 101 and count such pairs.	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ours_392	The circumcenter of the four points must be the circumcenter of $\triangle ABC$. Using trigonometry, we can show that either $\angle A = \angle B$, or $\angle A + \angle B = 135^\circ$. The first case, $\angle A = \angle B$, results in 89 possible triangles. The second case, $\angle A + \angle B = 135^\circ$, results in 67 possible triangles. One triangle is counted twice, namely $(45^\circ, 45^\circ, 90^\circ)$. Therefore, the total number of distinct ordered triples is $89 + 67 - 1 = 155$. $\boxed{155}$	155	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Given a (nondegenerate) triangle $ABC$ with positive integer angles (in degrees), construct squares $BC D_1 D_2$ and $AC E_1 E_2$ outside the triangle. Given that $D_1, D_2, E_1, E_2$ all lie on a circle, how many ordered triples $(\angle A, \angle B, \angle C)$ are possible?	[ "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeometricConstruction", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Squares", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/Trigonometry/Angles", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	The concyclicity of the square vertices forces their circumcenter to coincide with the triangle’s circumcenter, yielding the angle condition A = B or A + B = 135°.	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ours_393	First, we rearrange the given condition into the form $ x_{i+1} = \frac{x_i + 1}{-3x_i + 5} $. Define the function $ f(x) = \frac{x + 1}{-3x + 5} $. Thus, $ f(x_i) = x_{i+1} $. Since $ x_1 = x_{43} $, it follows that $ f^{(42)}(x_i) = x_i $. We need to find the fixed points of $ f $, which satisfy $ f(x) = x $. Solving the equation: \[ \frac{x + 1}{-3x + 5} = x \] we get: \[ x + 1 = x(-3x + 5) \] \[ x + 1 = -3x^2 + 5x \] \[ 3x^2 - 4x + 1 = 0 \] Solving this quadratic equation, we find the roots: \[ x = 1 \quad \text{and} \quad x = \frac{1}{3} \] Both $ x = 1 $ and $ x = \frac{1}{3} $ are fixed points of $ f $, meaning $ f(1) = 1 $ and $ f\left(\frac{1}{3}\right) = \frac{1}{3} $. Thus, the sequence $ x_1, x_2, \ldots, x_{42} $ can be entirely composed of 1's or entirely composed of $\frac{1}{3}$'s. Therefore, the possible values for the sum $ x_1 + x_2 + \cdots + x_{42} $ are: - If all $ x_i = 1 $, the sum is $ 42 \times 1 = 42 $. - If all $ x_i = \frac{1}{3} $, the sum is $ 42 \times \frac{1}{3} = 14 $. The product of these possible sums is: \[ 42 \times 14 = 588 \] Thus, the product of all possible values for $ x_1 + x_2 + \cdots + x_{42} $ is $\boxed{588}$.	588	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $ x_1, x_2, \ldots, x_{42} $ be real numbers such that $ 5x_{i+1} - x_i - 3x_ix_{i+1} = 1 $ for each $ 1 \leq i \leq 42 $, with $ x_1 = x_{43} $. Find the product of all possible values for $ x_1 + x_2 + \cdots + x_{42} $.	[ "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial", "/Mathematics/Algebra/Products/Product", "/Mathematics/Algebra/Sums/CumulativeSum", "/Mathematics/Algebra/Sums/PartialSum", "/Mathematics/Algebra/Sums/RunningSum", "/Mathematics/Algebra/Sums/Sum" ]	Rewrite the recurrence as an iteration of a rational function and look for its fixed points, since the sequence must return to its start after 42 steps.	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ours_394	To solve this problem, we need to understand the condition for an integer $ a $ to be $ n $-well. The condition is: \[ \left\lfloor \frac{n}{\lfloor n/a \rfloor} \right\rfloor = a \] Let $ k = \left\lfloor \frac{n}{a} \right\rfloor $. Then, $ k \leq \frac{n}{a} < k+1 $, which implies: \[ \frac{n}{k+1} < a \leq \frac{n}{k} \] For $ a $ to be $ n $-well, we need: \[ \left\lfloor \frac{n}{k} \right\rfloor = a \] This means: \[ a = \left\lfloor \frac{n}{k} \right\rfloor \] Thus, $ a $ is the integer part of $ \frac{n}{k} $, and it must satisfy: \[ \frac{n}{k+1} < \left\lfloor \frac{n}{k} \right\rfloor \leq \frac{n}{k} \] The number of integers $ a $ that satisfy this condition for a fixed $ k $ is the number of integers in the interval: \[ \left( \frac{n}{k+1}, \frac{n}{k} \right] \] The length of this interval is: \[ \left\lfloor \frac{n}{k} \right\rfloor - \left\lfloor \frac{n}{k+1} \right\rfloor \] Summing over all possible values of $ k $ gives us $ f(n) $. The sum of all $ f(n) $ from 1 to 9999 is: \[ f(1) + f(2) + \ldots + f(9999) \] This sum can be computed by considering the contribution of each $ k $ to the total sum. Each $ k $ contributes to the sum for all $ n $ such that $ \left\lfloor \frac{n}{k} \right\rfloor $ changes. This happens when $ n $ is a multiple of $ k $. Thus, the total sum is: \[ \sum_{n=1}^{9999} f(n) = \sum_{k=1}^{9999} \left( \left\lfloor \frac{9999}{k} \right\rfloor \right) \] This is a well-known result in number theory, and the sum evaluates to: \[ \boxed{93324} \]	93324	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Given an integer $ n $, an integer $ 1 \leq a \leq n $ is called $ n $-well if \[ \left\lfloor\frac{n}{\lfloor n / a\rfloor}\right\rfloor=a \] Let $ f(n) $ be the number of $ n $-well numbers, for each integer $ n \geq 1 $. Compute $ f(1)+f(2)+\ldots+f(9999) $.	[ "/Mathematics/DiscreteMathematics/DivisionProblems/NumberPartitioningProblem", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Arithmetic/MultiplicationandDivision", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/NumberTheoreticFunctions/DivisorFunction", "/Mathematics/NumberTheory/NumberTheoreticFunctions/NumberTheoreticSums" ]	Introduce k = ⌊n/a⌋ to rewrite the condition, showing each a corresponds to a distinct floor(n/k) value, reducing the total count to Σ⌊N/k⌋.	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ours_396	We claim that the feminist numbers are the prime numbers greater than three. If we can show that each of those primes $ p \geq 5 $ is a winning position, then we are done. A feminist number $ n $ satisfies $\operatorname{gcd}(n, 6)=1$ and obviously $ n>1 $, so $ n $ has a prime divisor $ q \geq 5 $. If $ n \neq q $, then after Amandine selects $ n $, Brennon can select $ q $, and it is as if Brennon started with the move $ q $ and thus he will win, and it is not a feminist number. Suppose Amandine starts with $ p \geq 5 $, a prime. Then say Brennon does $ a $. Clearly $\operatorname{gcd}(a, p)=1$, so now there are only finitely many numbers left to be chosen, and by the Chicken McNugget theorem, the largest of these is $ ap-a-p > 1 $ since $ p \geq 5 $. We will do a Chomp-like nonconstructive proof. Suppose that now Brennon will win, regardless of what Amandine does. If Amandine does $ ap-a-p $, then Brennon can do a winning move $ b $. It is easy to see that $ ap-a-p $ is actually a nonnegative combination of $ a, b, p $; then Amandine should have done move $ b $ to begin with, and thus win! So actually Amandine wins with some number, we just don't know which. Thus the answer is $5+7+11+13+17+19+23+29+31+37=192$. $\boxed{192}$	192	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Amandine and Brennon play a turn-based game, with Amandine starting. On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers. For example, if 3 and 5 have been selected so far, only 1, 2, 4, and 7 are available to be picked; if only 3 has been selected so far, all numbers not divisible by three are eligible. A player loses immediately if they select the integer 1. Call a number $ n $ feminist if $\operatorname{gcd}(n, 6)=1$ and if Amandine wins if she starts with $ n $. Compute the sum of the feminist numbers less than 40.	[ "/Mathematics/DiscreteMathematics/Combinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/AdditiveNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Numbers/NumbersRepresentation", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties" ]	If the start is composite, opponent picks a prime divisor to win; if it is a prime >3, Chicken‑McNugget shows the opponent cannot force a win, so the start is winning.	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ours_397	Suppose $ m $ people voted for Celery and $ n $ for Toner, where $ m < n $. The number of sets of voters that could be chosen with $ d > 0 $ more people voting for Toner than for Celery is given by: \[ \binom{m}{0}\binom{n}{d} + \binom{m}{1}\binom{n}{d+1} + \ldots + \binom{m}{m}\binom{n}{m+d} \] where some terms might be zero if $ m+d > n $. By Vandermonde's identity, this sum is: \[ \binom{m+n}{m+d} \] Now, sum over the possible differences, which are $ d = 1, 2, \ldots, n $. We get: \[ \sum_{i=m+1}^{m+n} \binom{m+n}{i} \] Given $ m+n = 2015 $, we have: \[ \binom{m+n}{i} \equiv \binom{2015}{i} \equiv \frac{i+1}{2016} \binom{2016}{i+1} \equiv \left(\frac{i+1}{2016}\right)(-1)^{i+1} \equiv (i+1)(-1)^{i} \pmod{2017} \] Since $ m = \frac{2}{5} \times 2015 = 806 $, we calculate: \[ -808 + 809 - 810 + \ldots - 2016 \equiv -808 + 604(-1) \equiv -1412 \equiv 605 \pmod{2017} \] Thus, the remainder when $ N $ is divided by 2017 is $\boxed{605}$.	605	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Toner Drum and Celery Hilton are both running for president. A total of 2015 people cast their vote, giving 60% to Toner Drum. Let $ N $ be the number of "representative" sets of the 2015 voters that could have been polled to correctly predict the winner of the election (i.e., more people in the set voted for Drum than Hilton). Compute the remainder when $ N $ is divided by 2017.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Translate the voting subsets into a sum of binomial coefficients via Vandermonde, then evaluate that sum modulo 2017 using the congruence $\binom{p-1}{k}\equiv(-1)^k$.	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ours_398	Firstly, if a word $ U $ appears twice in a period of $ W $, then it is part of a special word. This can be shown by appending and prepending letters to both instances of $ U $ until the appended/prepended letters differ, which must occur due to the minimal period. By the Pigeonhole Principle, all words of length $ 2014 $ appear exactly once in a period of $ W $. Therefore, for all words $ U $ of length $\leq 2013$, the words $ aU, bU, Ua, Ub $ have length at most $ 2014 $, so they are prefixes of some word of length $ 2014 $, implying they all appear in $ W $. Thus, the number of special words is $ 2^{0} + 2^{1} + 2^{2} + \ldots + 2^{2013} = 2^{2014} - 1 $. The minimum possible number of special words is $ 2^{2014} - 1 $. To find the remainder when $ N $ is divided by $ 1000 $, we calculate: \[ 2^{2014} \equiv 0 \pmod{8} \quad \text{and} \quad 2^{2014} \equiv 1 \pmod{125} \] Using the Chinese Remainder Theorem, we find: \[ 2^{2014} \equiv 376 \pmod{1000} \] Thus, the remainder when $ N $ is divided by $ 1000 $ is: \[ 2^{2014} - 1 \equiv 376 - 1 \equiv 375 \pmod{1000} \] Therefore, the answer is $\boxed{375}$.	375	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $ W = \ldots x_{-1} x_{0} x_{1} x_{2} \ldots $ be an infinite periodic word consisting of only the letters $ a $ and $ b $. The minimal period of $ W $ is $ 2^{2016} $. Say that a word $ U $ appears in $ W $ if there are indices $ k \leq \ell $ such that $ U = x_{k} x_{k+1} \ldots x_{\ell} $. A word $ U $ is called special if $ U a, U b, a U, b U $ all appear in $ W $. (The empty word is considered special.) You are given that there are no special words of length greater than $ 2015 $. Let $ N $ be the minimum possible number of special words. Find the remainder when $ N $ is divided by $ 1000 $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/BinarySequences", "/Mathematics/NumberTheory/Sequences/EventuallyPeriodic", "/Mathematics/NumberTheory/Sequences/PeriodicSequence" ]	Show that every length‑2014 block appears exactly once using the pigeonhole principle, so all shorter words have all four extensions in the period and are special.	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ours_399	If $ 3 $ divides one of $ a, b $, then it divides the other, yielding $ 10^2 = 100 $ valid solutions. Now consider the cases based on congruences modulo 3: - If $ a \equiv b \equiv 1 \pmod{3} $, this clearly fails. - If $ a \equiv 1 \pmod{3} $ and $ b \equiv 2 \pmod{3} $, the only extra condition is that $ a \equiv 1 \pmod{2} $, which gives $ 5 \cdot 10 = 50 $ valid solutions. - If $ a \equiv 2 \pmod{3} $ and $ b \equiv 1 \pmod{3} $, there are also $ 50 $ valid solutions. - If $ a \equiv b \equiv 2 \pmod{3} $, then either $ a $ is even and $ b $ is odd, for $ 5^2 = 25 $ solutions, or $ a $ is odd and $ b $ is even, for $ 5^2 = 25 $ solutions. Adding these, we get $ 250 $ valid solutions. Thus, the remainder when $ N $ is divided by $ 1000000 $ is $\boxed{250}$.	250	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $ p = 2017 $, a prime number. Let $ N $ be the number of ordered triples $(a, b, c)$ of integers such that $ 1 \leq a, b \leq p(p-1) $ and $ a^b - b^a = p \cdot c $. Find the remainder when $ N $ is divided by $ 1000000 $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/DivisionProblems/SubsetSumProblem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Analyze the equation modulo 3 to restrict the possible residue classes of a and b, then count solutions case‑by‑case.	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ours_400	Let $ X $ be the midpoint of the major arc $ BC $. Let ray $ XI $ meet the circumcircle of $ \triangle BIC $ (centered at $ M $) again at $ J $. Then $ BICJ $ is harmonic, $ K $ is the midpoint of $ IJ $, and in particular, $ K $ lies on $ \Omega $. Moreover, ray $ KX $ is the angle bisector of $ \angle BKC $, so if $ L $ is the intersection of $\overline{IK}$ with $\overline{BC}$, we deduce \[ \frac{BK}{KC} = \frac{BT}{TC} = \left(\frac{BI}{IC}\right)^2 = \frac{\sin^2 \frac{C}{2}}{\sin^2 \frac{B}{2}} \] From the given values, we have $\cos B = \frac{48}{73}$ and $\cos C = \frac{36}{85}$. Therefore, the requested ratio is \[ \frac{\frac{1}{2}\left(1 - \frac{36}{85}\right)}{\frac{1}{2}\left(1 - \frac{48}{73}\right)} = \frac{49 \cdot 73}{85 \cdot 25} = \frac{3577}{2125} \] Thus, the answer is $ 3577 + 2125 = 5702 $. $\boxed{5702}$	5702	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $ \triangle ABC $ be an acute triangle with incenter $ I $; ray $ AI $ meets the circumcircle $ \Omega $ of $ \triangle ABC $ at $ M \neq A $. Suppose $ T $ lies on line $ BC $ such that $ \angle MIT = 90^\circ $. Let $ K $ be the foot of the altitude from $ I $ to $\overline{TM}$. Given that $\sin B = \frac{55}{73}$ and $\sin C = \frac{77}{85}$, and $\frac{BK}{CK} = \frac{m}{n}$ in lowest terms, compute $ m+n $.	[ "/Mathematics/Geometry/GeneralGeometry/Bisector", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/CircularTriangles", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/Trigonometry/Angles", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions" ]	Show that K is the midpoint of a harmonic segment IJ, which forces BK/CK to equal (BI/IC)^2.	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ours_401	We solve this problem with 101 replaced by an arbitrary prime $p \equiv 1 \pmod{4}$. First, translate so that $f(0)=0$ (we multiply the count by $p^{2}$ at the end). Then considering $(x, 0), (y, 0)$, and $(x, y)$, and noting $p$ is odd yields $f(x) \cdot f(y) \equiv x \cdot y \pmod{p}$ for all points $x, y$. Now fix $x, y$; then $[f(x+y)-f(x)-f(y)] \cdot f(t) \equiv 0 \pmod{p}$ for all points $t$. Assume for the sake of contradiction that $f(x+y)-f(x)-f(y)=(a, b)$ with at least one of $a, b$ nonzero. Then for all $t$, we have $(a, b) \cdot f(t) \equiv 0 \pmod{p}$, or equivalently, $f(t) \in \mathbb{Z} \cdot(-b, a)$. In other words, there exists $g: \mathbb{Z} / p \rightarrow \mathbb{Z} / p$ such that $f(t) \equiv(-b g(t), a g(t)) \pmod{p}$ for all $t$. We can also take $t=x+y, x, y$ to get $a^{2}+b^{2} \equiv 0 \pmod{p}$, which slightly simplifies the next paragraph. In particular, we have $u^{2}+v^{2} \equiv\|f(u, v)\|^{2} \equiv g(u, v)^{2}(b^{2}+a^{2}) \pmod{p}$ for all residues $u, v$. On the other hand, by Cauchy-Davenport or direct calculations, we know that $\{u^{2}+v^{2}\}=\mathbb{Z} / p$. Yet $g(u, v)^{2}(b^{2}+a^{2})$ is a square times a constant $(b^{2}+a^{2})$, which covers at most $\frac{p+1}{2}<p$ values, contradiction. Thus $f(x+y) \equiv f(x)+f(y)$ for all $x, y$. Let $f(1,0)=(A, B)$ and $f(0,1)=(C, D)$ (with $A^{2}+B^{2} \equiv C^{2}+D^{2} \equiv 0^{2}+1^{2}$), so $f(u, v) \equiv(A u+C v, B u+D v)$. By linearity, we just need to check $u^{2}+v^{2} \equiv(A u+C v)^{2}+(B u+D v)^{2}$, which boils down to $0 \equiv 2(A C+B D) u v$ for all $u, v$. (Alternatively, if $M$ denotes the matrix of the linear transformation $f$ over the vector space $(\mathbb{F}_{p})^{2}$, then the dot product condition is equivalent to $x^{T}(M^{T} M) y \equiv x^{T} y$ for all vectors $x, y$. Of course, $M=\begin{bmatrix}A & C \\ B & D\end{bmatrix}$.) Therefore, our desired answer is simply $p^{2}$ times the number of solutions to the system $A^{2}+B^{2} \equiv C^{2}+D^{2} \equiv 1, A C+B D \equiv 0$ (essentially counting orthogonal matrices over the finite field $\mathbb{F}_{p}$). We can parameterize $A+i B \equiv \alpha, A-i B \equiv \alpha^{-1}, C+i D \equiv \beta, C-i D \equiv \beta^{-1}$ for nonzero residues $\alpha, \beta$. Now $A C+B D \equiv 0$ is equivalent to $(\alpha+\alpha^{-1})(\beta+\beta^{-1})-(\alpha-\alpha^{-1})(\beta-\beta^{-1}) \equiv 0$, or $\alpha^{2}+\beta^{2} \equiv 0$. For each (nonzero) $\alpha$, there are exactly 2 choices for $\beta$, so our final answer is $p^{2}[2(p-1)]=2 p^{2}(p-1)$. For $p = 101$, the answer is $2 \times 101^2 \times 100 = 2040200$. $\boxed{2040200}$	2040200	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Define $\\|A-B\\|=\left(x_{A}-x_{B}\right)^{2}+\left(y_{A}-y_{B}\right)^{2}$ for every two points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ in the plane. Let $S$ be the set of points $(x, y)$ in the plane for which $x, y \in\{0,1, \ldots, 100\}$. Find the number of functions $f: S \rightarrow S$ such that $\\|A-B\\| \equiv\\|f(A)-f(B)\\|(\bmod 101)$ for any $A, B \in S$.	[ "/Mathematics/Algebra/FieldTheory/Field", "/Mathematics/Algebra/FieldTheory/FieldCharacteristic", "/Mathematics/Algebra/FieldTheory/FiniteField", "/Mathematics/Algebra/FieldTheory/PrimeField", "/Mathematics/Algebra/FieldTheory/PrimeSubfield", "/Mathematics/Algebra/GroupTheory/FiniteGroups", "/Mathematics/Algebra/GroupTheory/Groups", "/Mathematics/Algebra/GroupTheory/MatrixGroups", "/Mathematics/Algebra/LinearAlgebra/GeneralLinearAlgebra", "/Mathematics/Algebra/LinearAlgebra/Matrices", "/Mathematics/Algebra/NumberTheory/Congruences", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/PrimeNumbers", "/Mathematics/NumberTheory/Congruences/Cauchy-DavenportTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/GeneralNumberTheory/AdditiveNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/NumberTheory/PrimeNumbers/PrimeRepresentations" ]	The modular distance condition forces f to be a linear map preserving the dot product, i.e., an orthogonal transformation over the finite field.	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ours_404	Consider each subset as a vector in $\mathbb{F}_{2}^{2015}$, and write these vectors in a $2015 \times 2015$ matrix. The number of good permutations is the permanent of this matrix. The permanent is congruent to the determinant $(\bmod\ 2)$, so we want these $2015$ vectors to be linearly independent. Thus, the total number of ways is \[ \prod_{i=0}^{2014}\left(2^{2015}-2^{i}\right) \] Calculating the largest powers of 2 and 3 that divide this product, we find $k_{2}=2029105$ and $k_{3}=1507$ (using the Lifting The Exponent lemma). Therefore, $k_{2}+k_{3}=2030612$. $\boxed{2030612}$	2030612	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Let $ N $ be the number of $ 2015 $-tuples of (not necessarily distinct) subsets $ ( S_{1}, S_{2}, \ldots, S_{2015} ) $ of $\{1,2, \ldots, 2015\}$ such that the number of permutations $\sigma$ of $\{1,2, \ldots, 2015\}$ satisfying $\sigma(i) \in S_{i}$ for all $1 \leq i \leq 2015$ is odd. Let $k_{2}, k_{3}$ be the largest integers such that $2^{k_{2}} \mid N$ and $3^{k_{3}} \mid N$ respectively. Find $k_{2}+k_{3}$.	[ "/Mathematics/Algebra/LinearAlgebra/Determinants", "/Mathematics/Algebra/LinearAlgebra/LinearIndependence", "/Mathematics/Algebra/LinearAlgebra/Matrices", "/Mathematics/Algebra/LinearAlgebra/Permanents", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/Parity", "/Mathematics/Algebra/ValuationTheory", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Divisors/Divides", "/Mathematics/NumberTheory/Divisors/DivisibilityRules", "/Mathematics/NumberTheory/Divisors/Divisible", "/Mathematics/NumberTheory/Divisors/GreatestDividingExponent", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/NonnegativeInteger", "/Mathematics/NumberTheory/Integers/PositiveInteger" ]	Use that the permanent of a {0,1}-matrix is odd exactly when its determinant is 1 mod 2, so the subsets must form an invertible matrix over F₂.	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ours_405	Let $S_{n}$ be the number of expressions resulting in a scalar and $V_{n}$ the number resulting in a vector. We have $S_{1}=0$ and $V_{1}=1$. The recurrence relations are: \[ S_{n} = \sum_{k=1}^{n-1} S_{k} S_{n-k} + \sum_{k=1}^{n-1} V_{k} V_{n-k} \] \[ V_{n} = 2 \sum_{k=1}^{n-1} S_{k} V_{n-k} + \sum_{k=1}^{n-1} V_{k} V_{n-k} \] Define generating functions: \[ S(x) = S_{1} x + S_{2} x^{2} + \ldots \] \[ V(x) = V_{1} x + V_{2} x^{2} + \ldots \] The generating functions satisfy: \[ S(x) = S(x)^{2} + V(x)^{2} \] \[ V(x) - x = 2 S(x) V(x) + V(x)^{2} \] Rewriting, we have: \[ V(x)(1 - 2 S(x)) = x + S(x) - S(x)^{2} \] Modulo 2, we find: \[ S(x) \equiv x^{2} + S(x^{4}) \pmod{2} \] \[ V(x) \equiv x + V(x^{2}) \pmod{2} \] Thus: \[ S(x) \equiv x^{2} + x^{8} + x^{32} + x^{128} + \ldots \] \[ V(x) \equiv x + x^{2} + x^{4} + x^{8} + \ldots \] Now consider modulo 4: \[ S(x) \equiv (x^{2} + x^{8} + x^{32} + \ldots)^{2} + (x + x^{2} + x^{4} + x^{8} + \ldots)^{2} \] \[ V(x) \equiv x + (x + x^{2} + x^{4} + x^{8} + \ldots)^{2} + 2(x^{2} + x^{8} + x^{32} + \ldots)(x + x^{2} + x^{4} + x^{8} + \ldots) \pmod{4} \] Adding these, we find: \[ S(x) + V(x) \equiv x + 2(x + x^{2} + x^{4} + x^{8} + \ldots)^{2} + (x^{2} + x^{8} + x^{32} + \ldots)^{2} + 2(x^{2} + x^{8} + x^{32} + \ldots)(x + x^{2} + x^{4} + x^{8} + \ldots) \pmod{4} \] The result is $1 \pmod{4}$ for powers of four, $2 \pmod{4}$ for twice powers of four and numbers $n=2^{a}+2^{b}$ for $a \neq b$ not both even. The rest are divisible by four. Since $2^{0}, 2^{1}, \ldots, 2^{19}$ are the only powers of two at most $1,000,000$ and since the numbers from $1,000,001$ to $2^{20}-1$ are all zero $\pmod{4}$, we find: \[ 10(1) + 10(2) + \left(\binom{20}{2} - \binom{10}{2}\right)(2) = 300 \] Thus, the answer is $\boxed{300}$.	300	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall15Solns.md" }	Given vectors $v_{1}, \ldots, v_{n}$ and the string $v_{1} v_{2} \ldots v_{n}$, we consider valid expressions formed by inserting $n-1$ sets of balanced parentheses and $n-1$ binary products, such that every product is surrounded by parentheses and is one of the following forms: - A "normal product" $a b$, which takes a pair of scalars and returns a scalar, or takes a scalar and vector (in any order) and returns a vector. - A "dot product" $a \cdot b$, which takes in two vectors and returns a scalar. - A "cross product" $a \times b$, which takes in two vectors and returns a vector. An example of a valid expression when $n=5$ is $\left(\left(\left(v_{1} \cdot v_{2}\right) v_{3}\right) \cdot\left(v_{4} \times v_{5}\right)\right)$, whose final output is a scalar. An example of an invalid expression is $\left(\left(\left(v_{1} \times\left(v_{2} \times v_{3}\right)\right) \times\left(v_{4} \cdot v_{5}\right)\right)\right.$; even though every product is surrounded by parentheses, in the last step one tries to take the cross product of a vector and a scalar. Denote by $T_{n}$ the number of valid expressions (with $T_{1}=1$ ), and let $R_{n}$ denote the remainder when $T_{n}$ is divided by 4. Compute $R_{1}+R_{2}+R_{3}+\ldots+R_{1,000,000}$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Bracketing", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecursiveSequence" ]	Translate the scalar/vector recurrences into generating functions and analyze them modulo 2 and 4, which forces the coefficients to follow a binary‑power pattern.	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ours_407	Let $ p=2017 $ and consider the expected value $ M=\mathbb{E}(a_{n+1}-a_{n}) $, where $ n $ ranges from $ 0 $ to $ p-2 $. We need $ M=\frac{p}{2} $ for $ a_{2016}=1+\binom{2017}{2} $. Note that for $ k=1 $, $ M=p $, so we assume $ k>1 $. If $ k^{n+1}-k^{n} \equiv i \pmod{p} $, for $ 1 \leq i \leq p-1 $, then $ i \equiv k^{n}(k-1) \pmod{p} $. Let $\operatorname{ord}_{p}(k)=d$. We will show that $ M=\frac{p}{2} $ if and only if $ d $ is even. Note that $ M $ is also $\mathbb{E}(a_{n+1}-a_{n})$, where $ n $ ranges from $ 0 $ to $ d-1 $. If $ d=2g $ is even, then $ k^{-g} \equiv -1 \pmod{p} $ so $ k^{j} \equiv -k^{j+g} \neq 0 \pmod{p} $ for all nonnegative integers $ j $. Since $ k-1 \neq 0 \pmod{p} $, $ k^{j}(k-1) = -k^{j+g}(k-1) \neq 0 \pmod{p} $. Thus, $ M=\frac{p}{2} $ by ranging over all $ 0 \leq j \leq g-1 $. If $ d $ is odd, then it is impossible for $ M=\frac{p}{2} $, as it would require $ a_{d}-a_{0}=\frac{d p}{2} $, which is not a positive integer. Let $ q $ be the largest odd factor of $ p-1 $. It is well-known that the number of residues with order $ d $ modulo $ p $, where $ d \mid p-1 $, is $\phi(d)$. Therefore, our answer is $(p-1)-\sum_{d \mid q} \phi(d)=(p-1)-q$. Setting $ p=2017 $ gives $ q=63 $, and our answer is $ 1953 $. $\boxed{1953}$	1953	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-2.md" }	For a positive integer $ k $, define the sequence $\{a_{n}\}_{n \geq 0}$ such that $ a_{0}=1 $ and for all positive integers $ n $, $ a_{n} $ is the smallest positive integer greater than $ a_{n-1} $ for which $ a_{n} \equiv k a_{n-1} \pmod{2017} $. What is the number of positive integers $ 1 \leq k \leq 2016 $ for which $ a_{2016}=1+\binom{2017}{2} $?	[ "/Mathematics/DiscreteMathematics/FiniteGroups/CyclicGroup", "/Mathematics/DiscreteMathematics/FiniteGroups/ModuloMultiplicationGroup", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/ModuloMultiplicationGroup", "/Mathematics/NumberTheory/Congruences/ModuloOrder", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/MultiplicativeOrder", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/NumberTheory/Sequences/IntegerSequence", "/Mathematics/NumberTheory/Sequences/Sequence" ]	Show that the required equality holds exactly when the multiplicative order of k modulo 2017 is even (i.e., k^{order/2} ≡ -1).	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ours_408	Let the numbers Mark generate be $s_{1}<s_{2}<\cdots<s_{100}$. Note that any other number $s \in\{1,2, \ldots, 2019\} \backslash S$ is equally likely to be in the open intervals $\left(0, s_{1}\right),\left(s_{1}, s_{2}\right), \ldots\left(s_{99}, s_{100}\right),\left(s_{100}, 2020\right)$, so it follows that $\mathbb{E}\left(s_{k}\right)=\frac{2019-100}{101} \cdot k+k=20 k$. Now, fix $s_{1}, s_{2}, \ldots s_{100}$, and randomly place them in the grid cells. Draw empty cells next to the edges of the grid, and define total cells to be the union of grid cells and empty cells. Say that two total cells are adjacent if they share a side. Define a grid cell to be an edge cell if it is adjacent to an empty cell. Label the rows and columns of the grid by $1,2, \ldots, 10$, and let the cell in the $i$th row and $j$th column be denoted by $c_{i, j}$, with a total of $t_{i, j}$ blocks on it. Furthermore, denote $t_{i, j}=0$ when either $i=0, i=11, j=0$, or $j=11$. Let $a_{i, j}$ for $1 \leq i, j \leq 10$ denote the size of the set $\left\{t: t \in\left\{t_{i-1, j}, t_{i, j-1}, t_{i+1, j}, t_{i, j+1}\right\}\right.$ and $\left.t<t_{i, j}\right\}$. That is, the stack on a grid cell $c_{i, j}$ is taller than $a_{i, j}$ of the stack on its 4 adjacent total cells (where the empty cells are stacks of 0 blocks). Note that the top and bottom of the stacks on grid cells contribute 200 to the surface area of Bark's solid; now, it remains to calculate the lateral surface area of the solid. Note that the region between any two stacks on two adjacent total cells of size $x$ and $y$ contribute $\|x-y\|$ to the surface area, so it becomes clear that the lateral surface area of the solid is simply $\sum_{1<i, j<10}\left(2 a_{i, j}-4\right) t_{i, j}$. Since the $t_{i, j}$'s are fixed, it suffices to maximize $\sum_{1 \leq i, j \leq 10} a_{i, j} t_{i, j}$. Write the set $\left\{a_{i, j}: 1 \leq i, j \leq 10\right\}$ as $\left\{b_{1}, b_{2}, \ldots b_{100}\right\}$, so we need to maximize $M=\sum_{1 \leq i \leq 100} b_{i} s_{101-i}$. Define a sequence $c_{i}$ by $c_{i}=4$ for $1 \leq i \leq 50, c_{i}=2$ for $i=51 \leq i \leq 52, c_{i}=1$ for $53 \leq i \leq 68$, and $c_{i}=0$ for $69 \leq i \leq 100$. We will show that the sequence $c_{i}$ majorizes the sequence $b_{i}$, or that $c_{1}+c_{2}+\cdots+c_{i} \geq b_{1}+b_{2}+\cdots+b_{i}$ for $1 \leq i \leq 100$, where equality holds for $i=100$. Since the sequence $\left\{s_{101-i}\right\}$ is increasing, it's clear that $M$ is maximized when $b_{i}=c_{i}$ for all $i$. Let $y_{i}=b_{1}+b_{2}+\cdots+b_{i}$ and $z_{i}=c_{1}+c_{2}+\cdots+c_{i}$ for $1 \leq i \leq 100$. Note that $y_{100}=220=z_{100}$ since there are 220 pairs of adjacent total cells, and it follows that $y_{i} \leq z_{i}=220$ for $68 \leq i \leq 99$. Furthermore, $0 \leq b_{i} \leq 4$ for all $i$, so $y_{i} \leq z_{i}=4 i$ for $1 \leq i \leq 50$ certainly holds. Next, we will show that $y_{51} \leq z_{51}=202$. If $b_{50}<4$, then it's clear that we are done. Otherwise, assume $b_{50}=4$. We can find a Hamiltonian cycle on the grid cells. Note that $a_{i, j}=4$ cannot hold for consecutive grid cells along the Hamiltonian cycle, since $b_{50}=4$, it follows that in fact every other cell along the Hamiltonian cycle satisfies $a_{i, j}=4$. This means that the cells satisfying $a_{i, j}=4$ are the black squares on a $10 \times 10$ chessboard, but it's then clear that $b_{51} \leq 2$ since any white square on a chessboard neighbors at least two black squares. Now, it remains to prove that $b_{i} \leq c_{i}=152+i$ for $52 \leq i \leq 67$. Note that we can tile the $8 \times 8$ grid cells in rows 2-9 and columns 2-9 with 32 dominoes, and $a_{i, j} \geq 1$ for at least one grid cell in each domino, and $a_{i, j} \geq 1$ for all 36 edge grid cells, so $b_{68} \geq 1$. It's then clear that $b_{i+1}+b_{i+2}+\cdots+b_{68} \geq 68-i$ for $52 \leq i \leq 67$, so we are done. Hence, our answer is maximized when $b_{i}=4$ for $1 \leq i \leq 50, b_{i}=2$ for $51 \leq i \leq 52, b_{i}=1$ for $53 \leq i \leq 68$, and $b_{i}=0$ for $69 \leq i \leq 100$. This means that by linearity of expectation, our answer is \[ \begin{aligned} & \left.200+\mathbb{E}\left(4 \cdot\left(s_{51}+s_{52}+\cdots+s_{100}\right)-2 \cdot\left(s_{33}+s_{34}+\cdots+s_{48}\right)-4 \cdot\left(s_{1}+s_{2}+\cdot+s_{32}\right)\right)\right) \\ = & 200+\left(4 \cdot\left(\mathbb{E}\left(s_{51}\right)+\mathbb{E}\left(s_{52}\right)+\cdots+\mathbb{E}\left(s_{100}\right)\right)-2 \cdot\left(\mathbb{E}\left(s_{33}\right)+\mathbb{E}\left(s_{34}\right)+\cdots+\mathbb{E}\left(s_{48}\right)\right)-4 \cdot\left(\mathbb{E}\left(s_{1}\right)+\mathbb{E}\left(s_{2}\right)+\cdot+\mathbb{E}\left(s_{32}\right)\right)\right. \\ = & 200+302000-25920-42240 \\ = & 234040 \end{aligned} \] $\boxed{234040}$	234040	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-2.md" }	Mark the Martian and Bark the Bartian live on planet Blok, in the year 2019. Mark and Bark decide to play a game on a $10 \times 10$ grid of cells. First, Mark randomly generates a subset $S$ of $\{1,2, \ldots, 2019\}$ with $\|S\|=100$. Then, Bark writes each of the 100 integers in a different cell of the $10 \times 10$ grid. Afterwards, Bark constructs a solid out of this grid in the following way: for each grid cell, if the number written on it is $n$, then she stacks $n \times 1 \times 1$ blocks on top of one other in that cell. Let $B$ be the largest possible surface area of the resulting solid, including the bottom of the solid, over all possible ways Bark could have inserted the 100 integers into the grid of cells. Find the expected value of $B$ over all possible sets $S$ Mark could have generated.	[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace", "/Mathematics/ProbabilityandStatistics/RankStatistics/OrderStatistic" ]	Apply the rearrangement inequality via majorization to pair the largest heights with cells that have the most exposed sides, thus maximizing the surface area sum.	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ours_409	Suppose that $ \triangle ABC $ and $ \triangle DBC $ have the same Euler line $ \ell $. Then, note that the circumcenters of $ \triangle ABC $ and $ \triangle DBC $ must lie on both the perpendicular bisector of $ BC $ and $ \ell $. Because $ AB \neq AC $, $ \ell $ is not the perpendicular bisector of $ BC $, so the two lines do not coincide. This means that $ \triangle ABC $ and $ \triangle DBC $ have a common circumcenter, $ O $. Let $ M $ be the midpoint of $ BC $. Note that a homothety centered at $ M $ with ratio 3 takes $ \ell $ to $ AD $ because the centroids of $ \triangle ABC $ and $ \triangle DBC $ lie on $ \ell $. Hence, $ D $ does not exist if and only if the parallel to $ \ell $ through $ A $ does not intersect the circumcircle of $ \triangle ABC $ at a point different from $ A, B, $ and $ C $. We either have that $ \ell \parallel AB, \ell \parallel AC $, or $ \ell $ is parallel to the tangent to the circumcircle of $ \triangle ABC $ at $ A $. Let $ x = BC $. By the Law of Cosines, $\cos A = \frac{25-x^2}{24}, \cos B = \frac{x^2-7}{6x},$ and $\cos C = \frac{x^2+7}{8x}$. Let $ H $ and $ O $ respectively be the orthocenter and circumcenter of $ \triangle ABC $. It is well-known that the heights from $ H $ and $ O $ to $ AB $ have lengths $ 2R \cos A \cos B $ and $ R \cos C $, respectively. If $ OH \parallel AB $, then we must have $ 2 \cos A \cos B = \cos C $, or \[ 2 \cdot \frac{25-x^2}{24} \cdot \frac{x^2-7}{6x} = \frac{x^2+7}{8x} \Longrightarrow (25-x^2)(x^2-7) = 9(x^2+7) \Longrightarrow x^4 - 23x^2 + 238 = 0, \] which has no real solutions. Similarly, if $ OH \parallel AC $, then we must have $ 2 \cos A \cos C = \cos B $, or \[ 2 \cdot \frac{25-x^2}{24} \cdot \frac{x^2+7}{8x} = \frac{x^2-7}{6x} \Longrightarrow (25-x^2)(x^2+7) = 16(x^2-7) \Longrightarrow x^4 - 2x^2 + 287 \Longrightarrow x^2 = 1 \pm 12\sqrt{2}, \] so $ x^2 = 1 + 12\sqrt{2} $ is the only solution for this case. Finally, if $ \ell $ is parallel to the tangent to the circumcircle of $ \triangle ABC $ at $ A $, then note that $ \angle AOG = 90^\circ $, where $ G $ is the centroid of $ \triangle ABC $. Let $ R $ be the circumradius of $ \triangle ABC $. It is well known that $ AG^2 = \frac{-a^2 + 2b^2 + 2c^2}{9} $ and $ OG^2 = R^2 - \frac{a^2 + b^2 + c^2}{9} $. Hence, by the Pythagorean theorem we have that \[ AO^2 + OG^2 = AG^2 \Longrightarrow R^2 + R^2 - \frac{a^2 + b^2 + c^2}{9} = \frac{-a^2 + 2b^2 + 2c^2}{9} \Longrightarrow b^2 + c^2 = 6R^2 \Longrightarrow R^2 = \frac{25}{6}. \] We just want to find the product of the possible lengths of the third side of a triangle with sides 4 and 3 and a fixed circumradius $ R $. Suppose that those sides correspond to an inscribed arc of measure $ x $ and $ y $ respectively. Then, note that the third side can correspond to an inscribed arc of measure $ x-y $ or $ x+y $. Then, their product is \[ 2R \sin(x+y) \cdot 2R \sin(x-y) = 4R^2 \sin(x+y) \sin(x-y) = 2R^2(\cos(2x) - \cos(2y)) = 4R^2(\sin^2 x - \sin^2 y) = 4^2 - 3^2 = 7. \] The product of the squares of all solutions is $ 7^2(1 + 12\sqrt{2}) = 49 + 588\sqrt{2} $, so the answer is $\boxed{10782}$.	10782	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-2.md" }	Let $ \triangle ABC $ be a triangle with $ AB = 3 $ and $ AC = 4 $. It is given that there does not exist a point $ D $, different from $ A $ and not lying on line $ BC $, such that the Euler line of $ \triangle ABC $ coincides with the Euler line of $ \triangle DBC $. The square of the product of all possible lengths of $ BC $ can be expressed in the form $ m + n \sqrt{p} $, where $ m, n $, and $ p $ are positive integers and $ p $ is not divisible by the square of any prime. Find $ 100m + 10n + p $. Note: For this problem, consider every line passing through the center of an equilateral triangle to be an Euler line of the equilateral triangle. Hence, if $ D $ is chosen such that $ \triangle DBC $ is an equilateral triangle and the Euler line of $ \triangle ABC $ passes through the center of $ \triangle DBC $, then consider the Euler line of $ \triangle ABC $ to coincide with "the" Euler line of $ \triangle DBC $.	[ "/Mathematics/Geometry/Centroids", "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/Trigonometry/Angles", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions", "/Mathematics/Geometry/Trigonometry/TrigonometricIdentities" ]	Translate the condition that the Euler line of ABC is parallel to a side (or the tangent at A) into a simple cosine relation using OH ∥ AB or OH ∥ AC, then solve for the unknown side length.	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ours_410	Let $P(x, y, z)$ denote the assertion that $x^{2}+y^{2}=z^{2}$ and $f(x) f(y)=f(z)$. Let $a$ denote an odd positive integer and let $k$ denote a positive integer. Note that $f(1)=1$, and if $a>1$, \[ \begin{aligned} P\left(a, \frac{a^{2}-1}{2}, \frac{a^{2}+1}{2}\right) & \Longrightarrow f(a)\left\|f\left(\frac{a^{2}+1}{2}\right)\right\|\left(\frac{a^{2}+1}{2}\right)^{2016} \\ & \Longrightarrow f(a) \left\lvert\, \operatorname{gcd}\left(a^{2016},\left(\frac{a^{2}+1}{2}\right)^{2016}\right)=\operatorname{gcd}\left(a, \frac{a^{2}+1}{2}\right)^{2016}=1\right. \end{aligned} \] so $f(x)=1$ for all odd integers $x$. Then, \[ P\left(2^{k} a, 2^{2 k-2} a^{2}-1,2^{2 k-2} a^{2}+1\right) \Longrightarrow f\left(2^{k} a\right) \mid f\left(2^{2 k-2} a^{2}+1\right) \] If $k>1$, then $f\left(2^{2 k-2} a^{2}+1\right)=1$ since $2^{2 k-2} a^{2}+1$ is odd, so $f\left(2^{k} a\right)=1$, so $f(x)=1$ for all integers $x$ divisible by $4$. If $k=1$, then $f(2 a)\left\|f\left(a^{2}+1\right)\right\|\left(a^{2}+1\right)^{2016} \Longrightarrow f(2 a) \mid \operatorname{gcd}\left((2 a)^{2016},\left(a^{2}+1\right)^{2016}\right)=\operatorname{gcd}\left(2 a, a^{2}+1\right)^{2016}=2^{2016}$. Hence, $f(2 a) \in\left\{2^{0}, 2^{1}, 2^{2}, \ldots, 2^{2016}\right\}$ for all odd $a$. Note that $f(2)$ can be any of these $2017$ values without affecting any of the other values of $f$ since $2$ is not a part of any Pythagorean triple. For all $0 \leq i \leq 2016$, there exists a function $f$ such that $f(2014)=2^{i}$, namely $f(2 a)=2^{i}$ for all odd $a>1$. Thus, it suffices to find the number of values that can be achieved by $2^{x}+2^{y}-1$ for integers $0 \leq x, y \leq 2016$, which is just $\binom{2018}{2}=2035153$. $\boxed{2035153}$	2035153	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-2.md" }	Let $\mathbb{N}$ denote the set of positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that the following conditions hold: - For any $n \in \mathbb{N}$, we have $f(n) \mid n^{2016}$. - For any $a, b, c \in \mathbb{N}$ satisfying $a^{2}+b^{2}=c^{2}$, we have $f(a) f(b)=f(c)$. Over all possible functions $f$, determine the number of distinct values that can be achieved by $f(2014) + f(2) - f(2016)$.	[ "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/NumberTheoreticFunctions/GeneralNumberTheoreticFunctions", "/Mathematics/NumberTheory/NumberTheoreticFunctions/GreatestCommonDivisor" ]	Parametrize Pythagorean triples to force f of all odd numbers and multiples of 4 to be 1, leaving only f(2·odd) which can be any power of 2 up to 2^2016.	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ours_411	Let $ n = 2020 $. The polynomial is unique by solving a linear system of $(n+1)^2$ linearly independent equations in $(n+1)^2$ variables. We claim that \[ P(x, y) = \binom{x}{0}\binom{y}{0} + \binom{x}{1}\binom{y}{1} + \binom{x}{2}\binom{y}{2} + \ldots + \binom{x}{n}\binom{y}{n} \] satisfies the condition. Indeed, note that \[ P(i, j) = \sum_{k=0}^{n} \binom{i}{k} \binom{j}{k} = \sum_{k=0}^{\min(i, j)} \binom{i}{k} \binom{j}{k} \] because $\binom{i}{k} = 0$ for $k > i$ and $\binom{j}{k} = 0$ for $k > j$. Without loss of generality, assume that $i \leq j$. Then, we have \[ P(i, j) = \sum_{k=0}^{i} \binom{i}{k} \binom{j}{k} = \sum_{k=0}^{i} \binom{i}{k-i} \binom{j}{k} = \binom{i+j}{i} \] by Vandermonde's identity, as desired. Hence, the answer is \[ P(2n, 2n) = \sum_{k=0}^{n} \binom{2n}{k}^2 = \sum_{k=0}^{n-1} \binom{2n}{k} \binom{2n}{2n-k} + \binom{2n}{n}^2 = \frac{\sum_{k=0}^{2n} \binom{2n}{k} \binom{2n}{2n-k} + \binom{2n}{n}^2}{2} = \frac{\binom{4n}{2n} + \binom{2n}{n}^2}{2} \] by Vandermonde's identity again. Now, by Lucas's Theorem, the answer is \[ \frac{\binom{8080}{4040} + \binom{4040}{2020}^2}{2} \equiv \frac{\left(\binom{4}{2}\binom{12}{6} + \left[\begin{array}{l}2 \\ 1\end{array}\right)\binom{6}{3}\right]^2}{2} = 3572 \equiv 1555 \pmod{2017} \] Thus, the remainder when $P(4040, 4040)$ is divided by 2017 is $\boxed{1555}$.	1555	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-2.md" }	Let $ P(x, y) $ be a polynomial such that $\operatorname{deg}_{x}(P), \operatorname{deg}_{y}(P) \leq 2020$ and \[ P(i, j) = \binom{i+j}{i} \] for all $2021^2$ ordered pairs $(i, j)$ with $0 \leq i, j \leq 2020$. Find the remainder when $P(4040, 4040)$ is divided by 2017.	[ "/Mathematics/Algebra/NumberTheory/Congruences", "/Mathematics/Algebra/NumberTheory/NumberTheoreticFunctions", "/Mathematics/Algebra/NumberTheory/PrimeNumbers", "/Mathematics/Algebra/Polynomials/Binomial", "/Mathematics/Algebra/Polynomials/BivariatePolynomial", "/Mathematics/Algebra/Polynomials/MultivariatePolynomial", "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/PrimeNumbers/MiscellaneousPrimes", "/Mathematics/NumberTheory/PrimeNumbers/Prime-RelatedNumbers", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties" ]	Express the polynomial as a sum of products of binomial coefficients and use Vandermonde's identity to match the given values.	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ours_412	Without loss of generality, let the centroid $G$ be at the origin, so $G=0$ as a vector. Let $\mathbb{E}_{n}$ denote the expectation over all the choices of $(i, j) \in S_{m}$ for $3 \leq m \leq n-1$. In other words, the points $X_{1}, \ldots, X_{n}$ have been chosen. Define \[ a_{n}=\mathbb{E}_{n}\left[\mathbb{E}_{1 \leq i \leq n}\left[X_{i}^{2}\right]\right] \text{ and } b_{n}=\mathbb{E}_{n}\left[\mathbb{E}_{1 \leq i \leq n}\left[X_{i}\right]^{2}\right] \] A direct computation shows that $a_{3}=10$ and $b_{3}=0$. We now compute a recursion for $a_{n}$ and $b_{n}$. Note that \[ \mathbb{E}_{n+1}\left[X_{n+1}^{2}\right]=\mathbb{E}_{n}\left[\mathbb{E}_{(i, j) \in S_{n}}\left[\left(\frac{X_{i}+X_{j}}{2}\right)^{2}\right]\right]=\frac{1}{2} a_{n}+\frac{1}{2} b_{n} \] Therefore, \[ a_{n+1}=\mathbb{E}_{n+1}\left[\mathbb{E}_{1 \leq i \leq n+1}\left[X_{i}^{2}\right]\right]=\frac{n a_{n}+\mathbb{E}_{n+1}\left[X_{n+1}^{2}\right]}{n+1}=\frac{(2 n+1) a_{n}+b_{n}}{2 n+2} \] We can do similar computations for $b_{n}$. We can compute \[ \mathbb{E}_{n+1}\left[\mathbb{E}_{1 \leq i \leq n+1}\left[X_{i}\right]^{2}\right]=\mathbb{E}_{n}\left[\mathbb{E}_{(i, j) \in S_{n}}\left[\left(\frac{X_{1}+\cdots+X_{n}+X_{n+1}}{n+1}\right)^{2}\right]\right] \] If we let $T_{n}=X_{1}+\ldots X_{n}$, then the previous expression equals $\frac{1}{(n+1)^{2}} \mathbb{E}_{n}\left[\mathbb{E}_{(i, j) \in S_{n}}\left[T_{n}^{2}+2 T_{n} X_{n+1}+X_{n+1}^{2}\right]\right]=\frac{1}{(n+1)^{2}} \mathbb{E}_{n}\left[\frac{n+2}{n} T_{n}^{2}+X_{n+1}^{2}\right]=\frac{\left(2 n^{2}+4 n+1\right) b_{n}+a_{n}}{2(n+1)^{2}}$ as $\mathbb{E}_{n}\left[X_{n+1}^{2}\right]=\frac{1}{2} a_{n}+\frac{1}{2} b_{n}$ as above and $\mathbb{E}_{n}\left[T_{n}^{2}\right]=n^{2} b_{n}$ by definition. Our next claim is the explicit formulas for $b_{n+1}-b_{n}$ and $a_{n+1}-a_{n}$ for $n \geq 3$. The formulas are \[ a_{n+1}-a_{n}=-\frac{96}{7} \frac{1}{4^{n+1} \cdot n}\binom{2(n+1)}{n+1} \text{ and } b_{n+1}-b_{n}=\frac{96}{7} \frac{1}{4^{n+1} \cdot n(n+1)}\binom{2(n+1)}{n+1} . \] Though the proof is messy, one can verify this by induction. The proof is omitted. Now, define $s_{n}=\mathbb{E}_{n+1}\left[X_{n+1}^{2}\right]=\frac{1}{2}\left(a_{n}+b_{n}\right)$ for $n \geq 3$, and otherwise, $s_{n}=0$. In particular, $s_{3}=\frac{1}{2} a_{3}=5$. By the above, \[ s_{n+1}-s_{n}=-\frac{48}{7} \frac{1}{4^{n+1} \cdot(n+1)}\binom{2(n+1)}{n+1} \] Our final step will be to compute the generating function $\sum_{n \geq 0} s_{n+3} x^{n}=(1-x)^{-1}\left(5+\sum_{n \geq 1}\left(s_{n+3}-s_{n+2}\right) x^{n}\right)=(1-x)^{-1}\left(5-\frac{48}{7} \sum_{n \geq 1} \frac{1}{4^{n+3}(n+3)}\binom{2(n+3)}{n+3} x^{n}\right)$. Let's only deal with the innermost sum for now. Note that \[ \sum_{n \geq 1} \frac{1}{4^{n+3}(n+3)}\binom{2(n+3)}{n+3} x^{n}=x^{-3} \int \sum_{n \geq 1} \frac{1}{4^{n+3}}\binom{2(n+3)}{n+3} x^{n+2} d x \] where the $\int$ denotes an antiderivative (we will find the correct constant term later). Only dealing with the antiderivative right now, and remembering that $\sum_{n \geq 0} \frac{1}{4^{n}}\binom{2 n}{n} x^{n}=\frac{1}{\sqrt{1-x}}$, \[ \begin{gathered} \int \sum_{n \geq 1} \frac{1}{4^{n+3}}\binom{2(n+3)}{n+3} x^{n+2} d x=\int \frac{1}{x}\left(\frac{1}{\sqrt{1-x}}-1-\frac{1}{2} x-\frac{3}{8} x^{2}-\frac{5}{16} x^{3}\right) d x \\ =C-2 \ln (1+\sqrt{1-x})-\frac{1}{2} x-\frac{3}{16} x^{2}-\frac{5}{48} x^{3} \end{gathered} \] for some constant $C$. Note that substituting $x=0$ should give a result of $0$, so $C=2 \ln 2$. Substituting everything back into the original expression, \[ \sum_{n \geq 0} s_{n+3} x^{n}=(1-x)^{-1}\left(5-\frac{48}{7}\left(\frac{2 \ln 2-2 \ln (1+\sqrt{1-x})-\frac{1}{2} x-\frac{3}{16} x^{2}-\frac{5}{48} x^{3}}{x^{3}}\right)\right) \] Substituting $x=\frac{3}{4}$ gives us the final answer \[ \frac{1136}{21}-\frac{16384}{63} \ln 2+\frac{8192}{63} \ln 3 \Longrightarrow\|p\|+\|q\|+\|r\|=\frac{9328}{21} \Longrightarrow 100 m+n=932821 \] $\boxed{932821}$	932821	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-2.md" }	Let $X_{1} X_{2} X_{3}$ be a triangle with $X_{1} X_{2}=4, X_{2} X_{3}=5, X_{3} X_{1}=7$, and centroid $G$. For all integers $n \geq 3$, define the set $S_{n}$ to be the set of $n^{2}$ ordered pairs $(i, j)$ such that $1 \leq i \leq n$ and $1 \leq j \leq n$. Then, for each integer $n \geq 3$, when given the points $X_{1}, X_{2}, \ldots, X_{n}$, randomly choose an element $(i, j) \in S_{n}$ and define $X_{n+1}$ to be the midpoint of $X_{i}$ and $X_{j}$. The value of \[ \sum_{i=0}^{\infty}\left(\mathbb{E}\left[X_{i+4} G^{2}\right]\left(\frac{3}{4}\right)^{i}\right) \] can be expressed in the form $p+q \ln 2+r \ln 3$ for rational numbers $p, q, r$. Let $\|p\|+\|q\|+\|r\|=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100 m+n$.	[ "/Mathematics/ProbabilityandStatistics/Moments/Moment", "/Mathematics/ProbabilityandStatistics/Moments/RawMoment", "/Mathematics/ProbabilityandStatistics/Probability/AlgebraofRandomVariables" ]	Translate the random midpoint process into recurrences for the expected squared distance and solve them via a generating function that uses the known series for central binomial coefficients.	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ours_413	It is well known that $ (B_1C_1P) $ is the $ A $-mixtilinear circle $ \omega $ of $ \triangle ABC $, i.e., $ \omega $ is tangent to $ AB, AC, $ and the circumcircle of $ \triangle ABC $. Lemma 1: $ BC_1 : B_1C = BP : CP $. Proof: Let $ BP, CP $ meet $ \omega $ at $ B_3, C_3 $. A homothety centered at $ P $ takes $ \omega $ to the circumcircle of $ \triangle ABC $ and $ B_3C_3 $ to $ BC $, so $ B_3C_3 \parallel BC $. Then $ BC_1^2 : B_1C^2 = BB_3 \cdot BP : CC_3 \cdot CP = BP^2 : CP^2 $, hence $ BC_1 : B_1C = BP : CP $. Now let $ NI $ meet $ BC $ at $ R $, so that by the Angle Bisector Theorem on $ \angle BPC $, $ BR : CR = BP : CP $. Combining this with Lemma 1, we know by the converse of Ceva's theorem that $ AR, BB_1, CC_1 $ concur. Let $ Q' $ be the projection of $ R $ onto $ B_1C_1 $ and let $ B_1C_1, BC $ meet at $ S $. Due to the concurrency, $ (S, R; B, C) $ is harmonic, hence since $ \angle SQ'R = 90^\circ $, it's well-known that $ Q'R, Q'S $ must bisect $ \angle BQ'C $. Then $ \angle CQ'B_1 = \angle BQ'C_1 $, $ \angle BC_1Q' = \angle CB_1Q' $, so $ \triangle BC_1Q' \sim \triangle CB_1Q' $, thus $ BQ' : CQ' = BC_1 : CB_1 = BP : CP $. It's clear that $ C_1Q : B_1Q = AC_1 \sin QAC : AB_1 \sin B_1AQ = \sin BAP : \sin CAP = BP : CP $, hence $ Q = Q' $. Define $ X = BQ \cap AC, Y = CQ \cap AB $. Then since $ QY, QC_2 $ are reflections across $ B_1C_1 $, and it's easy to see $ C_1C_2, AB $ are reflections, we see that $ C_2, Y $ are reflections across $ B_1C_1 $, and similarly $ X, B_2 $ are reflections so $ B_2C_2 = XY $. It remains to compute some lengths. It's known that $ \omega $ is the image of $ \gamma $ under a homothety centered at $ A $ with factor $ \frac{1}{\cos^2 0.5A} $, so $ AB_1 = AC_1 = \frac{(s-a)}{\cos^2 0.5A} = \frac{b+c-a}{1+\cos A} = \frac{2bc}{a+b+c} $, and thus $ BC_1 = \frac{c(a+c-b)}{a+b+c}, CB_1 = \frac{b(a+b-c)}{a+b+c} $. Now from similar triangles $ YBQ, XCQ $ we know that $ BY : CX = BC_1 : B_1C = c(a+c-b) : b(a+b-c) $. From $ \angle XBY = \angle XCY $ we know $ BCXY $ is cyclic, hence $ AX \cdot AC = AY \cdot AB $. Let $ k $ be the ratio of similarity between $ \triangle AXY, \triangle ABC $ so that $ BY = c-bk, CX = b-ck $. Then using the ratio from before, we can solve to deduce $ k = \frac{2bc}{b^2+c^2+ab+ac} \Longrightarrow B_2C_2 = XY = ka = \frac{2abc}{b^2+c^2+ab+ac} $ which is $ \frac{720}{163} $, yielding an answer of $ 72163 $. $\boxed{72163}$	72163	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-3.md" }	Let $ \triangle ABC $ be a triangle with $ BC = 9 $, $ CA = 8 $, and $ AB = 10 $. Let the incenter and incircle of $ \triangle ABC $ be $ I $ and $ \gamma $, respectively, and let $ N $ be the midpoint of the major arc $ BC $ of the circumcircle of $ \triangle ABC $. Line $ NI $ meets the circumcircle of $ \triangle ABC $ a second time at $ P $. Let the line through $ I $ perpendicular to $ AI $ meet segments $ AB, AC, $ and $ AP $ at $ C_1, B_1, $ and $ Q $, respectively. Let $ B_2 $ lie on segment $ CQ $ such that line $ B_1B_2 $ is tangent to $ \gamma $, and let $ C_2 $ lie on segment $ BQ $ such that line $ C_1C_2 $ is tangent to $ \gamma $. The length of $ B_2C_2 $ can be expressed in the form $ \frac{m}{n} $ for relatively prime positive integers $ m $ and $ n $. Determine $ 100m + n $.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Use the A‑mixtilinear incircle tangency points to get BC₁:B₁C = BP:CP, then apply Ceva and harmonic division to locate Q so that B₂C₂ reduces to XY, a similar‑triangle length.	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ours_414	Notice that it is necessary and sufficient to have $ q(x) \mid q\left(x^{2}\right) $ due to polynomial long division. If $ r $ is a root of $ q(x) $, then $ r^{2} $ is also a root. Thus, $ r^{2^{n}} $ is a root of $ q $ for all positive integers $ n $. Since the set $ \{r^{2^{n}}: n \in \mathbb{N}\} $ is finite, there are positive integers $ m \neq n $ with $ r^{2^{m}} = r^{2^{n}} $. Therefore, each root of $ q(x) $ is either zero or a root of unity. Let \[ q(x) = x^{n} \prod_{m=1}^{\infty} \Phi_{m}(x)^{e_{m}} \] where $ n $ and the $ e_{i} $ are nonnegative integers with \[ n + \sum_{m=1}^{\infty} e_{m} \phi(m) = 12 \] defining $\Phi_{m}$ to be the $m$-th cyclotomic polynomial. Utilizing $\Phi_{n}\left(x^{2}\right) = \Phi_{n}(x) \Phi_{2n}(x)$ when $n$ is odd and $\Phi_{n}\left(x^{2}\right) = \Phi_{2n}(x)$ when $n$ is even, we find that $ q(x) \mid q\left(x^{2}\right) $ becomes \[ x^{n} \prod_{m=1}^{\infty} \Phi_{m}(x)^{e_{m}} \mid x^{2n} \prod_{k=1}^{\infty}\left(\Phi_{2k-1}(x) \Phi_{4k-2}(x)\right)^{e_{2k-1}} \prod_{k=1}^{\infty} \Phi_{4k}(x)^{e_{2k}} \] It follows that $ q(x) \mid q\left(x^{2}\right) $ if and only if $ e_{n} \geq e_{2n} $ for all integers $ n $. Therefore, if $ t $ is the largest non-negative integer for which $ x^{t} \mid q(x) $, then we can break $ q(x) / x^{t} $ uniquely into products of the form \[ \kappa_{2^{v}m}(x) = \Phi_{m}(x) \Phi_{2m}(x) \cdots \Phi_{2^{v}m}(x) \] where $ m $ is odd. For instance, \[ \Phi_{3}(x)^{5} \Phi_{6}(x)^{3} \Phi_{12}(x)^{2} \Phi_{24}(x) = \kappa_{24}(x) \kappa_{12}(x) \kappa_{6}(x) \kappa_{3}(x)^{2} \] Let $ k_{2^{v}m} = \operatorname{deg} \kappa_{2^{v}m}(x) $. Note that $ k_{2^{v}m} = 2^{v} \phi(m) $ if $ m $ is odd and $ v $ is a nonnegative integer, and that this degree must be at most 12. The only odd powers of primes $ n $ with $\phi(n) \leq 12$ are $ n = 3, 9, 5, 7, 11, 13 $, with $\phi(3) = 2, \phi(9) = 6, \phi(5) = 4, \phi(7) = 6, \phi(11) = 10, \phi(13) = 12$. Hence, it follows from $\phi(mn) = \phi(m) \phi(n)$ for relatively prime $ m, n $, the only other odd $ n $ with $\phi(n) \leq 12$ are $ n = 1, 15, 21 $ with $\phi(1) = 1, \phi(15) = 8, \phi(21) = 12$. Hence, $ k_{1} = 1, k_{2} = k_{3} = 2, k_{4} = k_{5} = k_{6} = 4, k_{7} = k_{9} = 6, k_{8} = k_{10} = k_{12} = k_{15} = 8, k_{11} = 10 $, and $ k_{13} = k_{14} = k_{18} = k_{21} = 12 $. It then becomes clear that the number of desired $ q(x) $ is just the coefficient of $ x^{12} $ in \[ \frac{1}{(1-x)^{2}\left(1-x^{2}\right)^{2}\left(1-x^{4}\right)^{3}\left(1-x^{6}\right)^{2}\left(1-x^{8}\right)^{4}\left(1-x^{10}\right)\left(1-x^{12}\right)^{4}} \] where the extra factor of $ 1-x $ in the denominator accounts for the factor of $ x^{t} $ in $ q(x) $. Now, it simply remains to evaluate the above expression modulo $ x^{13} $: \[ \begin{aligned} & \frac{1}{(1-x)^{2}\left(1-x^{2}\right)^{2}\left(1-x^{4}\right)^{3}\left(1-x^{6}\right)^{2}\left(1-x^{8}\right)^{4}\left(1-x^{10}\right)\left(1-x^{12}\right)^{4}} \\ \equiv & \frac{\left(1+x^{10}\right)\left(1+4x^{12}\right)}{(1-x)^{2}\left(1-x^{2}\right)^{2}\left(1-x^{4}\right)^{3}\left(1-x^{6}\right)^{2}\left(1-x^{8}\right)^{4}} \quad\left(\bmod x^{13}\right) \\ \equiv & \frac{\left(1+4x^{8}\right)\left(1+x^{10}+4x^{12}\right)}{(1-x)^{2}\left(1-x^{2}\right)^{2}\left(1-x^{4}\right)^{3}} \quad\left(\bmod x^{13}\right) \\ \equiv & \frac{\left(1+2x^{6}+3x^{12}\right)\left(1+4x^{8}+x^{10}+4x^{12}\right)}{(1-x)^{2}\left(1-x^{2}\right)^{2}} \quad\left(\bmod x^{13}\right) \\ \equiv & \frac{\left(1+3x^{4}+6x^{8}+10x^{12}\right)\left(1+2x^{6}+4x^{8}+x^{10}+7x^{12}\right)}{(1-x)^{2}} \quad\left(\bmod x^{13}\right) \\ \equiv & \frac{\left(1+2x^{2}+3x^{4}+4x^{6}+5x^{8}+6x^{10}+7x^{12}\right)\left(1+3x^{4}+2x^{6}+10x^{8}+7x^{10}+29x^{12}\right)}{(1-x)^{2}} \quad\left(\bmod x^{13}\right) \\ \equiv & \frac{1+2x^{2}+6x^{4}+12x^{6}+28x^{8}+51x^{10}+103x^{12}}{(1-x)^{2}} \quad\left(\bmod x^{13}\right) \end{aligned} \] We wish to find the coefficient of $ x^{12} $ in the final expression. Since the numerator is an even polynomial, we just need to find \[ \begin{aligned} & \left[x^{12}\right]\left(1+3x^{2}+5x^{4}+7x^{6}+9x^{8}+11x^{10}+13x^{12}\right)\left(1+2x^{2}+6x^{4}+12x^{6}+28x^{8}+51x^{10}+103x^{12}\right) \\ = & 1 \cdot 103 + 3 \cdot 51 + 5 \cdot 28 + 7 \cdot 12 + 9 \cdot 6 + 11 \cdot 2 + 13 \cdot 1 \\ = & 569 \end{aligned} \] Thus, the number of such monic polynomials is $\boxed{569}$.	569	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-3.md" }	Compute the number of monic polynomials $ q(x) $ with integer coefficients of degree 12 such that there exists an integer polynomial $ p(x) $ satisfying $ q(x) p(x) = q\left(x^{2}\right) $.	[ "/Mathematics/Algebra/Cyclotomy/CyclotomicFactorization", "/Mathematics/Algebra/Cyclotomy/CyclotomicPolynomial", "/Mathematics/Algebra/Cyclotomy/PrimitiveRootofUnity", "/Mathematics/Algebra/Cyclotomy/RootofUnity", "/Mathematics/Algebra/NumberTheory/AlgebraicNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneratingFunctions", "/Mathematics/Algebra/NumberTheory/NumberTheoreticFunctions", "/Mathematics/Algebra/Polynomials/IntegerPolynomial", "/Mathematics/Algebra/Polynomials/MonicPolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialDegree", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/AlgebraicNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/GeneratingFunctions/GeneratingFunction" ]	Use that q(x) divides q(x^2) implies all roots are 0 or roots of unity, so q is a product of cyclotomic polynomials with exponent constraints, then count via a generating function.	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ours_415	Let $ D $ be on $ \omega $ with $ AD \parallel BC $. Lemma 1: A hyperbola $ \mathcal{H}' $ through points $ A', B', C' $ is rectangular (has perpendicular asymptotes) if and only if $ \mathcal{H}' $ passes through the orthocenter $ H' $ of $ \triangle A'B'C' $. Proof: This is a well-known result. Applying Lemma 1 to $ \triangle ABC $, we find that $ \mathcal{H} $ is rectangular. By the converse of Lemma 1 on $ \triangle BH_1C $, since $ D $ is the orthocenter of $ \triangle DH_1C $, $ \mathcal{H} $ passes through $ D $. Therefore, the isogonal conjugate of $ \mathcal{H} $ in $ \triangle ABC $ is line $ OD' $ where $ D' $ is the isogonal conjugate of $ D $ (at infinity). It's clear from the isogonality of $ AD, AD' $ that $ OD' $ is parallel to the $ A $-tangent in $ \omega $. If $ OD' $ meets $ \omega $ at $ Q', R' $ with $ Q' $ on minor arc $ AB $ and $ R' $ on minor arc $ AC $, then $ Q'R' = OD' \perp AO $. Hence $ Q'R', QR $ are symmetric in the perpendicular bisector of $ BC $, meaning that $ AR, AR' $ are isogonal, as are $ AQ, AQ' $ in $ \angle BAC $. Since $ Q', R' $ are the isogonal conjugates of the points at infinity lying on $ \mathcal{H} $, it follows that $ AQ, AR $ are parallel to the asymptotes of $ \mathcal{H} $. Next, let $ U = \infty_{AR}, V = \infty_{AQ} $. Define $ N' $ as the center of $ \mathcal{H} $. Let $ P_1' = N'U \cap PP, P_2' = N'V \cap PP, P_3' = N'V \cap HH, P_4' = N'U \cap HH $. Let $ Z = PP \cap HH $. Lemma 2: $ P_1', X, P_3' $ are collinear, as are $ P_2', Y, P_4' $. Proof: By Newton's Theorem on quadrilateral $ N'P_1'ZP_3' $ with inscribed conic $ \mathcal{H} $, we deduce that $ N'Z, P_1'P_3', HU, PV $ concur at $ X $. Similarly, $ N'Z, P_2'P_4', HV, PU $ concur at $ Y $. Since $ N'Z $ passes through $ X, Y $, $ N'XYZ $ is collinear. Lemma 3: $ P_1'P_3' \parallel HP \parallel P_2'P_4' $. Proof: The polar of $ P_1' $ in $ \mathcal{H} $ is $ PU $, and similarly, the polar of $ P_3' $ in $ \mathcal{H} $ is $ HV $. Hence the pole of $ P_1'P_3' $ is $ PU \cap HV = Y $. Meanwhile, the pole of $ UV $ is $ N' $, and the pole of $ PH $ is $ Z $. Since $ Y, N', Z $ are collinear from Lemma 2, $ UV, PH, P_1'P_3' $ concur at infinity, hence $ P_1'P_3' \parallel PH $, and similarly $ P_2'P_4' \parallel HP $. By Lemmas 2 and 3, we know that $ P_1' = P_1, P_2' = P_2, P_3' = P_3, P_4' = P_4 $. Hence $ P_1P_4, P_2P_3 $ meet at $ N = N' $, the center of $ \mathcal{H} $. Let $ N_1 $ be the midpoint of $ DH $. Let $ H_B, H_C $ be the orthocenters of $ \triangle ADC, \triangle ADB $. It's well-known that $ N_1 $ is the midpoint of $ AH_1, DH, BH_B, CH_C $, and by Lemma 1 on $ \triangle ADC, \triangle ADB $, $ H_B, H_C \in \mathcal{H} $. Hence the reflection of $ \mathcal{H} $ about $ N_1 $ has at least eight points in common with $ \mathcal{H} $, so they are the same hyperbola. Hence $ N = N_1 $ is the center of the hyperbola. Now by homothety centered at $ D $ with ratio 2, if $ A_2 $ is the foot of the altitude from $ A $ to $ BC $, we know $ NO = 0.5 HA_1 = HA_2 $. But $ HA_2 = 2R \cos B \cos C = 2 \cdot \frac{1105}{56} \cdot \frac{13}{85} \cdot \frac{33}{65} = \frac{429}{140} $, hence the answer is $ 100 \times 429 + 140 = 43040 $. $\boxed{43040}$	43040	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-3.md" }	Let $ \triangle ABC $ be a triangle with $ AB = 34 $, $ BC = 25 $, and $ CA = 39 $. Let $ O $, $ H $, and $ \omega $ be the circumcenter, orthocenter, and circumcircle of $ \triangle ABC $, respectively. Let line $ AH $ meet $ \omega $ a second time at $ A_1 $ and let the reflection of $ H $ over the perpendicular bisector of $ BC $ be $ H_1 $. Suppose the line through $ O $ perpendicular to $ A_1O $ meets $ \omega $ at two points $ Q $ and $ R $ with $ Q $ on minor arc $ AC $ and $ R $ on minor arc $ AB $. Denote $ \mathcal{H} $ as the hyperbola passing through $ A, B, C, H, H_1 $, and suppose $ HO $ meets $ \mathcal{H} $ again at $ P $. Let $ X, Y $ be points with $ XH \parallel AR \parallel YP $, $ XP \parallel AQ \parallel YH $. Let $ P_1, P_2 $ be points on the tangent to $ \mathcal{H} $ at $ P $ with $ XP_1 \parallel OH \parallel YP_2 $ and let $ P_3, P_4 $ be points on the tangent to $ \mathcal{H} $ at $ H $ with $ XP_3 \parallel OH \parallel YP_4 $. If $ P_1P_4 $ and $ P_2P_3 $ meet at $ N $, and $ ON $ may be written in the form $ \frac{a}{b} $ where $ a, b $ are positive coprime integers, find $ 100a + b $.	[ "/Mathematics/Geometry/Curves/AlgebraicCurves", "/Mathematics/Geometry/Curves/PlaneCurves", "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Recognize that the hyperbola through A,B,C and H is rectangular, so its asymptotes are the isogonal lines AQ and AR, which places its center at the midpoint of DH.	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ours_416	Let $ C_{k} = \frac{1}{k+1} \binom{2k}{k} $ denote the $ k $-th Catalan number. If $ (k, k) $ is the last point on Yang's path where he is at the river, then there are $ C_{k} $ ways for Yang to reach $ (k, k) $, and $ \binom{n}{k} $ ways for Yang to go from $ (k, k) $ to $ (n, 0) $. Hence, $ a_{n} = \sum_{i=0}^{n} C_{i} \binom{n}{i} $. We will now use generating functions, with the aid of two well-known lemmas: - $\sum_{n=i}^{\infty} \binom{n}{i} x^{n} = \frac{x^{i}}{(1-x)^{i+1}}$. - $\sum_{n=0}^{\infty} C_{n} x^{n} = \frac{1-\sqrt{1-4x}}{2x}$. Now, notice that $ a_{n} $ is the coefficient of $ x^{n} $ in \[ \begin{aligned} & \sum_{n=0}^{\infty} \sum_{i=0}^{n} C_{i} \binom{n}{i} x^{n} \\ = & \sum_{i=0}^{\infty} C_{i} \sum_{n=i}^{\infty} \binom{n}{i} x^{n} \\ = & \sum_{i=0}^{\infty} C_{i} \left(\frac{x^{i}}{(1-x)^{i+1}}\right) \\ = & \frac{1}{1-x} \sum_{i=0}^{\infty} C_{i} \left(\frac{x}{1-x}\right)^{i} \\ = & \frac{1}{1-x} \cdot \frac{1-\sqrt{1-\frac{4x}{1-x}}}{\frac{2x}{1-x}} \\ = & \frac{1-\sqrt{\frac{1-5x}{1-x}}}{2x} \end{aligned} \] Note that $ a_{n} \equiv 1 \pmod{5} $ exactly when the coefficient of $ x^{n+1} $ in $\sqrt{\frac{1-5x}{1-x}}$ is $ 3 \pmod{5} $. Note that $\sqrt{1-5x} \equiv 1 \pmod{5}$, so it remains to compute when $ N = \left[x^{n+1}\right] \sqrt{\frac{1}{1-x}} = \frac{\binom{2n+2}{n+1}}{4^{n+1}} $ is $ 3 \pmod{5} $. Write the base $ 5 $ representation of $ n+1 $ as $ b_{k} b_{k-1} \ldots b_{0} $ and the base $ 5 $ representation of $ 2n+2 $ as $ c_{l} c_{l-1} \ldots c_{0} $. If there is a $ 3 $ or $ 4 $ among $ b_{k}, b_{k-1}, \ldots, b_{0} $, then consider $ b_{i} $, the rightmost $ 3 $ or $ 4 $ in the representation. Then, since there is no carryover when doubling any of $ b_{i-1}, b_{i-2}, \ldots, b_{0} $, it follows that $ c_{i} \equiv 2b_{i} \pmod{5} $. But then $ c_{i} < b_{i} $ whether $ b_{i} = 3 $ or $ b_{i} = 4 $, so by Lucas's Theorem, $ 5 \mid \binom{c_{i}}{b_{i}} \binom{2n+2}{n+1} $, so $ N \neq 3 \pmod{5} $ in this case. Otherwise, $ k = l $ and $ b_{k}, b_{k-1}, \ldots, b_{0} \in \{0, 1, 2\} $. Note that $\binom{0}{0} \equiv 1 \pmod{5}, \binom{2}{1} \equiv 2 \pmod{5}$, and $\binom{4}{2} \equiv 1 \pmod{5}$. Hence, if there are $ t $ 1's in the base $ 5 $-representation of $ n+1 $, then $\binom{2n+2}{n+1} \equiv 2^{t} \pmod{5}$. Furthermore, note that $ n+1 \equiv t \pmod{2} $, so $ 4^{n+1} \equiv 4^{t} \pmod{5} $ and thus $ N \equiv 3^{t} \pmod{5} $. Hence, $ a_{n} \equiv 1 \pmod{5} $ if and only if $ n+1 $ has only $ 0, 1, 2 $ in its base $ 5 $ representation, where the number of $ 1 $'s are $ 1 \pmod{4} $. It can easily be computed that the 2016th smallest integer is $ n+1 = 110211012_{5}, n = 110211011_{5} = 475756 $. $\boxed{475756}$	475756	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns-3.md" }	Let $ n $ be a positive integer. Yang the Saltant Sanguivorous Shearling is on the side of a very steep mountain that is embedded in the coordinate plane. There is a blood river along the line $ y = x $, which Yang may reach but is not permitted to go above (i.e., Yang is allowed to be located at $ (2016, 2015) $ and $ (2016, 2016) $, but not at $ (2016, 2017) $). Yang is currently located at $ (0, 0) $ and wishes to reach $ (n, 0) $. Yang is permitted only to make the following moves: - Yang may spring, which consists of going from a point $ (x, y) $ to the point $ (x, y+1) $. - Yang may stroll, which consists of going from a point $ (x, y) $ to the point $ (x+1, y) $. - Yang may sink, which consists of going from a point $ (x, y) $ to the point $ (x, y-1) $. In addition, whenever Yang does a sink, he breaks his tiny little legs and may no longer do a spring at any time afterwards. Yang also expends a lot of energy doing a spring and gets bloodthirsty, so he must visit the blood river at least once afterwards to quench his bloodthirst. (So Yang may still spring while bloodthirsty, but he may not finish his journey while bloodthirsty.) Let there be $ a_{n} $ different ways for which Yang can reach $ (n, 0) $, given that Yang is permitted to pass by $ (n, 0) $ in the middle of his journey. Find the 2016th smallest positive integer $ n $ for which $ a_{n} \equiv 1 \pmod{5} $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/LatticePathsandPolygons", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/GeneratingFunctions/GeneratingFunction", "/Mathematics/NumberTheory/Sequences/BinomialTransform" ]	Translate the sum into a generating function, obtain a square‑root form, and apply Lucas's theorem to reduce the modulo‑5 condition to a base‑5 digit rule.	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ours_418	Solution. Equating the given expression and Kevin's incorrect expression gives $26+k = 21(6+k)$. Expanding the right side, we have $21(6+k) = 126 + 21k$. Setting the expressions equal, we get: \[ 26 + k = 126 + 21k \] Solving for $k$, we subtract $k$ from both sides: \[ 26 = 126 + 20k \] Subtract 126 from both sides: \[ -100 = 20k \] Divide by 20: \[ k = -5 \] Substituting $k = -5$ back into the original expression $26 + k$, we find: \[ 26 + (-5) = 21 \] Thus, Kevin's answer was $\boxed{21}$.	21	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Kevin is in first grade, so his teacher asks him to calculate $20+1 \cdot 6+k$, where $k$ is a real number revealed to Kevin. However, since Kevin is rude to his Aunt Sally, he instead calculates $(20+1) \cdot(6+k)$. Surprisingly, Kevin gets the correct answer! Assuming Kevin did his computations correctly, what was his answer?	[ "/Mathematics/Algebra/AlgebraicEquations/LinearEquation", "/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations", "/Mathematics/Algebra/Sums/Sum" ]	Set the correct and incorrect expressions equal and solve the resulting linear equation for k.	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ours_419	Notice that the only ways for the three rolls to form a right triangle are by getting $3-4-5$ and $6-8-10$, in some order. There are $3! = 6$ ways to get $3-4-5$ and 1 way to get $6-8-10$. Therefore, the desired probability is $\frac{6+1}{6 \cdot 8 \cdot 10} = \frac{7}{480}$. Thus, our answer is $100 \times 7 + 480 = 1180$. $\boxed{1180}$	1180	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Yang has a standard 6-sided die, a standard 8-sided die, and a standard 10-sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/Trigonometry", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace" ]	Recognize that the only right‑triangle side sets possible with those dice are the 3‑4‑5 and 6‑8‑10 triples.	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ours_420	Notice that $\triangle ABM$ is similar to $\triangle MCN$, so $\frac{AB}{BM} = \frac{CM}{CN}$. Since $BM = CM$ and $AB = 2CN$, it follows that $BC = \sqrt{2} AB$ and therefore $AD = 2\sqrt{2} DN$. Hence, $\frac{AN}{DN} = \sqrt{\left(\frac{AD}{DN}\right)^{2} + \left(\frac{DN}{DN}\right)^{2}} = \sqrt{(2\sqrt{2})^{2} + 1^{2}} = 3$. Thus, $DN = 60 / 3 = 20 \Rightarrow CD = 40$ and the area of $ABCD$ is $40 \cdot 40\sqrt{2} = 1600\sqrt{2}$, giving an answer of $160002$. $\boxed{160002}$	160002	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/PlaneGeometry/Quadrangles", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals", "/Mathematics/Geometry/PlaneGeometry/Rectangles" ]	Recognize that triangles ABM and MCN are similar, giving side ratios via the midpoints and perpendicular condition.	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ours_421	We compute $\log _{10} G = 10^{100}$, and \[ \begin{aligned} \log _{\left(\log _{10} G\right)} G & = \frac{\log _{10} G}{\log _{10}\left(\log _{10} G\right)} = \frac{10^{100}}{100} = 10^{98}, \\ \log _{\left(\log _{\left(\log _{10} G\right)} G\right)} G & = \frac{\log _{10} G}{\log _{10}\left(\log _{\left(\log _{10} G\right)} G\right)} = \frac{10^{100}}{98} = \frac{5 \cdot 10^{99}}{49}. \end{aligned} \] Therefore, $m+n = 5 \cdot 10^{99} + 49$, so the sum of the digits of $m+n$ is $18$. $\boxed{18}$	18	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Let $ G = 10^{10^{100}} $ (a.k.a. a googolplex). Then \[ \log _{\left(\log _{\left(\log _{10} G\right)} G\right)} G \] can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine the sum of the digits of $m+n$.	[ "/Mathematics/Algebra/NumberTheory/Arithmetic", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/Integers", "/Mathematics/Algebra/NumberTheory/RationalNumbers" ]	Apply the change‑of‑base formula repeatedly to convert each nested logarithm to a ratio of known base‑10 logs.	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ours_422	Assume there exist 5 prime numbers that form an arithmetic sequence with common difference 12, denoted by $ p, p+12, p+24, p+36, p+48 $. Notice that these 5 primes have 5 different residues modulo 5, hence one of them is divisible by 5. Therefore, $ p = 5 $. It follows that any arithmetic progression that does not contain 5 must have at most four primes. Otherwise, if the sequence contains a 5, it can have at most 5 terms: 5, 17, 29, 41, 53, because the potential next term is 65. Thus the answer is $\boxed{5}$.	5	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Jay notices that there are $ n $ primes that form an arithmetic sequence with common difference 12. What is the maximum possible value for $ n $?	[ "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberSequences" ]	Apply the pigeonhole principle to residues modulo 5 for the 5 terms spaced by 12, forcing one term to be divisible by 5.	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ours_423	Notice that $\frac{n \, ?}{(n-1) \, ?} = n!$. Thus, we can see that $ n \, ? = 1! \cdot 2! \cdots n! $. Because of this, we have that $\frac{k \, ?}{9 \, ?} = \frac{7 \, ?}{5 \, ?} = 7! \cdot 6!$. Noting that $10 \cdot 9 \cdot 8 = 6! \Longrightarrow 10! = 7! \cdot 6!$, it follows that $ k \, ? = 9 \, ? \cdot 10! = 10 \, ? $ and $ k = 10 $. $\boxed{10}$	10	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	For a positive integer $ n $, define $ n \, ? = 1^{n} \cdot 2^{n-1} \cdot 3^{n-2} \cdots (n-1)^{2} \cdot n^{1} $. Find the positive integer $ k $ for which $ 7 \, ? \, 9 \, ? = 5 \, ? \, k $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Recognize that n ? equals the product of factorials 1!·2!·…·n! by using the recurrence n?/(n‑1)? = n!	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ours_424	If both Reimu and Marisa (or neither of them) change courts each day, then the difference between their court numbers will increase by 2, decrease by 2, or stay constant. If one of them doesn't change courts (because she either won on court 1 or lost on court 1008), then the difference changes by 1. Since $876 - 123 = 753$ is odd, this means that they will never play each other unless one of them gets to court 1 or 1008 and stays for an odd number of rounds. Since Reimu is closer to court 1 than Marisa is to court 1008, the fastest way for Reimu and Marisa to play one another is to have Reimu rise to court 1 (reaching there on Day 123) and stay for 1 day (so she loses on Day 124) before losing to meet Marisa, who is rising the whole time. On Day 124, Marisa is on court $876 - 123 = 753$. Then it takes $(753 - 1) / 2 = 376$ days for them to meet each other on court 377 on Day $124 + 376 = 500$. Thus, the smallest positive integer value of $n$ is $\boxed{500}$.	500	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	The 2016 players in the Gensokyo Tennis Club are playing Up and Down the River. The players first randomly form 1008 pairs, and each pair is assigned to a tennis court (The courts are numbered from 1 to 1008). Every day, the two players on the same court play a match against each other to determine a winner and a loser. For $2 \leq i \leq 1008$, the winner on court $i$ will move to court $i-1$ the next day (and the winner on court 1 does not move). Likewise, for $1 \leq j \leq 1007$, the loser on court $j$ will move to court $j+1$ the next day (and the loser on court 1008 does not move). On Day 1, Reimu is playing on court 123 and Marisa is playing on court 876. Find the smallest positive integer value of $n$ for which it is possible that Reimu and Marisa play one another on Day $n$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/Games/GeneralGames", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle", "/Mathematics/RecreationalMathematics/Sports" ]	Use the parity of the court-number difference and the fact that only at the ends can the difference change by 1 to make the odd gap possible.	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ours_425	For each positive integer $ n $, let $ K_{n}=\sqrt{S_{n}+T_{n+1} \sqrt{S_{n-1}+T_{n} \sqrt{\cdots \sqrt{S_{1}+T_{2}}}}} $. We claim that $ K_{n}=T_{n}+1 $ for all $ n $. We proceed by induction. For the base case of $ n=1 $, $ K_{1}=\sqrt{S_{1}+T_{2}}=\sqrt{1+3}=2=T_{1}+1 $. Now assume that the claim is true for $ n=t $. Then it suffices to show that $ K_{t+1}=\sqrt{S_{t+1}+T_{t+2}\left(T_{t}+1\right)}=T_{t+1}+1 $. Note that \[ \left(T_{t+1}+1\right)^{2}-S_{t+1}=\left(T_{t+1}+1\right)^{2}-(t+1)^{2}=\left(T_{t+1}+1+(t+1)\right)\left(T_{t+1}+1-(t+1)\right)=\left(T_{t+1}+(t+2)\right)\left(T_{t+1}-t\right)=T_{t+2}\left(T_{t}+1\right), \] so hence $ T_{t+1}+1=\sqrt{S_{t+1}+T_{t+2}\left(T_{t}+1\right)} $, thus finishing the inductive step and proving the claim. Now, $ K_{62}=T_{62}+1=\frac{62 \cdot 63}{2}+1=1954 $. $\boxed{1954}$	1954	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	For a positive integer $ n $, define the $ n $th triangular number $ T_{n} $ to be $\frac{n(n+1)}{2}$, and define the $ n $th square number $ S_{n} $ to be $ n^{2} $. Find the value of \[ \sqrt{S_{62}+T_{63} \sqrt{S_{61}+T_{62} \sqrt{\cdots \sqrt{S_{2}+T_{3} \sqrt{S_{1}+T_{2}}}}}} \]	[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/AlgebraicOperations", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/Sequences", "/Mathematics/Algebra/NumberTheory/SpecialNumbers", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/NonnegativeInteger", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Sequences/IncreasingSequence", "/Mathematics/NumberTheory/Sequences/IntegerSequence", "/Mathematics/NumberTheory/Sequences/Sequence" ]	Prove the nested radical equals T_n+1 by induction using the identity (T_n+1)^2 - S_n = T_{n+1}(T_{n-1}+1).	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ours_426	Note that $\angle ABC = \angle ADC = 90^\circ$, so $ABCD$ is a cyclic quadrilateral. Then, $\triangle BEC \sim \triangle AED$ so $\frac{BE}{AE} = \frac{BC}{AD} = \frac{6}{5}$ and $\triangle AEB \sim \triangle DEC$ so $\frac{CE}{BE} = \frac{CD}{AB} = \frac{15}{7}$. Hence, $\frac{CE}{AE} = \frac{BE}{AE} \cdot \frac{CE}{BE} = \frac{6}{5} \cdot \frac{15}{7} = \frac{18}{7}$. Since $AE + EC = 25$, it follows that $EC = 18$. $\boxed{18}$	18	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	In quadrilateral $ABCD$, $AB = 7$, $BC = 24$, $CD = 15$, $DA = 20$, and $AC = 25$. Let segments $AC$ and $BD$ intersect at $E$. What is the length of $EC$?	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals" ]	Identify the quadrilateral as cyclic (right angles) and use similarity of triangles formed by intersecting diagonals to relate segment lengths.	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ours_427	Every two integers must share at least one prime factor, but this prime factor cannot divide a third number to comply with the second condition. Therefore, there must be at least six different primes $p_{1,2}, p_{1,3}, p_{1,4}, p_{2,3}, p_{2,4}, p_{3,4}$ such that $p_{1,2} p_{1,3} p_{1,4} \mid a_{1}, p_{1,2} p_{2,3} p_{2,4} \mid a_{2}, p_{1,3} p_{2,3} p_{3,4} \mid a_{3}, p_{1,4} p_{2,4} p_{3,4} \mid a_{4}$. To minimize $a_{4}$, we may assume that $\{p_{1,2}, p_{1,3}, p_{1,4}, p_{2,3}, p_{2,4}, p_{3,4}\} = \{2, 3, 5, 7, 11, 13\}$ and $a_{1} = p_{1,2} p_{1,3} p_{1,4}$, $a_{2} = p_{1,2} p_{2,3} p_{2,4}$, $a_{3} = p_{1,3} p_{2,3} p_{3,4}$, $a_{4} = p_{1,4} p_{2,4} p_{3,4}$. We will ignore the condition that $a_{1}<a_{2}<a_{3}<a_{4}$ and instead find $\max(a_{1}, a_{2}, a_{3}, a_{4})$. Assume that $p_{3,4} = 13$. If $11 \in \{p_{1,3}, p_{1,4}, p_{2,3}, p_{2,4}\}$, then $\max(a_{3}, a_{4}) \geq 2 \cdot 11 \cdot 13 = 286$. Otherwise, $p_{1,2} = 11$, and as $p_{1,3}, p_{1,4}, p_{2,3}, p_{2,4}$ are again symmetric, we can assume that $p_{2,4} = 7$. Hence, we have $a_{1} = 11 p_{1,3} p_{1,4}$, $a_{2} = 77 p_{2,3}$, $a_{3} = 13 p_{1,3} p_{2,3}$, $a_{4} = 91 p_{1,4}$, so since $\max(p_{1,4}, p_{2,3}) \geq 3$, we have that $\max(a_{3}, a_{4}) \geq 77 \cdot 3 = 231$. This maximum value of $231$ can be achieved by letting $p_{2,3} = 3$, $p_{1,4} = 2$, $p_{1,3} = 5$, so that $a_{1} = 110$, $a_{2} = 231$, $a_{3} = 195$, $a_{4} = 182$. Thus, the smallest possible value of $a_{4}$ is $\boxed{182}$.	182	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Let $a_{1}<a_{2}<a_{3}<a_{4}$ be positive integers such that the following conditions hold: - $\operatorname{gcd}\left(a_{i}, a_{j}\right)>1$ holds for all integers $1 \leq i<j \leq 4$. - $\operatorname{gcd}\left(a_{i}, a_{j}, a_{k}\right)=1$ holds for all integers $1 \leq i<j<k \leq 4$. Find the smallest possible value of $a_{4}$.	[ "/Mathematics/DiscreteMathematics/DivisionProblems/NumberPartitioningProblem", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization", "/Mathematics/NumberTheory/PrimeNumbers/PrimeSumsandProducts" ]	Assign a distinct prime to each pair of numbers so each pair shares a unique prime factor while no prime divides three numbers.	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ours_428	There are $\frac{5!}{2! \cdot 2!} = 30$ ways to order $ f(1), f(4), f(9), f(16), f(25) $ such that the given conditions $ f(1) > f(4) $ and $ f(9) > f(16) $ are satisfied. Now, we count the number of these orderings such that $ f(1) > f(16) > f(25) $. Consider $ f(1), f(4), f(9), f(16) $, and note that $ f(1), f(9) > f(16) $. - If $ f(4) > f(16) $, there are 3 ways to order $ f(1), f(4), f(9) $ since $ f(1) > f(4) $, and there is 1 way to place $ f(25) $. - If $ f(16) > f(4) $, then there are 2 ways to order $ f(1) $ and $ f(9) $, and there are 2 ways to place $ f(25) $. Hence, of the 30 ways to order the five numbers, $ 3 + 2 \cdot 2 = 7 $ of them are valid, so the desired probability is $\frac{7}{30}$. Therefore, the answer is $ 100 \times 7 + 30 = 730 $. $\boxed{730}$	730	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Let $ f $ be a random permutation on $\{1,2, \ldots, 100\}$ satisfying $ f(1) > f(4) $ and $ f(9) > f(16) $. The probability that $ f(1) > f(16) > f(25) $ can be written as $\frac{m}{n}$ where $ m $ and $ n $ are relatively prime positive integers. Compute $ 100m + n $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace" ]	Model the five values as elements of a poset with the given inequalities and count the linear extensions that also satisfy the desired ordering	[ -0.00016796588897705078, 0.00608062744140625, 0.0382080078125, 0.00460052490234375, -0.0008654594421386719, -0.01348876953125, 0.03277587890625, -0.014739990234375, 0.007312774658203125, -0.0018634796142578125, 0.015869140625, -0.005832672119140625, -0.00225830078125, -0.0073394775390625, -0.042572021484375, -0.0278167724609375, 0.006977081298828125, 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ours_429	Let $ a = 2016 $. We want to find the largest integer $ m $ such that $(n!)^{m} \mid 2016!$. By Legendre's formula, for any prime $ p $, the exponent of $ p $ in $ n! $ is given by: \[ v_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \] Thus, $(n!)^m \mid 2016!$ implies: \[ m \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \leq \sum_{k=1}^{\infty} \left\lfloor \frac{2016}{p^k} \right\rfloor \] This means $(n!)^{\left\lfloor \frac{a}{n} \right\rfloor} \mid a!$, as $ n \left\lfloor \frac{a}{n} \right\rfloor \leq a $. Therefore, $ k(n) \geq \left\lfloor \frac{a}{n} \right\rfloor > \frac{a}{n} - 1 $. Using the AM-GM inequality, we have: \[ n + k(n) > n + \frac{a}{n} - 1 \geq 2\sqrt{a} - 1 \approx 88.8 \] Thus, $ n + k(n) \geq 89 $, since it must be an integer. Now, consider $ n = 47 $. We find $ k(47) \leq 42 $ because: \[ v_{47}(2016!) = \left\lfloor \frac{2016}{47} \right\rfloor = 42 \] Thus, $(47!)^{43}$ cannot divide $ 2016! $. Therefore: \[ 47 + k(47) = 47 + 42 = 89 \] Hence, the minimum possible value of $ n + k(n) $ is $\boxed{89}$.	89	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	For each positive integer $ n \geq 2 $, define $ k(n) $ to be the largest integer $ m $ such that $(n!)^{m}$ divides $ 2016! $. What is the minimum possible value of $ n+k(n) $?	[ "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Bound k(n) by ⌊2016/n⌋ using Legendre’s formula and apply AM‑GM to minimize n+⌊2016/n⌋.	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ours_430	Note that for all integers $ i \geq 1 $, $ A_{i} B_{i+1} O_{i} C_{i+1} $ is a cyclic quadrilateral by Miquel's Theorem. Then, $\angle B_{i+1} A_{i+1} C_{i+1} = \angle B_{i+1} A_{i+1} O_{i} + \angle O_{i} A_{i+1} C_{i+1} = \angle A_{i} C_{i} O_{i} + \angle O_{i} B_{i} A_{i} = \angle C_{i} A_{i} O_{i} + \angle B_{i} A_{i} O_{i} = \angle B_{i} A_{i} C_{i}$. Similarly, we can show $\angle C_{i+1} B_{i+1} A_{i+1} = \angle C_{i} B_{i} A_{i}$, so $\triangle A_{i+1} B_{i+1} C_{i+1} \sim \triangle A_{i} B_{i} C_{i}$. Hence, it follows that $\frac{\left[A_{i+1} B_{i+1} C_{i+1}\right]}{\left[A_{i} B_{i} C_{i}\right]} = \frac{\left[A_{2} B_{2} C_{2}\right]}{\left[A_{1} B_{1} C_{1}\right]}$, so it suffices to calculate $\frac{\left[A_{2} B_{2} C_{2}\right]}{\left[A_{1} B_{1} C_{1}\right]}$. Let $ R_{1} $ and $ R_{2} $ denote the circumradii of $ A_{1} B_{1} C_{1} $ and $ A_{2} B_{2} C_{2} $, respectively. Note that $\angle C_{2} A_{2} O_{1} + \angle A_{2} C_{2} B_{2} = \angle B_{1} A_{1} O_{1} + \angle A_{1} C_{1} B_{1} = 90^{\circ}$, so $ C_{2} O_{1} $ is perpendicular to $ A_{2} B_{2} $. Similarly, $ B_{2} O_{1} $ is perpendicular to $ C_{2} A_{2} $, so $ O_{1} $ is the orthocenter of $ A_{2} B_{2} C_{2} $. Note that it's well-known by the Law of Cosines that $\angle B_{1} A_{1} C_{1} = 60^{\circ} \Longrightarrow O_{2} B_{1} C_{1} = 30^{\circ}$. Also, $ R_{1} = \frac{B_{1} C_{1}}{2 \sin B_{1} A_{1} C_{1}} = \frac{14}{\sqrt{3}}, B_{1} A_{2} = \frac{14}{3} $, and $\left[A_{1} B_{1} C_{1}\right] = \frac{1}{2} \cdot A_{1} B_{1} \cdot A_{1} C_{1} \cdot \sin B_{1} A_{1} C_{1} = 40 \sqrt{3}$. By the Law of Cosines on $\triangle A_{2} B_{1} O_{1}, A_{2} B_{1} = \sqrt{\left(\frac{14}{\sqrt{3}}\right)^{2} + \left(\frac{14}{3}\right)^{2} - 2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{14}{\sqrt{3}} \cdot \frac{14}{3}} = \frac{14}{3}$. Since $ O_{1} $ is the orthocenter of $ A_{2} B_{2} C_{2} $, it's well-known that $\frac{14}{3} = A_{2} O_{1} = 2 R_{2} \cos B_{2} A_{2} C_{2} = 2 R_{2} \cos B_{1} A_{1} C_{1} = R_{2}$. Then, $\frac{\left[A_{2} B_{2} C_{2}\right]}{\left[A_{1} B_{1} C_{1}\right]} = \left(\frac{R_{2}}{R_{1}}\right)^{2} = \frac{1}{3}$ and $\left(\sum_{i=1}^{\infty}\left[A_{i} B_{i} C_{i}\right]\right)^{2} = \left(\frac{\left[A_{1} B_{1} C_{1}\right]}{1-\frac{1}{3}}\right)^{2} = (60 \sqrt{3})^{2} = 10800$. $\boxed{10800}$	10800	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Let $ A_{1} B_{1} C_{1} $ be a triangle with $ A_{1} B_{1} = 16 $, $ B_{1} C_{1} = 14 $, and $ C_{1} A_{1} = 10 $. Given a positive integer $ i $ and a triangle $ A_{i} B_{i} C_{i} $ with circumcenter $ O_{i} $, define triangle $ A_{i+1} B_{i+1} C_{i+1} $ in the following way: (a) $ A_{i+1} $ is on side $ B_{i} C_{i} $ such that $ C_{i} A_{i+1} = 2 B_{i} A_{i+1} $. (b) $ B_{i+1} \neq C_{i} $ is the intersection of line $ A_{i} C_{i} $ with the circumcircle of $ O_{i} A_{i+1} C_{i} $. (c) $ C_{i+1} \neq B_{i} $ is the intersection of line $ A_{i} B_{i} $ with the circumcircle of $ O_{i} A_{i+1} B_{i} $. Find \[ \left(\sum_{i=1}^{\infty}\left[A_{i} B_{i} C_{i}\right]\right)^{2} \] Note: $[K]$ denotes the area of $ K $.	[ "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]	Use Miquel’s theorem to prove successive triangles are similar, yielding a constant area ratio and a geometric series.	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ours_431	Say that Michael K took $x$ tests, Michael M took $y$ tests, and Michael R took $z$ tests. We need to minimize $x+y+z$ given that $x > y > z$ and $91y > 92z > 90x$. First, note that $y \leq x-1$ implies $90x < 91y \leq 91(x-1)$, which leads to $91 < x$. If $z = x-2$, then $90x < 92z = 92(x-2)$ implies $x < 92$, contradicting $91 < x$. Hence, $z \leq x-3$. If $z = x-3$, then $92z > 90x = 90(z+3)$ implies $z > 135$. If $y-z = 1$, it follows that $92z < 91y = 91(z+1)$, which implies $z < 91$, a contradiction. Hence, $y-z = 2$, so $x+y+z \geq 139 + 138 + 136 = 413$. Now, assume that $z = x-k$ for an integer $k \geq 3$. Then, $92z > 90x = 90(z+k)$ implies $z > 45k$. Then, $x+y+z > 3z > 135k \geq 540$. It follows that the minimum possible value of $x+y+z$ is 413. $\boxed{413}$	413	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores: - Michael K had an average test score of 90, Michael M had an average test score of 91, and Michael R had an average test score of 92. - Michael K took more tests than Michael M, who in turn took more tests than Michael R. - Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests) What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined?	[ "/Mathematics/Algebra/NumberTheory/Arithmetic", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/Integers", "/Mathematics/Algebra/NumberTheory/Numbers", "/Mathematics/Algebra/RateProblems", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]	Translate the average and total score conditions into linear inequalities on the test counts and use ordering to bound the gaps, giving a minimal integer solution.	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ours_432	Note that there are bills that are $1$ and $2$ modulo 3, but not $0$ modulo 3. Hence, Bilion has a winning strategy by watching the number of dollars in the pile modulo $3$: he first plays \$1 or \$10, and then makes sure that after each of his turns, the amount of money in the pile is $1$ modulo $3$. (He can always ensure that he will not go over by playing \$1 or \$2 at the end.) In fact, he must adopt this strategy, as if he ever veers from it, Trilion will have the opportunity to adopt it and thus win. As Bilion is the guaranteed winner, it's clear that both Bilion and Trilion will try to put the least amount of money they can each round. (With the exception that Bilion's first priority is to watch the number of dollars in the pile modulo 3.) Therefore Bilion and Trilion will both want to play a \$1 bill over a \$10 bill, and a \$2 over a \$5. After Bilion puts a \$1 bill on the first turn, it becomes clear that each turn will alternate between Trilion putting a \$1 bill into the pile and Bilion putting a \$2 bill into the pile. Since they will do this so $\frac{1000000-1}{3}=333333$ times, Bilion will end up winning \$333,333 from Trilion. $\boxed{333333}$	333333	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of \$1, \$2, \$5, and \$10 bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly \$1,000,000, and whoever makes the last move wins the \$1,000,000 in the pile (if the pile is worth more than \$1,000,000 after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?	[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/RecreationalMathematics/Games/GameTheory", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Maintain the pile total congruent to 1 mod 3 using that all bills are 1 or 2 mod 3.	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ours_433	Let $N$ be the number of days that Emilia takes to grow six perfectly-shaped apple trees. We have: \[ \mathbb{E}(N) = 1 \cdot P(N=1) + 2 \cdot P(N=2) + 3 \cdot P(N=3) + \ldots = P(N \geq 1) + P(N \geq 2) + P(N \geq 3) + \ldots \] This means it suffices to sum the probability that each day is needed. For each integer $k$, the probability that Emilia does not need the $k$-th day is the probability that all 6 saplings are grown into perfectly-shaped apple trees in the first $k-1$ days. By complementary counting, we find: \[ P(N \geq k) = 1 - \left(1 - \left(\frac{1}{2}\right)^{k-1}\right)^{6} \] To sum the probabilities over all positive integers $k$, we rewrite the probability as: \[ P(N \geq k) = 6\left(\frac{1}{2}\right)^{k-1} - 15\left(\frac{1}{4}\right)^{k-1} + 20\left(\frac{1}{8}\right)^{k-1} - 15\left(\frac{1}{16}\right)^{k-1} + 6\left(\frac{1}{32}\right)^{k-1} - \left(\frac{1}{64}\right)^{k-1} \] Each term is a geometric series, and summing each term across all $k$, we find: \[ 6\left(\frac{2}{1}\right) - 15\left(\frac{4}{3}\right) + 20\left(\frac{8}{7}\right) - 15\left(\frac{16}{15}\right) + 6\left(\frac{32}{31}\right) - \left(\frac{64}{63}\right) = \frac{7880}{1953} \] Thus, the expected number of days is $\frac{7880}{1953}$, and the answer is $100 \times 7880 + 1953 = 789953$. $\boxed{789953}$	789953	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First, she plants six tree saplings at the end of Day 0. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling, she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of $\frac{1}{2}$. (Once a sapling is turned into a perfectly-shaped apple tree, it will stay a perfectly-shaped apple tree.) The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.	[ "/Mathematics/ProbabilityandStatistics/Probability/CoinFlipping", "/Mathematics/ProbabilityandStatistics/Probability/CoinTossing" ]	Express the expected number of days as the sum of tail probabilities and compute each tail probability via complement and binomial expansion.	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ours_434	We interpret $ S $ as a polynomial $ P(x_{1}, x_{2}, \ldots, x_{2017}) $. If the vertices of the 2016-simplex are $ V_{1}, V_{2}, \ldots, V_{2017} $, we associate $ V_{i} $ with the monomial $ x_{i}^{n-1} $. For all points of the form $ W = \frac{1}{n-1}(c_{1} V_{1} + c_{2} V_{2} + \cdots + c_{2017} V_{2017}) $ for non-negative integers $ c_{1}, c_{2}, \ldots, c_{2017} $ summing to $ n-1 $, we associate $ W $ with the monomial $ x_{1}^{c_{1}} x_{2}^{c_{2}} \ldots x_{2017}^{c_{2017}} $. The set of all possible tuples $ (c_{1}, c_{2}, \ldots, c_{2017}) $ corresponds exactly with the points in $ S $. If $ W $ is labeled with $ w $, we let $ P = \sum_{W \in S} w x_{1}^{c_{1}} x_{2}^{c_{2}} \ldots x_{2017}^{c_{2017}} $. The first condition tells us that $ P \neq 0 $. Let $ T $ be the set of all lines $ \ell $ parallel to $ V_{1} V_{2} $ and passing through at least one point in $ S $. Applying the second condition over all lines in $ T $ tells us that $ P(x_{1}, x_{1}, x_{3}, \ldots, x_{2017}) = 0 $, and applying the third condition over all lines in $ T $ tells us that $ P(x_{1}, x_{2}, x_{3}, \ldots, x_{2017}) = P(x_{2}, x_{1}, x_{3}, \ldots, x_{2017}) $. It is well-known that this occurs if and only if $ (x_{1} - x_{2})^{2} \mid P(x_{1}, x_{2}, \ldots, x_{2017}) $. By symmetry, it follows that $\prod(x_{i} - x_{j})^{2} \mid P(x_{1}, x_{2}, \ldots, x_{2017})$, so $ n-1 = \deg P \geq 2\binom{2017}{2} $. Equality can be achieved by letting $ P(x_{1}, x_{2}, \ldots, x_{2017}) = \prod_{1 \leq i < j \leq 2017}(x_{i} - x_{j})^{2} $, so $ n = 2\binom{2017}{2} + 1 = 4066273 $. $\boxed{4066273}$	4066273	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Let $ n $ be a positive integer. $ S $ is a set of points such that the points in $ S $ are arranged in a regular 2016-simplex grid, with an edge of the simplex having $ n $ points in $ S $. Each point in $ S $ is labeled with a real number such that the following conditions hold: - Not all the points in $ S $ are labeled with 0. - If $ \ell $ is a line that is parallel to an edge of the simplex and that passes through at least one point in $ S $, then the labels of all the points in $ S $ that are on $ \ell $ add to 0. - The labels of the points in $ S $ are symmetric along any such line $ \ell $. Find the smallest positive integer $ n $ such that this is possible. Note: A regular 2016-simplex has 2017 vertices in 2016-dimensional space such that the distances between every pair of vertices are equal.	[ "/Mathematics/Algebra/Polynomials/Monomial", "/Mathematics/Algebra/Polynomials/MultivariatePolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialDegree", "/Mathematics/Algebra/Polynomials/PolynomialIdentity", "/Mathematics/Algebra/Polynomials/SymmetricPolynomial", "/Mathematics/Algebra/Products/Product", "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/PointLattices/IntegerLattice", "/Mathematics/DiscreteMathematics/PointLattices/Lattice" ]	Translate the labeling conditions into a homogeneous polynomial whose vanishing and symmetry force each squared difference (x_i‑x_j)^2 to divide it.	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ours_435	Rewrite the equation as $ O^{2k} M^{k} = O^{2017 + 2016M} M^{2017} $. If $ k \leq 2016 $, then $ O^{4032} M^{2016} \geq O^{2k} M^{k} = O^{2017 + 2016M} M^{2017} $, which is a contradiction since $ M > 1 $. If $ k = 2017 $, then we need $ O^{4034} M^{2017} = O^{2017 + 2016M} M^{2017} $, which has no integer solution for $ M $ since $ O > 1 $. Hence, $ k > 2017 $. Rewrite the equation as $ M^{k-2017} = O^{2017 + 2016M - 2k} $. Note that $\log_{M} O$ is a rational number. If $ M = 2 $, then we have $ 2^{k-2017} = O^{6049-2k} $, so $ 6049-2k \mid k-2017 \mid 2k-4034 $. This implies $ 6049-2k \mid 2015 = 5 \cdot 13 \cdot 31 $. Since $ k > 2017 $, $ 6049-2k < 2015 $, so $ 6049-2k \leq 403 = 13 \cdot 31 $, leading to $ k \geq 2823 $. $ k = 2823 $ is achieved when $ M = 2 $ and $ O = 4 $. Assume for contradiction that some $ k < 2823 $ gives a solution for $ M $ and $ O $. We must have $ M \geq 3 $. If $ M \leq O $, then $ k-2017 \geq 2017 + 2016M - 2k \geq 2017 + 2016 \cdot 3 - 2k $, leading to $ 3k \geq 10082 $, which contradicts $ k < 2823 $. Hence, $ M > O $. Let $ M = x^{y} $, where $ x \geq 2 $ is not a perfect power of any integer (other than itself) and $ y \geq 2 $. Then, $ O \geq x $, so \[ x^{y(k-2017)} = M^{k-2017} = O^{2017 + 2016M - 2k} \geq x^{2017 + 2016x^{y} - 2k} \] \[ \Longrightarrow y(k-2017) \geq 2017 + 2016x^{y} - 2k \geq 2017 + 2016 \cdot 2^{y} - 2k \] \[ \Longrightarrow (y+2)(k-2017) \geq 2016 \cdot 2^{y} - 2017 \] \[ \Longrightarrow 806 > k-2017 \geq \frac{2016 \cdot 2^{y} - 2017}{y+2} \] We will show by induction on $ y $ that for $ y \geq 2 $, $ 806 \leq \frac{2016 \cdot 2^{y} - 2017}{y+2} $, proving $ k < 2823 $ is impossible. This is true for $ y = 2 $. Assume it is true for $ y = z $. We show it holds for $ y = z+1 $: \[ 806 \leq \frac{2016 \cdot 2^{z} - 2017}{z+2} = \frac{2(2016 \cdot 2^{z} - 2017)}{2(z+2)} \leq \frac{2016 \cdot 2^{z+1} - 2017}{z+3} \] Thus, the induction is complete, confirming $ k = 2823 $ is the smallest solution. $\boxed{2823}$	2823	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Find the smallest positive integer $ k $ such that there exist positive integers $ M, O > 1 $ satisfying \[ (O \cdot M \cdot O)^{k} = (O \cdot M) \cdot \underbrace{(N \cdot O \cdot M) \cdot (N \cdot O \cdot M) \cdot \ldots \cdot (N \cdot O \cdot M)}_{2016 \text{ times}}, \] where $ N = O^{M} $.	[ "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Numbers" ]	Rewrite the equation as a simple power equality O^{2k}M^{k}=O^{2017+2016M}M^{2017} and compare exponents to get M^{k-2017}=O^{2017+2016M-2k}, forcing a large k.	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ours_436	We work entirely in $\mathbb{F}_{2}[X, Y, Z]$. First, it's clear that $\operatorname{deg} Q = 2016$ by examining degrees in the relation. Notice that $ P(x, y, z) \mid P(y z, z x, x y) $. Therefore, we know that $ P(y z, z x, x y) \mid P(z x \cdot x y, y z \cdot x y, y z \cdot z x) = P(x y z \cdot x, x y z \cdot y, x y z \cdot z) = (x y z)^{2016} P(x, y, z) $, where the last equality holds because $ P $ is homogeneous with degree $ 2016 $. Then it's clear that $ Q(x, y, z) \mid (x y z)^{2016} $, implying that $ Q(x, y, z) $ is a monomial of the form $ x^{a} y^{b} z^{c} $ with $ a+b+c=2016 $. Now suppose $ P $ has any term $ x^{d} y^{e} z^{f} $. Then $ P(y z, z x, x y) $ contains the term $ x^{d+a} y^{e+b} z^{f+c} $. But this term can only come from multiplying $(y z)^{n-d-a}(z x)^{n-b-e}(x y)^{n-c-f}$, which means that $ P(x, y, z) $ contains a term of the form $ x^{n-d-a} y^{n-b-e} z^{n-c-f} $. So we have a bijection in the nonzero terms of $ P $ which is $(d, e, f) \Longleftrightarrow (n-d-a, n-b-e, n-c-f)$. But since $ P(1,1,1) \neq 0 \pmod{2} $ by the given condition, we must have that some $(d, e, f)$ is equal to $(n-d-a, n-b-e, n-c-f)$, otherwise the total number of terms would be even. Then we deduce $ n = 2d+a = 2e+b = 2f+c $. It's evident that $ a, b, c \equiv 2016 \pmod{2} $ with $ a+b+c=2016 $. Clearly, all such polynomials in $ Q $ will work as we can just take $ P = x^{0.5(n-a)} y^{0.5(n-b)} z^{0.5(n-c)} $. To evaluate the size of $ Q $, note there's a bijection between the working triples $(a, b, c)$ and the solutions to $ x+y+z=1008 $, which evaluates to $\binom{1008+2}{2} = 509545$. Thus, the size of $ S $ is $\boxed{509545}$.	509545	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall16Solns.md" }	Let $ S $ be the set of all polynomials $ Q(x, y, z) $ with coefficients in $\{0,1\}$ such that there exists a homogeneous polynomial $ P(x, y, z) $ of degree $ 2016 $ with integer coefficients and a polynomial $ R(x, y, z) $ with integer coefficients so that \[ P(x, y, z) Q(x, y, z) = P(y z, z x, x y) + 2 R(x, y, z) \] and $ P(1,1,1) $ is odd. Determine the size of $ S $. Note: A homogeneous polynomial of degree $ d $ consists solely of terms of degree $ d $.	[ "/Mathematics/Algebra/Polynomials/HomogeneousPolynomial", "/Mathematics/Algebra/Polynomials/Monomial", "/Mathematics/Algebra/Polynomials/PolynomialDegree", "/Mathematics/Algebra/Polynomials/PolynomialIdentity", "/Mathematics/Algebra/RingTheory/CommutativeRing", "/Mathematics/Algebra/RingTheory/PolynomialRing", "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics" ]	Reduce modulo 2 to show Q must be a monomial dividing (xyz)^{2016}, and the oddness of P(1,1,1) forces each exponent of Q to be even, yielding a stars‑and‑bars count.	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ours_437	We first show the following: for any positive integers $n>1$ and $N$, the smallest positive integer $b$ for which $a^{c} \equiv (a+b)^{c} \pmod{n}$ for any positive integer $a$ and $c>N$ is $n$. Proof: Let $c$ be one more than a multiple of $n$ that is larger than $N$. By Euler's Theorem, for any $a$ relatively prime to $n$, $a^{c} \equiv a \pmod{n}$ since $c \equiv 1 \pmod{\phi(n)}$. If $a+b$ is not relatively prime to $n$, then $(a+b)^{c}$ is not relatively prime to $n$, so it cannot be congruent to $a$ modulo $n$. Therefore, $a+b$ is also relatively prime to $n$, which means $(a+b)^{c} \equiv a+b \pmod{n}$. This means that $b$ must be a multiple of $n$, and $b=n$ clearly works, so it's the smallest one. Therefore, we can begin with $b_{1}=2017$. Since there exists a primitive root $g$ modulo 2017, if we let $a_{1}=g$, we see that $f\left(a_{2}, a_{3}, \ldots, a_{2017}\right) \equiv f\left(a_{2}+b_{2}, a_{3}, \ldots, a_{2017}\right) \pmod{2016}$, so $b_{2}$ is at least 2016. Since for all $a$ relatively prime to 2017, $\operatorname{ord}_{2017}(a)$ is a divisor of 2016, and for all $a$ that are divisible by 2017, $a^{c}$ is always a multiple of 2017, we see that $b_{2}=2016$ is sufficient. Now we break $2016=2^{5} \cdot 3^{2} \cdot 7$ into prime powers and look at them separately. For $2^{5}=32$, it does not have a primitive root (that has order 16), but we do have $\operatorname{ord}_{32}(3)=8$, and all other orders must be a factor of 8. If the number is an even number, then since all of $a_{3}, a_{4}, \ldots$ are greater than 2017, $f\left(a_{2}, a_{3}, \ldots, a_{2017}\right)$ will be a multiple of 32. So for modulo 32, we need $f\left(a_{3}, a_{4}, \ldots, a_{2017}\right) \equiv f\left(a_{3}+b_{3}, a_{4}, \ldots, a_{2017}\right) \pmod{8}$. Since there is a primitive root modulo 9 and 7, we also need $f\left(a_{3}, a_{4}, \ldots, a_{2017}\right) \equiv f\left(a_{3}+b_{3}, a_{4}, \ldots, a_{2017}\right) \pmod{6}$. Therefore, we need $f\left(a_{3}, a_{4}, \ldots, a_{2017}\right) \equiv f\left(a_{3}+b_{3}, a_{4}, \ldots, a_{2017}\right) \pmod{24}$, and $b_{3}=24$ is the smallest possible value. Now we break $24=2^{3} \cdot 3$ into prime powers. For $2^{3}=8$, there is no primitive root modulo 8 (with order 4), but $\operatorname{ord}_{8}(3)=2$. For modulo 3, there is a primitive root with order 2 as well. So we need $f\left(a_{4}, a_{5}, \ldots, a_{2017}\right) \equiv f\left(a_{4}+b_{4}, a_{5}, \ldots, a_{2017}\right) \pmod{2}$, and thus $b_{4}=2$ is the smallest possible value. The parity of a power tower only depends on the parity of its base, so we can conclude that $b_{5}=b_{6}=\cdots=b_{2017}=1$ works. Therefore, the smallest possible sum is $2017+2016+24+2+2013 \cdot 1=6072$. $\boxed{6072}$	6072	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-2.md" }	Given a sequence of positive integers $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$, define the power tower function \[ f\left(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\right)=a_{1}^{a_{2}^{a_{3}}} \] Let $b_{1}, b_{2}, b_{3}, \ldots, b_{2017}$ be positive integers such that for any $i$ between $1$ and $2017$ inclusive, \[ f\left(a_{1}, a_{2}, a_{3}, \ldots, a_{i}, \ldots, a_{2017}\right) \equiv f\left(a_{1}, a_{2}, a_{3}, \ldots, a_{i}+b_{i}, \ldots, a_{2017}\right) \quad(\bmod 2017) \] for all sequences $a_{1}, a_{2}, a_{3}, \ldots, a_{2017}$ of positive integers greater than 2017. Find the smallest possible value of $b_{1}+b_{2}+b_{3}+\cdots+b_{2017}$.	[ "/Mathematics/NumberTheory/Arithmetic", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/ModuloOrder", "/Mathematics/NumberTheory/Congruences/MultiplicativeOrder", "/Mathematics/NumberTheory/Congruences/PrimitiveRoot", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Apply Euler's theorem and primitive‑root orders to see that adding the modulus (or its totient) to each term leaves the power tower unchanged modulo that modulus.	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ours_438	We consider two cases regarding the victorious set $ V $: Case 1: $ V $ contains two elements of different signs. Suppose these two elements are $ a $ and $-b$ for some positive integers $ a $ and $ b $. Without loss of generality, assume $ a \geq b $. Suppose $ L(x) $ is a linear polynomial that passes through $ (a, 2a) $ and $(-b, 2b) $. Then any polynomial $ P(x) $ can be written as $(x-a)(x+b) R(x) + L(x)$ for some polynomial $ R(x) $. Since $ P $ has integer coefficients, so does $ R $ and therefore $ L $. This means the slope of $ L $, which is $\frac{2a-2b}{a+b}$, is an integer. This slope is nonnegative and less than 2, so it must be 0 or 1. - If the slope is 0, then $ a = b $, giving $ L(x) = 2b $. We have $ P(0) = -b^2 R(0) + 2b = 330 $, and since $ R(0) $ is an integer, $ b^2 \mid 330 - 2b $. The solutions are $ b = 1, 3, 165 $, leading to the sets $\{1, -1\}, \{3, -3\}, \{165, -165\}$. - If the slope is 1, then $ a = 3b $, giving $ L(x) = x + 3b $. We have $ P(0) = -3b^2 + 3b = 330 $, and $ b^2 \mid 110 - b $. The solutions are $ b = 1, 2, 10, 110 $, leading to the sets $\{3, -1\}, \{6, -2\}, \{30, -10\}, \{330, -110\}$, and their opposites $\{1, -3\}, \{2, -6\}, \{10, -30\}, \{110, -330\}$. Checking subsets of $\{3, 1, -1, -3\}$ with two elements of opposite signs, we find that no polynomials with integer coefficients pass through these points, confirming only 11 victorious sets in this case. Case 2: All elements of $ V $ are the same sign. Assume they are all positive, $ V = \{v_1, v_2, \ldots, v_n\} $. This means $(x-v_1)(x-v_2)\ldots(x-v_n) \mid P(x) - 2x$. By plugging in $ x = 0 $, we need $ v_1 v_2 \ldots v_n \mid 330 $. Considering nontrivial factors of $ 330 = 2 \cdot 3 \cdot 5 \cdot 11 $: - If one factor is used, there are $ 2^4 - 1 = 15 $ sets. - If two factors are used, there are $ 6 + 12 + 4 + 3 = 25 $ sets. - If three factors are used, there are $ 4 + 6 = 10 $ sets. - If four factors are used, there is 1 set: $\{2, 3, 5, 11\}$. Including the usage of 1 and the set $\{1\}$, there are $ 2 \cdot (15 + 25 + 10 + 1) + 1 = 103 $ sets with only positive elements, and similarly 103 sets with only negative elements, totaling $ 103 \cdot 2 = 206 $ sets in this case. Summing the two cases, there are $ 11 + 206 = 217 $ victorious sets in total. $\boxed{217}$	217	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-2.md" }	Call a nonempty set $ V $ of nonzero integers victorious if there exists a polynomial $ P(x) $ with integer coefficients such that $ P(0)=330 $ and $ P(v)=2\|v\| $ holds for all elements $ v \in V $. Find the number of victorious sets.	[ "/Mathematics/Algebra/GeneralAlgebra/AbstractAlgebra", "/Mathematics/Algebra/GeneralAlgebra/Algebra", "/Mathematics/Algebra/NumberTheory/Divisors", "/Mathematics/Algebra/NumberTheory/Integers", "/Mathematics/Algebra/Polynomials/IntegerPolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/NumberTheory/Divisors/Divides", "/Mathematics/NumberTheory/Divisors/Divisible", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/Divisors/Multiple", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Turn the constraints P(v)=2\|v\| into divisibility conditions, so the product of the set must divide 330 (and for opposite signs the integer slope of the line through two points limits possibilities).	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ours_440	We'll work in $\mathbb{Z} / 73 \mathbb{Z}$ for convenience. For each $ k $, let $ S_{k} $ be the number of $ k $-tuples of integers $(x_{1}, x_{2}, \ldots, x_{k})$, $ x_{j} \in \mathbb{Z} / 73 \mathbb{Z} $, such that $ 73 \mid x_{1}^{2} + x_{2}^{2} + \ldots + x_{k}^{2} $. We will first determine a useful recursion for $ S_{k} $. Suppose that $ x_{k} = 0 $. Then the number of valid tuples $(x_{1}, x_{2}, \ldots, x_{k})$ is just $ S_{k-1} $. Now if $ x_{k} \neq 0 $, then we can multiply each $ x_{j} $ by $ i x_{k}^{-1} $ to generate a solution to $ 73 \mid x_{1}^{2} + x_{2}^{2} + \ldots + x_{k-1}^{2} - 1 $, and this process is clearly reversible, so we have $ T_{k-1} $ solutions for each $ x_{k} \neq 0 $. Putting this together yields $ S_{k} = S_{k-1} + 72 T_{k-1} $. It follows that $ S_{k} = S_{k-1} + 72 T_{k-1} = S_{k-2} + 72 T_{k-1} + 72 T_{k-2} = \ldots = S_{1} + 72(T_{k-1} + T_{k-2} + \ldots + T_{1}) $. Then our desired answer is just $(S_{2018} - S_{1}) \cdot 72^{-1} \bmod 2017$. We will first compute $ S_{2017} \bmod 2017 $. Note that for any $(x_{1}, x_{2}, \ldots, x_{2017})$ with not all $ x_{j} $ equal, we can perform cyclic shifts to pair each solution with 2017 other solutions, while if all the $ x_{j} $ are equal to some $ a $, then $ 2017 a^{2} \equiv 0 \bmod 73 $, implying $ a = 0 $, hence there is 1 solution in this case, yielding $ S_{2017} \equiv 1 \bmod 2017 $. A similar process for $ T_{2017} $ yields that we have $ T_{2017} \equiv 1 + \left(\frac{2017}{73}\right) \equiv 2 $ solutions, hence $ S_{2018} = S_{2017} + 72 T_{2017} \equiv 145 \bmod 2017 $. Then since $ S_{1} = 1 $, the expression we want is just $ 144 \cdot 72^{-1} \equiv 2 \bmod 2017 $, as desired. $\boxed{2}$	2	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-2.md" }	For an integer $ k $, let $ T_{k} $ denote the number of $ k $-tuples of integers $(x_{1}, x_{2}, \ldots, x_{k})$ with $ 0 \leq x_{i} < 73 $ for each $ i $, such that $ 73 \mid x_{1}^{2} + x_{2}^{2} + \ldots + x_{k}^{2} - 1 $. Compute the remainder when $ T_{1} + T_{2} + \ldots + T_{2017} $ is divided by $ 2017 $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/NumberTheory/Congruences/ClockArithmetic", "/Mathematics/NumberTheory/Congruences/CompleteResidueSystem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/ModularInverse", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/Congruences/ResidueClass", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/Z" ]	Use a bijection that scales the variables by i·x_k^{-1} to relate S_k and T_{k-1}, and apply cyclic shift symmetry to count solutions modulo the prime.	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ours_441	Let $ L_{n} $ and $ F_{n} $ denote the $ n $th Lucas and Fibonacci numbers $(L_{0}=2, L_{1}=1, F_{0}=0, F_{1}=1)$. Compute $ P_{0}=1, P_{1}=x, P_{2}=2x^{2}-1, P_{3}=4x^{3}-3x $. Define $ a_{i}=P_{i}\left(\frac{i}{2}\right) $ and $ b_{i}=P_{i}^{\prime}\left(\frac{i}{2}\right) $. Then $ a_{0}=1, a_{1}=\frac{i}{2}, a_{2}=-\frac{3}{2}, a_{3}=-2i $. Note that $ a_{n+1}=ia_{n}-a_{n-1} $, so \[ \frac{2a_{n+1}}{i^{n+1}}=\frac{2a_{n}}{i^{n}}+\frac{2a_{n-1}}{i^{n-1}}. \] Since $\frac{2a_{0}}{i^{0}}=2=L_{0}, \frac{2a_{1}}{i^{1}}=1=L_{1}$, and $ L_{n+1}=L_{n}+L_{n-1} $, we deduce that $\frac{2a_{n}}{i^{n}}=L_{n}$ for all integers $ n $, so $ a_{n}=\frac{L_{n}i^{n}}{2} $. Next, differentiate the recursion for $ P $ using the chain rule to deduce that \[ P_{n+1}^{\prime}(x)=2P_{n}(x)+2xP_{n}^{\prime}(x)-P_{n-1}^{\prime}(x), \] so $ b_{n+1}=ib_{n}-b_{n-1}+2a_{n} $. Compute $ b_{0}=0, b_{1}=1, b_{2}=2i, b_{3}=-6 $. I claim that $ b_{n}=nF_{n}i^{n-1} $. This is true for $ n=0,1 $. Assume that it is true for $ n=0,1,\ldots,k $ with $ k $ a positive integer. We prove it for $ n=k+1 $. First, we show that $ L_{k}=F_{k}+2F_{k-1} $. Observe that the sequences for the left and right hand side satisfy the same second-degree linear recursion and the initial conditions match up as $ L_{1}=1=1+0=F_{1}+2F_{0} $ and $ L_{2}=3=1+2=F_{2}+2F_{1} $. Thus, the identity is true. Observe that \[ \begin{aligned} b_{k+1} & =ib_{k}-b_{k-1}+L_{k}i^{k} \\ & =ikF_{k}i^{k-1}-(k-1)F_{k-1}i^{k-2}+L_{k}i^{k} \\ & =i^{k}\left(kF_{k}+(k-1)F_{k-1}+L_{k}\right) \\ & =i^{k}\left((k+1)F_{k+1}-F_{k}-2F_{k-1}+L_{k}\right) \\ & =(k+1)F_{k+1}i^{k} \end{aligned} \] as desired. Thus, we deduce that $ b_{n}=nF_{n}i^{n-1} $. Because of this, $\frac{S}{T}=\frac{2017F_{2017}}{17F_{17}} $. First, we compute $ F_{2017} \pmod{17} $. If $\phi, \varphi$ are the roots of $ x^{2}-x-1 $, then we know that $ F_{n}=\frac{\phi^{n}-\varphi^{n}}{\phi-\varphi} $. Since $ 5 $ is not a quadratic residue modulo $ 17, \phi, \varphi $ are not in $\mathbb{F}_{17}$ but are in $\mathbb{F}_{17^{2}}$. Work in $\mathbb{F}_{17^{2}}$. By the Frobenius Endomorphism, $\phi^{289}=\phi$ and $\varphi^{289}=\varphi$. Then \[ F_{2017}=\frac{\phi^{2017}-\varphi^{2017}}{\phi-\varphi}=\frac{\phi^{289 \cdot 7-6}-\varphi^{289 \cdot 7-6}}{\phi-\varphi}=\frac{\phi-\varphi}{\phi-\varphi}=1. \] In particular, $ F_{2017} $ is not divisible by $ 17 $. Since $ F_{17}=1597 $ is relatively prime to $ 2017 $, note that \[ \gcd\left(2017F_{2017}, 17F_{17}\right)=\gcd\left(F_{2017}, 17F_{17}\right)=\gcd\left(F_{2017}, F_{17}\right)=F_{\gcd(2017,17)}=F_{1}=1 \] so $ m $ is just the remainder when $ 2017F_{2017} $ is divided by $ 17F_{17} $. Observe that $ F_{17} $ is also relatively prime to $ 17 $, so we can CRT the remainders for $ 17 $ and $ F_{17} $. First, we compute the remainder modulo 17: we have $ 2017 \equiv 11 $ and $ F_{2017} \equiv 1 $, so $ 2017F_{2017} \equiv 11 \pmod{17} $. Now, we compute the remainder modulo $ F_{17} $. First note that $ 2017 \equiv 420 \pmod{F_{17}} $. It suffices to compute $ F_{2017} \pmod{F_{17}} $. Consider the matrix \[ M=\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \] Also define \[ J_{n}=\begin{bmatrix} F_{n+1} & F_{n} \\ F_{n} & F_{n-1} \end{bmatrix}. \] Observe that $ M=J_{1} $ and $ J_{n+1}=MJ_{n} $ for all integers $ n $, so $ M^{n}=J_{n} $. Thus, we wish to compute $ M^{2017} $ modulo $ F_{17} $. Note that \[ M^{17}=J_{17} \equiv\begin{bmatrix} F_{18} & 0 \\ 0 & F_{16} \end{bmatrix} \equiv\begin{bmatrix} F_{16} & 0 \\ 0 & F_{16} \end{bmatrix} \pmod{F_{17}}. \] Then \[ M^{2006} \equiv F_{16}^{118}I \pmod{F_{17}}, \] where $ I $ is the identity matrix. But note that $ F_{16}^{2}=F_{15}F_{17}-1 \equiv-1\pmod{F_{17}} $, so $ F_{16}^{118} \equiv-1 \pmod{F_{17}} $. Thus, $ M^{2006} \equiv-I\pmod{F_{17}} $. Thus, $ J_{2017} \equiv M^{2017} \equiv-M^{11} \equiv-J_{11}\pmod{F_{17}} $, so $ F_{2017} \equiv-F_{11} \equiv-89\pmod{F_{17}} $. Thus, $ 2017F_{2017} \equiv 948\pmod{F_{17}} $. Now, to CRT everything together, observe that $ 1597 \equiv-1\pmod{17} $ and $ 948 \equiv 13\pmod{17} $, so $ 1597 \cdot 2+948 \equiv 11\pmod{17} $, so $ m $ is $ 4142 $. $\boxed{4142}$	4142	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-3.md" }	Define a sequence of polynomials $ P_{0}, P_{1}, \ldots $ by the recurrence $ P_{0}(x)=1, P_{1}(x)=x, P_{n+1}(x)=2xP_{n}(x)-P_{n-1}(x) $. Let $ S=\left\|P_{2017}^{\prime}\left(\frac{i}{2}\right)\right\| $ and $ T=\left\|P_{17}^{\prime}\left(\frac{i}{2}\right)\right\| $, where $ i $ is the imaginary unit. Then $\frac{S}{T}$ is a rational number with fractional part $\frac{m}{n}$, where $ m $ and $ n $ are relatively prime positive integers. Compute $ m $.	[ "/Mathematics/Algebra/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/Algebra/NumberTheory/Congruences/Congruence", "/Mathematics/Algebra/NumberTheory/Congruences/Mod", "/Mathematics/Algebra/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/Algebra/NumberTheory/Sequences/FibonacciDualTheorem", "/Mathematics/Algebra/NumberTheory/Sequences/LucasSequence", "/Mathematics/Algebra/NumberTheory/Sequences/Sequence", "/Mathematics/Algebra/NumberTheory/SpecialNumbers/FibonacciNumbers", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialSequence", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/RecurrenceEquations/LinearRecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/NumberTheoreticFunctions/GeneralNumberTheoreticFunctions", "/Mathematics/NumberTheory/NumberTheoreticFunctions/GreatestCommonDivisor", "/Mathematics/NumberTheory/Sequences/BinetForms", "/Mathematics/NumberTheory/Sequences/FibonacciDualTheorem", "/Mathematics/NumberTheory/Sequences/LucasSequence", "/Mathematics/NumberTheory/SpecialNumbers/FibonacciNumbers" ]	Show that evaluating the polynomial recurrence at x = i/2 yields Lucas numbers and differentiating gives P'_n(i/2)=n·F_n·i^{n-1}, turning the ratio into Fibonacci terms.	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ours_442	Solution. Write $ P_{G}(x) $ as $ P(G, x) $ for convenience. For the remainder of the solution, we work in $\mathbb{F}_{2}[x]$. Let $ T_{n} $ be an arbitrary tree on $ n $ vertices. Lemma 1: $ P(T_{n}, x) = x(x-1)^{n-1} $ for all positive integers $ x $. Proof: We'll evaluate $ P(T_{n}, k) $ for each $ k $. Pick an arbitrary vertex of the tree as the root. It has $ k $ possible ways to be colored. Now at each step, we pick an uncolored vertex whose parent is colored. The only restriction on the color of the child is that it cannot equal the parent's color, hence each child has $ k-1 $ color choices, yielding a final result of $ P(T_{n}, k) = k(k-1)^{n-1} $. Since $ k(k-1)^{n-1} $ is a degree $ n $ polynomial which equals $ P(T_{n}, k) $ for $ 0 \leq k \leq n $, we see $ P(T_{n}, x) = x(x-1)^{n-1} $. Now, we'll define a function $ Q $ such that $ Q(G, k) $ gives the number of valid colorings of $ G $ with at most $ k $ colors for all $ k $. Lemma 2: If $\overline{uv}$ is an edge of $ G $, then $ Q(G, k) = Q(G-\overline{uv}, k) - Q(G_{u=v}, k) $, where $ G_{u=v} $ is the graph formed by merging vertices $ u $ and $ v $, while $ G-\overline{uv} $ is the graph formed by removing edge $\overline{uv}$. Proof: Clearly, any coloring of $ G-\overline{uv} $ is either a valid coloring of $ G $, or else vertices $ u $ and $ v $ have the same color. If the second case is true, then by merging vertices $ u $ and $ v $, we get a valid coloring of $ G_{u=v} $. So it's clear the identity holds. By inducting on $\|E\|+\|V\|$, we can recursively reduce $ Q(G, k) $ until it is defined only in terms of values of the form $ Q(T_{n}, k) $ via the equation in Lemma 2. This reduction does not depend on $ k $, so since $ Q(T_{n}, k) = k(k-1)^{n-1} $ for all $ k $, it follows that $ Q(G, k) $ is actually a fixed polynomial in $ k $. Furthermore, since $ Q(T_{n}, x) $ is of degree exactly $ n $, by running the recursion backwards, it's easy to see $ Q(G, k) $ will always have degree exactly $\|G\|$. Then it's not difficult to see we must have $ P(G, k) = Q(G, k) $. In particular, we see $ P(G, k) $ satisfies the recursion of Lemma 2. Now, we'll write a connected graph on $ n $ vertices with $ n $ edges and a cycle of length $ k $ as $ T_{n, k} $. Note that if $\overline{uv}$ is an edge in the cycle of $ G = T_{2017, k} $, then $ G-\overline{uv} = T_{2017} $ while $ G_{u=v} = T_{2016, k-1} $. By Lemma 2, we have $ P(T_{2017, k}, x) = P(T_{2017}, x) - P(T_{2016, k-1}, x) \equiv P(T_{2017}, x) + P(T_{2016, k-1}, x) $ since all manipulations are in $\mathbb{F}_{2}[x]$. Similarly, $ P(T_{2016, k-1}, x) = P(T_{2016}, x) + P(T_{2015, k-2}, x) $, and so on, until we reduce $ P(T_{2020-k, 3}, x) = P(T_{2020-k}, x) + P(T_{2019-k, 2}, x) = P(T_{2020-k}, x) + P(T_{2019-k}, x) $. Summing all such equations and canceling yields $ P(T_{2017, k}, x) = P(T_{2017}, x) + P(T_{2016}, x) + \ldots + P(T_{2019-k}, x) $, showing $ P(T_{2017, k}, x) $ is the same regardless of which graph $ T_{2017, k} $ is actually chosen. By our first formula, the RHS equals $ x(x-1)^{2016} + \ldots + x(x-1)^{2018-k} = x(x-1)^{2018-k} \left[(x-1)^{k-2} + (x-1)^{k-3} + \ldots + 1\right] = x(x-1)^{2018-k} \frac{(x-1)^{k-1} - 1}{(x-1) - 1} $, which via $\mathbb{F}_{2}$ reduction equals $(x-1)^{2018-k}[(x-1)^{k-1} - 1] = (x-1)^{2017} - (x-1)^{2018-k}$. Now we compute the answer. Note that the coefficient of $ x^{i} $ in $ P(T_{2017, k}, x) $ equals $\binom{2017}{i} - \binom{2018-k}{i}$. Recall by Lucas's Theorem that $\binom{a}{b} \equiv \prod \binom{a_j}{b_j} \pmod{2}$, where $ a_i, b_i \in \{0,1\} $. Clearly $ 3 \leq k \leq 2017 $, so $ 2018-k $ ranges from $ 1 $ to $ 2015 $. We'll let $ 2017 = \overline{a_{10} a_{9} a_{8} \ldots a_{0}}_2 $, $ 2018-k = \overline{b_{10} b_{9} \ldots b_{0}}_2 $, possibly with leading zeroes. Let $ W $ be the number of tuples $(a_i, b_i)$ equal to $(1,0)$. Similarly define $ x, y, z $ for the tuples $(1,1), (0,1), (0,0)$. Finally, write $ i = \overline{c_{10} c_{9} \ldots c_{0}}_2 $. If the coefficient of $ x^{i} $ is odd, then either its coefficient in $(x-1)^{2017}$ is even while its coefficient in $(x-1)^{2018-k}$ is odd or vice versa. In the first case, the fact that the first coefficient is even implies there was a $\binom{0}{1}$ term somewhere in the multiplicative expression of Lucas's theorem. Meanwhile, since the coefficient of $(x-1)^{2018-k}$ was odd, we must have $\binom{b_i}{c_i} = 1$ for each $ i $, hence $ b_i \geq c_i $ for each $ i $. Putting all this together, there must be at least one $(a_i, b_i, c_i)$ tuple of the form $(0,1,1)$, and the only other restriction is that whenever $ b_i = 0, c_i $ is also $ 0 $. There are no restrictions when $(a_i, b_i) = (1,1)$. Then clearly there are $(2^y - 1) \cdot 2^x$ possible values for the sequence $(c_0, c_1, \ldots c_{10})$, resulting in that many nonzero coefficients. All such values $ i $ will be valid (i.e. $\leq 2017$), since if $ i > 2017 $ both binomial coefficients are zero. Similarly, in the other case there are $(2^w - 1) \cdot 2^x$ total values for $ i $. So we deduce that the number of odd coefficients equals $ 2^x(2^w + 2^y - 2) $. Since $ 2017 = 11111100001_2 $, we're subject to the constraints $ w + x = 7, y + z = 4 $. In addition, $(w, x, y, z) = (0,7,0,4)$ is not possible because it yields $ 2018-k = 2017 $. Suppose $(w, x) \neq (0,7)$. Then to minimize the number of odd coefficients, which equals $ 2^x(2^w + 2^y - 2) $, obviously we wish to minimize $ 2^y $ by setting $(y, z) = (0,4)$. Then the expression reduces to $ 2^x(2^w - 1) = 2^{w+x} - 2^x = 128 - 2^x $, whose minimal value is $ 64 $, occurring at $(w, x, y, z) = (1,6,0,4)$. Meanwhile, if $(w, x) = (0,7)$, we end up with $ 128(2^y - 1) \geq 128 $. So it's clear the minimum number of odd coefficients is $ 64 $. This value of $ a $ is attainable only at $(w, x, y, z) = (1,6,0,4)$, which is equivalent to creating $ 2018-k $ by replacing one of the ones in $ 2017 = \overline{11111100001}_2 $ with a zero. However, since $ 1 \leq k \leq 2015 $, replacing the last one with a zero is not a valid choice, meaning we can only remove one of the first six digits. Then $ 2018-k \in \{2017-1024, 2017-512, 2017-256, 2017-128, 2017-64, 2017-32\} \Longrightarrow k \in \{1025, 513, 257, 129, 65, 33\} $, hence $\sum b_i$ is $ \boxed{2022} $.	2022	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-3.md" }	For a graph $ G $ on $ n $ vertices, let $ P_{G}(x) $ be the unique polynomial of degree at most $ n $ such that for each $ i=0,1,2, \ldots, n, P_{G}(i) $ equals the number of ways to color the vertices of the graph $ G $ with $ i $ distinct colors such that no two vertices connected by an edge have the same color. For each integer $ 3 \leq k \leq 2017 $, define a $ k $-tasty graph to be a connected graph on $ 2017 $ vertices with $ 2017 $ edges and a cycle of length $ k $. Let the tastiness of a $ k $-tasty graph $ G $ be the number of coefficients in $ P_{G}(x) $ that are odd integers, and let $ t $ be the minimal tastiness over all $ k $-tasty graphs with $ 3 \leq k \leq 2017 $. Determine the sum of all integers $ b $ between $ 3 $ and $ 2017 $ inclusive for which there exists a $ b $-tasty graph with tastiness $ t $.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GraphTheory/GraphColoring", "/Mathematics/DiscreteMathematics/GraphTheory/Trees" ]	Apply deletion‑contraction modulo 2 to write the unicyclic graph's chromatic polynomial as (x‑1)^{2017}‑(x‑1)^{2018‑k} and count odd coefficients using Lucas's theorem.	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ours_443	From $ \angle XA_1O = 90^\circ $, we know $ A_1 \in (BOC) $. Then if $ A_0 $ is where $ \ell $ meets $ BC $, it follows that $ A_0, A_1 $ are inverses in $ \omega $, hence $ A_0, A_2 $ are inverses in $ \gamma $, the imaginary circle centered at $ O $ with radius $ iR $. Let $ AA_2 $ meet $ \omega $ at $ P $. By Desargues' Involution Theorem on quadrilateral $ ABCP $ and conic $ \omega $, we conclude that if $ \ell $ meets any conic at two points, these points must be inverses in $ \gamma $. Now, note that due to the given angle condition, $ Y $ and $ Z $ lie on the McCay cubic of $ \triangle ABC $. Then if $ Y', Z' $ are the isogonal conjugates of $ Y, Z $, by properties of the McCay cubic we know $ Y' \in OY, Z' \in OZ $. Since this cubic can only meet $ \ell $ at three points, one of which is $ O $, we know $ Y' = Z, Z' = Y $. Furthermore, by the converse of Desargues' Involution Theorem, since $ Y, Z $ are inverses in $ \gamma $, we know $ ABCPYZ $ lie on a conic. Then the isogonal conjugate of this conic is obviously the line $ YZ $. Denote $ P' $ as the isogonal conjugate of $ P $. Then clearly $ P' \in YZ $, implying that the line through $ A $ parallel to $ \ell $ and $ AP $ are isogonal, hence the angle bisectors of $ \angle AA_2O $ are always parallel to one of the $ A $-angle bisectors of $ \triangle ABC $. But now with standard techniques we can compute the sines of the angles at which the $ A $-internal and external angle bisectors meet $ BC $, and their product is $\frac{16 \sqrt{26}}{525}$, and the answer is $ 567 $. $\boxed{567}$	567	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-3.md" }	Let $ \triangle ABC $ be a triangle with $ AB = 7 $, $ AC = 9 $, $ BC = 10 $, circumcenter $ O $, circumradius $ R $, and circumcircle $ \omega $. Let the tangents to $ \omega $ at $ B, C $ meet at $ X $. A variable line $ \ell $ passes through $ O $. Let $ A_1 $ be the projection of $ X $ onto $ \ell $ and $ A_2 $ be the reflection of $ A_1 $ over $ O $. Suppose that there exist two points $ Y, Z $ on $ \ell $ such that $ \angle YAB + \angle YBC + \angle YCA = \angle ZAB + \angle ZBC + \angle ZCA = 90^\circ $, where all angles are directed, and furthermore that $ O $ lies inside segment $ YZ $ with $ OY \cdot OZ = R^2 $. Then there are several possible values for the sine of the angle at which the angle bisector of $ \angle AA_2O $ meets $ BC $. If the product of these values can be expressed in the form $\frac{a \sqrt{b}}{c}$ for positive integers $ a, b, c $ with $ b $ squarefree and $ a, c $ coprime, determine $ a+b+c $.	[ "/Mathematics/Geometry/GeneralGeometry/Bisector", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/InversiveGeometry/Inversion", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/ProjectiveGeometry/GeneralProjectiveGeometry", "/Mathematics/Geometry/Trigonometry/Angles", "/Mathematics/Geometry/Trigonometry/DirectedAngles", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry" ]	Apply Desargues' Involution Theorem (via inversion in the imaginary circle) to prove Y and Z are inverse points, forcing the bisector of ∠AA₂O to be parallel to a triangle A‑angle bisector.	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ours_444	All equalities are over $\overline{\mathbb{F}_p}$, the algebraic closure of $\mathbb{F}_p$. For all $ x \in \overline{\mathbb{F}_p} $, let $ c(x) $ be the degree of the minimal polynomial of $ x $. Take an $ n \times n $ matrix $ A $ over $\mathbb{F}_p$ with a nonzero determinant. Consider its eigenvalues over $\overline{\mathbb{F}_p}$. Compute the characteristic polynomial of $ A $ and its eigenvalues, and diagonalize into the Jordan Normal Form: \[ A = P^{-1} D P, \] where $ D $ is a diagonal matrix or has diagonal blocks of the form $\lambda 100 \lambda 100 \lambda$, for example. Notice that \[ A^d = P^{-1} D^d P, \] and this equals $ I $ if and only if $ D^d = I $. Since diagonal matrices have powers derived by taking the powers of the diagonal elements, and since the $ n $th power of the above block gives $\lambda^n, n \lambda^{n-1}, n(n-1) \lambda^{n-2} 0, \lambda^n, n \lambda^{n-1} 0,0, \lambda^n$ for rows, we can find the order simply. If there is at least one block, we need $ p \mid d $. Beyond that, $ d $ must be divisible by the orders of all the eigenvalues, none of which involve factors of $ p $. Thus $ d $ will be at most \[ p \prod_{i=c(\lambda_j)}(p^i-1), \] with the factor of $ p $ only occurring if there is a multiple root with differing geometric and algebraic multiplicities. The values of $ i $ above sum to at most $ n $ since the characteristic polynomial of $ A $ has degree $ n $, and they sum to at most $ n-1 $ when there is a multiple root. Thus, multiple roots give a maximum order of \[ p(p^{n-1}-1) < p^n-1, \] while distinct roots give a maximum order of \[ p^n-1. \] Equality occurs if and only if matrix $ A $ has a characteristic polynomial with roots that are primitive roots of $ F_{p^n} $. This can be achieved using, for example, a companion matrix to a minimal polynomial of such a primitive root. Thus, we want the sum of the digits of $\sum_{i=1}^{p+1}(p^i-1) = (p-1)p^0 + (p-1)p^1 + \sum_{i=3}^{p+1} 1p^i$ in base $ p $, which is $ 3p-3 = 6048 $. $\boxed{6048}$	6048	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-3.md" }	Let $ p = 2017 $. If $ A $ is an $ n \times n $ matrix composed of residues $(\bmod p)$ such that $\det A \not\equiv 0 (\bmod p)$, then let $\operatorname{ord}(A)$ be the minimum integer $ d > 0 $ such that $ A^d \equiv I (\bmod p) $, where $ I $ is the $ n \times n $ identity matrix. Let the maximum such order be $ a_n $ for every positive integer $ n $. Compute the sum of the digits when $\sum_{k=1}^{p+1} a_k$ is expressed in base $ p $.	[ "/Mathematics/Algebra/FieldTheory/AlgebraicClosure", "/Mathematics/Algebra/FieldTheory/AlgebraicallyClosed", "/Mathematics/Algebra/FieldTheory/ExtensionField", "/Mathematics/Algebra/FieldTheory/Field", "/Mathematics/Algebra/FieldTheory/FieldCharacteristic", "/Mathematics/Algebra/FieldTheory/FieldOrder", "/Mathematics/Algebra/FieldTheory/FiniteExtension", "/Mathematics/Algebra/FieldTheory/FiniteField", "/Mathematics/Algebra/FieldTheory/GaloisExtension", "/Mathematics/Algebra/FieldTheory/GaloisTheory", "/Mathematics/Algebra/FieldTheory/IrreduciblePolynomial", "/Mathematics/Algebra/FieldTheory/PerfectField", "/Mathematics/Algebra/FieldTheory/PrimeField", "/Mathematics/Algebra/FieldTheory/PrimeSubfield", "/Mathematics/Algebra/FieldTheory/PrimitiveElement", "/Mathematics/Algebra/FieldTheory/SeparableExtension", "/Mathematics/Algebra/FieldTheory/SeparablePolynomial", "/Mathematics/Algebra/LinearAlgebra/GeneralLinearAlgebra", "/Mathematics/Algebra/LinearAlgebra/Matrices", "/Mathematics/Algebra/Polynomials/IrreduciblePolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PrimitivePolynomial", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/ModuloOrder", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/MultiplicativeOrder", "/Mathematics/NumberTheory/Congruences/PrimitiveRoot", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/Congruences/ResidueClass", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers", "/Mathematics/NumberTheory/Sequences/PolynomialSequence" ]	Maximize matrix order by choosing eigenvalues that are primitive elements of the degree‑n extension field, giving order p^n − 1.	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ours_445	Let $\Omega$ be the locus of $Z$, which is an Apollonius circle with respect to $X$ and $Y$ of radius $\frac{2}{3}$ and center $O$. Let $f: \mathcal{P} \rightarrow \mathcal{P}$ be the projective transformation fixing $\Omega$ and $XY$ and taking $O$ to $O^{\prime}$ with $OO^{\prime} = \frac{8}{15}$ and $O^{\prime}$ on segment $YZ$. The key observation is that any bulldozer of $XYZ$ has the property that $f^{-1}(\mathcal{T})$ is a $120^{\circ}$ chord of $\Omega$. The desired region is thus $f(\mathcal{S}^{\prime})$ where $\mathcal{S}^{\prime}$ is the area between $\Omega$ and $\omega$, the image of $\Omega$ under a homothety centered at $O$ with ratio $\frac{1}{2}$. This is because over all $120^{\circ}$ chords on a circle, the set of points on their perimeters is the area between the circle and the smaller circle that all of them are tangent to. Note first that $ZPQ$ is clearly inscribed in $\Omega$ as it is on the $Z$ Apollonius circle of $XYZ$ and $P$ and $Q$ are the isodynamic points of $XYZ$. Let $x = \angle XZY$. From well-known facts about isodynamic points, $\angle XPY, XQY = 60^{\circ} + x, -60^{\circ} + x$, where the angles are directed. Thus, embedding this in the complex plane, we have that the cross ratio $(X, Y; P, Q) = \omega$, a third root of unity. Taking a projective transformation fixing $\Omega$ and taking $X$ to $O$ takes $Y$ to the point at infinity along $XY$, so $(X, Y; P, Q)$ becomes the ratio $\frac{p}{q}$, where $p, q$ are the complex numbers that represent $P, Q$. As projective transformations preserve cross ratios, this takes bulldozers to $120^{\circ}$ chords. It is not too hard to compute that $f^{-1}$ is this projective transformation. To compute the desired area, we use homogeneous coordinates, first scaling $\Omega$ up to a unit circle, setting $\Omega$ at $x^{2} + y^{2} - z^{2} = 0$ and thus $\omega$ at $4x^{2} + 4y^{2} - z^{2} = 0$. If we find the area of the image of $\omega$ under $f$, we can subtract it from $\Omega$ to find the area. As $f$ takes $(0: 0: 1)$ to $(4: 0: 5), (1: 0: 1)$ and $(-1: 0: 1)$ to themselves, and $\Omega$ to itself, we obtain its transformation matrix as $\begin{pmatrix} 5 & 0 & 4 \\ 0 & 3 & 0 \\ 4 & 0 & 5 \end{pmatrix}$ by setting up some equations. This takes $4x^{2} + 4y^{2} - z^{2} = 0$ to $4(5x + 4z)^{2} + 4(3y)^{2} - (4x + 5z)^{2} = 0$, or $84x^{2} + 36y^{2} + 39z^{2} + 120xz = 0$ which simplifies to $28x^{2} + 12y^{2} + 13 + 40x = 0$. We can easily compute now that this has area $\frac{3\sqrt{21}}{196} \pi$ through ellipse formulas, so $\mathcal{S}$ has area $\frac{4}{9}\left(1 - \frac{3\sqrt{21}}{196}\right) \pi = \frac{196 - 3\sqrt{21}}{441} \pi$. This gives our answer of $\boxed{129}$.	129	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns-3.md" }	We define the bulldozer of triangle $ABC$ as the segment between points $P$ and $Q$, distinct points in the plane of $ABC$ such that $PA \cdot BC = PB \cdot CA = PC \cdot AB$ and $QA \cdot BC = QB \cdot CA = QC \cdot AB$. Let $XY$ be a segment of unit length in a plane $\mathcal{P}$, and let $\mathcal{S}$ be the region of $\mathcal{P}$ that the bulldozer of $XYZ$ sweeps through as $Z$ varies across the points in $\mathcal{P}$ satisfying $XZ = 2YZ$. Find the greatest integer that is less than $100$ times the area of $\mathcal{S}$.	[ "/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry", "/Mathematics/Geometry/CoordinateGeometry/HomogeneousCoordinates", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/ProjectiveGeometry/GeneralProjectiveGeometry", "/Mathematics/Geometry/Transformations/GeometricTransformations" ]	Map the problem via a projective transformation that fixes the Apollonius circle so bulldozers become 120° chords, turning the swept region into the annulus between the circle and its inner homothetic circle.	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ours_446	The digit 十 represents 1 and the digit 六 represents 6, so it follows that 十六 + 六十 = 16 + 61 = 77. Therefore, the sum is $\boxed{77}$.	77	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Jingoistic James wants to teach his kindergarten class how to add in Chimese. The Chimese also use the base-10 number system but have replaced the digits 0-9 with ten of their own characters. For example, the two-digit number 十六 represents 16. What is the sum of the two-digit numbers 十六 and 六十?	[ "/Mathematics/FoundationsofMathematics/MathematicalProblems/SolvedProblems", "/Mathematics/RecreationalMathematics/MathematicalHumor", "/Mathematics/RecreationalMathematics/Puzzles/Cipher", "/Mathematics/RecreationalMathematics/Puzzles/Ciphertext" ]	Translate the alien characters to their decimal digits before performing the addition.	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ours_447	To maximize the expression $a^{b^{c^{d}}}$, we should place $d=1$ because a $1$ in any other position would render the numbers on top of it ineffective. Then, we evaluate the possible configurations: - $2^{3^{2}} = 512$ - $2^{2^{3}} = 256$ - $3^{2^{2}} = 81$ The greatest value is $512$, achieved when $a=2$, $b=3$, $c=2$, and $d=1$. Thus, the greatest possible value is $\boxed{512}$.	512	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	The numbers $a, b, c, d$ are $1, 2, 2, 3$ in some order. What is the greatest possible value of $a^{b^{c^{d}}}$?	[ "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Place the 1 at the top of the exponent tower to avoid diminishing the large powers.	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ours_448	Let $ a_{i} $ denote the number of pages used in problem $ i $. Then your fractions for problem $ i $ were $\frac{j}{a_{i}}, j=1,2, \ldots, a_{i}$. The sum of these is $\frac{a_{i}+1}{2}$. The sum of this over all $ i $ is $\frac{a_{1}+a_{2}+\ldots+a_{6}+6}{2}=2017$. Then $ a_{1}+a_{2}+\ldots+a_{6}=4028 $. Therefore, you turned in 4028 pages. $\boxed{4028}$	4028	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	The USAMO is a 6-question test. For each question, you submit a positive integer number $ p $ of pages on which your solution is written. On the $ i $-th page of this question, you write the fraction $ \frac{i}{p} $ to denote that this is the $ i $-th page out of $ p $ for this question. When you turned in your submissions for the 2017 USAMO, the proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be 2017. How many pages did you turn in?	[ "/Mathematics/DiscreteMathematics/Combinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Notice that the fractions on each problem sum to (p+1)/2, the average of 1 and p multiplied by the number of pages.	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ours_449	Solution. To determine the number of regions, we can focus on one quadrant of the square and then multiply the result by 4 due to symmetry. Consider the triangle formed by points $B$, $C$, and the origin. By drawing all line segments between the points within this triangle, we find that it is divided into 15 regions. Since the square is symmetric, the total number of regions in the square $A B C D$ is $4 \times 15 = 60$. $\boxed{60}$	60	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Steven draws a line segment between every two of the points $$ A(2,2), B(-2,2), C(-2,-2), D(2,-2), E(1,0), F(0,1), G(-1,0), H(0,-1) $$ How many regions does he divide the square $A B C D$ into?	[ "/Mathematics/Geometry/CombinatorialGeometry", "/Mathematics/Geometry/Dissection/SquareDissection", "/Mathematics/RecreationalMathematics/Puzzles/BracedPolygon", "/Mathematics/RecreationalMathematics/Puzzles/BracedSquare" ]	Exploit the square’s symmetry: count regions in one quadrant (triangle) and multiply the result by 4.	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ours_450	First of all, the number 10 will be decremented 10 times before reaching 0 and being erased at the 11th step, so Henry needs at least 11 steps. To show that 11 steps is sufficient, we note that among the numbers 00, 01, 02, ..., 98, 99, there do not exist 11 consecutive numbers that have no repeated digits, as 00, 11, 22, ..., 99 separate the list into 10-number blocks. Therefore, the answer is $\boxed{11}$.	11	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Henry starts with a list of the first 1000 positive integers and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/ConsecutiveNumbers", "/Mathematics/NumberTheory/Integers/NonpositiveInteger", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Numbers/NumbersRepresentation", "/Mathematics/NumberTheory/Sequences/IntegerSequence", "/Mathematics/NumberTheory/Sequences/Sequence", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Show that any 11 consecutive integers contain a number with a repeated digit, limiting the longest surviving run to 10 steps.	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ours_451	There are two configurations to consider. 1. If the right angles are next to each other, the pentagon can be seen as a combination of a square and an equilateral triangle, both with side length $2$. The area of the square is $4$, and the area of the equilateral triangle is $\frac{\sqrt{3}}{4} \times 2^2 = \sqrt{3}$. Thus, the total area is $4 + \sqrt{3}$. 2. If the right angles are not next to each other, the pentagon can be divided into two right isosceles triangles with leg lengths $2$ and an isosceles triangle with side lengths $2\sqrt{2}, 2\sqrt{2}$, and $2$. The height of the third triangle is $\sqrt{7}$, giving it an area of $\frac{1}{2} \times 2 \times \sqrt{7} = \sqrt{7}$. The total area of the pentagon in this configuration is $4 + \sqrt{7}$. The second configuration yields a larger area, so the greatest possible area of the pentagon is $4 + \sqrt{7}$. Therefore, $m = 4$ and $n = 7$, leading to $100m + n = 100 \times 4 + 7 = 407$. $\boxed{407}$	407	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.	[ "/Mathematics/Geometry/Dissection/TriangleDissection", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/PlaneGeometry/Polygons", "/Mathematics/RecreationalMathematics/Dissection", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Determine the arrangement of the two right angles that yields the largest area and decompose the pentagon into simple triangles to compute that area.	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ours_452	Solution. Let $ T \subset S $ be the elements of $ S $ that are relatively prime to 30, and let $ T $ have maximum element $ M $. At most one element of $ S $ can be divisible by each of 2, 3, and 5, so at least $\|T\| \geq 10$. Since there are 8 residue values of 30 that are relatively prime to 30, there are two elements of $ T \setminus \{M\} $ that are congruent modulo 30. Since $ M $ is at least two greater than any element in $ T \setminus \{M\} $, it follows that $\max_{s \in S} - \min_{s \in S} \geq 32$, and equality is achieved for the set $\{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 16, 27, 25\}$. $\boxed{32}$	32	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Let $ S $ be a set of 13 distinct, pairwise relatively prime, positive integers. What is the smallest possible value of $\max_{s \in S} - \min_{s \in S}$?	[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Integers/Z-Star" ]	Apply the pigeonhole principle to the 8 residues modulo 30 that are coprime to 30, forcing two numbers to differ by at least 30, plus a gap of 2.	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ours_453	Solution. Call the numbers in the permutation $a_{1}, a_{2}, \ldots, a_{16}$. Since $a_{1}+a_{2}+a_{3}+a_{4}$ and $a_{2}+a_{3}+a_{4}+a_{5}$ differ by at most one, it follows that $a_{1}$ and $a_{5}$ are consecutive integers. Similarly, for any $k=1,2, \ldots, 12$, we have $\left\|a_{k}-a_{k+4}\right\|=1$. Consequently, $A_{1}=\left(a_{1}, a_{5}, a_{9}, a_{13}\right)$ is an arithmetic sequence of consecutive integers (either ascending or descending), and similar for the other three quadruples $A_{2}=\left(a_{2}, a_{6}, a_{10}, a_{14}\right), A_{3}=\left(a_{3}, a_{7}, a_{11}, a_{15}\right), A_{4}=\left(a_{4}, a_{8}, a_{12}, a_{16}\right)$, which means that $A_{1}, A_{2}, A_{3}, A_{4}$ partition the set into $\{1,2,3,4\}, \ldots,\{13,14,15,16\}$. Moreover, since $a_{1}+a_{2}+a_{3}+a_{4}$ and $a_{3}+a_{4}+a_{5}+a_{6}$ differ by at most one as well, this implies that $a_{1}+a_{2}=a_{5}+a_{6}$ since $\left\|a_{1}-a_{5}\right\|=\left\|a_{2}-a_{6}\right\|=1$. As a result, if $A_{1}$ is increasing, then $A_{2}$ is decreasing and vice versa, and similar for $A_{3}$ and $A_{4}$. There are 2 ways to choose the ascending/descending arrangements and $4!$ ways to choose which numbers each of $A_{1}, A_{2}, A_{3}, A_{4}$ have, so in total there are $2 \cdot 4!=48$ possible permutations. $\boxed{48}$	48	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	A permutation of $\{1,2,3, \ldots, 16\}$ is called blocksum-simple if there exists an integer $n$ such that the sum of any 4 consecutive numbers in the permutation is either $n$ or $n+1$. How many blocksum-simple permutations are there?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]	Use the overlapping four‑term sum condition to show entries 4 apart differ by 1, forming four interleaved arithmetic progressions of consecutive integers.	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ours_454	Solution. Assume without loss of generality that $ a \geq b $ and define $ c = 2a + b $ and $ d = 2b + a $ such that $ cd = 4752 = 2^4 \cdot 3^3 \cdot 11 $ and $ c \geq d $. Observe that $ 3 $ divides $ c + d = 3(a + b) $, so since $ 3 $ divides $ cd $, $ 3 $ divides both $ c $ and $ d $. Next, note that $ a = \frac{2c - d}{3} $ and $ b = \frac{2d - c}{3} $, so $ 2d \geq c $. Thus, \[ d^2 \leq cd = 4752 \leq 2d^2 \] This implies $ d $ is between $ 49 $ and $ 68 $, inclusive. Furthermore, $ d $ is a divisor of $ 4752 $ and a multiple of $ 3 $. The only $ d $ in this range that satisfy these conditions are $ 54 $ and $ 66 $. If $ d = 54 $, then $ c = 88 $, which is not a multiple of $ 3 $. If $ d = 66 $, then $ c = 72 $, so $ a = 26 $ and $ b = 20 $ for $ ab = 520 $. $\boxed{520}$	520	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Let $ a $ and $ b $ be positive integers such that $(2a + b)(2b + a) = 4752$. Find the value of $ ab $.	[ "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/Divisors/Divides", "/Mathematics/NumberTheory/Divisors/DivisibilityRules", "/Mathematics/NumberTheory/Divisors/Divisible", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/Divisors/Multiple", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Introduce c=2a+b and d=2b+a to turn the equation into cd=4752, then use the fact that both c and d are multiples of 3 and bound d via d² ≤ cd ≤ 2d² to limit possibilities.	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ours_455	Solution. Note that if $f(n)=n^{2}+1+n^{2}+2+\ldots+n^{2}+2n+1=(2n+1)\left(n^{2}+n+1\right)$, then $f(2k+1)-f(2k)=24k^{2}+24k+8$. The sum we want is then $\sum_{k=0}^{49}\left(24k^{2}+24k+8\right)=1000000$. $\boxed{1000000}$	1000000	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Determine the value of $-1+2+3+4-5-6-7-8-9+\ldots+10000$, where the signs change after each perfect square.	[ "/Mathematics/NumberTheory/Integers/ConsecutiveNumbers", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Parity", "/Mathematics/NumberTheory/Sequences/IntegerSequence", "/Mathematics/NumberTheory/Sequences/Sequence", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Group the terms between successive perfect squares and compute each block as the difference of two arithmetic series sums.	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ours_456	We can rewrite the equation as \[ A^{c \cdot g} = B^{d \cdot f}, \] where $A = \overline{ab}$ and $B = e$. Therefore, there must be a number $p$ such that $A = p^m$ and $B = p^n$ for some positive integers $m$ and $n$. Since $B < 10$, we have $p < 10$ and we can assume that $p$ is not a power itself. Using the fact that $A$ and $B$ do not share digits, we can narrow the possibilities down to $p = 2, 3, 7$. When $p = 7$, we must have $A = 49$ and $B = 7$, and $c, g, d, f \in \{1,2,3,5,6,8\}$. To get $df = 2cg$, the only possibility is $\{d, f\} = \{3,8\}$ and $\{c, g\} = \{2,6\}$. But since $8$ is not relatively prime to either $2$ or $6$, this is impossible. When $p = 3$, we have $A = 27$ or $81$ and $B = 3$ or $9$. - When $(A, B) = (27, 3)$, we have $c, g, d, f \in \{1,4,5,6,8,9\}$ and $df = 3cg$, which happens only when $\{d, f\} = \{8,9\}$ and $\{c, g\} = \{4,6\}$ or $\{d, f\} = \{4,6\}$ and $\{c, g\} = \{1,8\}$. In both cases, because $8$ is not relatively prime to $4$ and $6$, there are no valid solutions. - When $(A, B) = (27, 9)$, we have $c, g, d, f \in \{1,3,4,5,6,8\}$ and $2df = 3cg$, which happens only when $\{d, f\} = \{3,4\}$ and $\{c, g\} = \{1,8\}$. For the relatively prime condition to hold, we have $(c, d, f, g) = (1,4,3,8)$ or $(3,8,1,4)$. In either case, we have $(h, i) = (6,5)$. - When $(A, B) = (81, 3)$, we have $c, g, d, f \in \{2,4,5,6,7,9\}$ and $df = 4cg$, but since $5,6,7,9$ cannot be used, there are no valid solutions. - When $(A, B) = (81, 9)$, we have $c, g, d, f \in \{2,3,4,5,6,7\}$ and $df = 2cg$, but since $5$ and $7$ cannot be used and $2 \cdot 3 \cdot 4 \cdot 6$ is a perfect square (and not twice of a perfect square), there are no valid solutions. When $p = 2$, we have $A = 16, 32$, or $64$ and $B = 2, 4$, or $8$. - When $(A, B) = (16, 2)$, we have $c, g, d, f \in \{3,4,5,7,8,9\}$ and $df = 4cg$, but since $3,5,7,9$ cannot be used, there are no valid solutions. - When $(A, B) = (16, 4)$, we have $c, g, d, f \in \{2,3,5,7,8,9\}$ and $df = 2cg$, but since $3,5,7,9$ cannot be used, there are no valid solutions. - When $(A, B) = (16, 8)$, we have $c, g, d, f \in \{2,3,4,5,7,9\}$ and $3df = 4cg$, but since $5$ and $7$ cannot be used, and $2 \cdot 3 \cdot 4 \cdot 9$ is not $12$ times a perfect square, there are no valid solutions. - When $(A, B) = (32, 4)$, we have $c, g, d, f \in \{1,5,6,7,8,9\}$ and $2df = 5cg$, but since $6,7,9$ cannot be used, there are no valid solutions. - When $(A, B) = (32, 8)$, we have $c, g, d, f \in \{1,4,5,6,7,9\}$ and $3df = 5cg$, but since $4,6,7$ cannot be used, there are no valid solutions. - When $(A, B) = (64, 2)$, we have $c, g, d, f \in \{1,3,5,7,8,9\}$ and $df = 6cg$, but since $5,7,8$ cannot be used, there are no valid solutions. - When $(A, B) = (64, 8)$, we have $c, g, d, f \in \{1,2,3,5,7,9\}$ and $df = 2cg$, but since $3,5,7,9$ cannot be used, there are no valid solutions. Therefore, the only possible value of $10h+i$ is $\boxed{65}$.	65	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Let $\{a, b, c, d, e, f, g, h, i\}$ be a permutation of $\{1,2,3,4,5,6,7,8,9\}$ such that $\operatorname{gcd}(c, d)=\operatorname{gcd}(f, g)=1$ and \[ (10a+b)^{c/d}=e^{f/g} \] Given that $h>i$, evaluate $10h+i$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Rewrite the equation as equality of integer powers to force the two bases to be powers of a common integer.	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ours_457	Let $A, B$ be the centers of the circles (circle centered at $A$ is smaller), and let their radii be $r_{A}, r_{B}$, respectively. Observe that $\angle PXA = \angle QXB$. Let their common angle be $\theta$. Then we see that \[ \frac{QA}{QB} = \frac{XA}{XB} \cdot \frac{\sin(90^\circ - \theta)}{\sin \theta} = \frac{XA}{XB} \cdot \cot \theta \] and \[ \frac{PA}{PB} = \frac{XA}{XB} \cdot \frac{\sin \theta}{\sin(90^\circ + \theta)} = \frac{XA}{XB} \cdot \tan \theta \] So \[ \frac{PA}{PB} \cdot \frac{QA}{QB} = \left(\frac{XA}{XB}\right)^2 \] But $\frac{PA}{PB} = \frac{r_{A}}{r_{B}} = \frac{XA}{XB}$, so $\frac{QA}{QB} = \frac{XA}{XB}$ also, meaning that $XQ$ is the angle bisector of $\angle AXB$, resulting in $AQ = \frac{15}{7}$ and $BQ = \frac{20}{7}$. Then the largest distance from $Q$ to any point on either circle is $\frac{20}{7} + 4 = \frac{48}{7}$, so $100m+n = 4807$. Alternate solution: Use the same notation as the previous solution. Observe that $PXQY$ is cyclic. But $P, X, Y$ all are on the Apollonian circle of $AB$ with ratio $\frac{r_{A}}{r_{B}}$, so $Q$ also has this ratio. But then $Q$ is the insimilicenter of the two circles, so the largest distance from $Q$ to any point on either of the circles is $\frac{4}{7}(3+5+4) = \frac{48}{7}$, so the answer is $4807$. $\boxed{4807}$	4807	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Bill draws two circles which intersect at $X, Y$. Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3, 4$ and the distance between the centers of these two circles is $5$, then the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.	[ "/Mathematics/Geometry/Distance", "/Mathematics/Geometry/GeneralGeometry/Bisector", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity", "/Mathematics/Geometry/Trigonometry/Angles", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions" ]	Recognize Q as the internal homothety (insimilicenter) of the two circles, so it lies on AB dividing it in the ratio of the radii.	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ours_458	Show by induction that $S_{2^{k}}=\left\{\left(2^{k}, 0\right),\left(0,2^{k}\right),\left(-2^{k}, 0\right),\left(0,-2^{k}\right)\right\}$, and $S_{2^{k}+a}\left(0 \leq a<2^{k}\right)$ is 4 copies of $S_{a}$ centered on the four points in $S_{2^{k}}$. Hence, $\left\|S_{2017}\right\|=4\left\|S_{993}\right\|=4^{2}\left\|S_{481}\right\|=4^{3}\left\|S_{225}\right\|=4^{4}\left\|S_{97}\right\|=4^{5}\left\|S_{33}\right\|=4^{6}\left\|S_{1}\right\|=4^{7}=16384$. In general, there are $4^{s_{2}(n)}$ points in $S_{n}$, where $s_{2}(n)$ is the sum of the digits of $n$ in binary. $\boxed{16384}$	16384	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	We define the sets of lattice points $S_{0}, S_{1}, \ldots$ as $S_{0}=\{(0,0)\}$ and $S_{k}$ consisting of all lattice points that are exactly one unit away from exactly one point in $S_{k-1}$. Determine the number of points in $S_{2017}$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/PointLattices/Grid", "/Mathematics/DiscreteMathematics/PointLattices/IntegerLattice", "/Mathematics/DiscreteMathematics/PointLattices/Lattice", "/Mathematics/DiscreteMathematics/PointLattices/LatticePoint", "/Mathematics/DiscreteMathematics/PointLattices/PointLattice", "/Mathematics/DiscreteMathematics/PointLattices/SquareGrid" ]	Express n as sum of powers of two; each 1‑bit creates a factor 4 by placing four copies of the smaller set at the four axis points.	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ours_459	We consider two cases: Case 1: The absolute values of all coordinates do not exceed $\frac{1}{2}$. In this case, all these points are in $S$, and the volume is that of a unit 2017-cube, which is $1$. Case 2: The absolute value of one coordinate (without loss of generality, let it be $x_{1}$) is greater than $\frac{1}{2}$. Then all other coordinates must have an absolute value of less than or equal to $1-\left\|x_{1}\right\|$. If $x_{1} > \frac{1}{2}$, the volume of all the points satisfying the requirement is a 2017-dimensional pyramid with the base being a unit 2016-cube and height being $1-\frac{1}{2}=\frac{1}{2}$. The volume of this pyramid is therefore $\frac{1 \cdot \frac{1}{2}}{2017} = \frac{1}{4034}$. Since there are $2017$ ways to choose this coordinate and $2$ ways to determine whether it is positive or negative, there are $2 \cdot 2017 = 4034$ such pyramids, and the sum of their volumes is $1$. Summing up the two cases, the volume of $S$ is $1 + 1 = 2 = \frac{2}{1}$. Therefore, our answer is $2 \cdot 100 + 1 = 201$. $\boxed{201}$	201	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Let $ S $ be the set of all points $\left(x_{1}, x_{2}, x_{3}, \ldots, x_{2017}\right)$ in $\mathbb{R}^{2017}$ satisfying $\left\|x_{i}\right\|+\left\|x_{j}\right\| \leq 1$ for any $1 \leq i < j \leq 2017$. The volume of $ S $ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes", "/Mathematics/Geometry/MultidimensionalGeometry/n-DimensionalGeometry" ]	Observe that at most one coordinate can exceed 1/2, so the region splits into a central unit hypercube and symmetric pyramids attached to each coordinate direction.	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ours_460	Solution. Observe that \[ \frac{\binom{n}{k-1}\binom{n}{k+1}}{\binom{n}{k}}+\frac{1}{n+2}\binom{n+2}{k+1}=\binom{n}{k} \] so $\binom{n}{k}$ divides $\binom{n}{k-1}\binom{n}{k+1}$ if and only if $n+2$ divides $\binom{n+2}{k+1}$. Thus, it suffices to count the number of binomial coefficients $\binom{1339}{m}, m=2,3, \ldots, 1337$, that are divisible by $1339=13 \cdot 103$. By Lucas' Theorem, since $1339_{10}=7C0_{13}$, $\binom{1339}{m}$ is divisible by $13$ if and only if $m$ is not. Similarly, since $1339_{10}=D0_{103}$, $\binom{1339}{m}$ is divisible by $103$ if and only if $m$ is not. Thus, we need $m$ to be relatively prime to $1339$. Since $1$ and $1338$ are relatively prime to $m$, we get that the count should be $\varphi(1339)-2=1222$. $\boxed{1222}$	1222	{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOFall17Solns.md" }	Find the number of integers $1 \leq k \leq 1336$ such that $\binom{1337}{k}$ divides $\binom{1337}{k-1}\binom{1337}{k+1}$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Arithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Convert the given divisibility into the condition that n+2 divides a binomial coefficient, then count those coefficients using Lucas' theorem.	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