ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Suppose $a$, $b$, and $c$ are real numbers that satisfy the equations:

$$a + b + c = 5$$

and

$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5}.$$ 

Find the greatest possible value of $a^3 + b^3 + c^3$.
Answer: 125
Problem: Nine points form a grid of size $3 \times 3$. How many triangles can be formed with $3$ vertices at these points?
Answer: 76
Problem: In an $n \times m$ grid, 1 row and 1 column are colored blue, and the rest of the cells are white. If precisely $\frac{1}{2010}$ of the cells in the grid are blue, how many values are possible for the ordered pair $(n, m)$?
Answer: 96
Problem: In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$.
Answer: 84
Problem: Evaluate $1 \oplus 2 \oplus \dots \oplus 987654321$ where $\oplus$ is bitwise exclusive OR.

($A \oplus B$ in binary has an $n$-th digit equal to $1$ if the $n$-th binary digits of $A$ and $B$ differ and $0$ otherwise. For example, $5 \oplus 9 = 0101_{2} \oplus 1001_{2} = 1100_2 = 12$ and $6 \oplus 7 = 110_2 \oplus 111_2 = 001_2 = 1$.)
Answer: 1
Problem: Given that $20N^2$ is a divisor of $11!$, what is the greatest possible integer value of $N$?
Answer: 72
Problem: Let $ABC$ be a right-angled triangle with $\angle B = 90^{\circ}$. The length of the altitude $BD$ is equal to $12$. What is the minimum possible length of $AC$, given that both $AC$ and the perimeter of triangle $ABC$ are integers?
Answer: 25
Problem: Line $DE$ cuts through triangle $ABC$, with $DF$ parallel to $BE$. Given that $BD = DF = 10$ and $AD = BE = 25$, find $BC$.
Answer: 50
Problem: A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\frac{\pi}{6}$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? The volume of a tetrahedron is defined as one-third the area of its base times its height.
Answer: 320
Problem: If $f(x)$ is a monic quartic polynomial such that $f(-1) = -1$, $f(2) = -4$, $f(-3) = -9$, and $f(4) = -16$, find $f(1)$.
Answer: 23
Problem: In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.

[asy]
size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
 label("$A$", (0,23), NW);
 label("$B$", (23, 23), NE);
 label("$C$", (23,0), SE);
 label("$D$", (0,0), SW);
 draw((0,6)--(23,6));
 draw((0,19)--(23,19));
 draw((5,0)--(5,23));
 draw((12,0)--(12,23));
 label("13", (17/2, 21));
 label("111",(35/2,25/2));
 label("37",(17/2,3));
 label("123",(2.5,12.5));[/asy]
Answer: 180
Problem: For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\sum_{n=1}^{2022} f(n) \mod 1000$.
Answer: 242
Problem: In a chess tournament, each of two players has only one game played. After 2 rounds, 5 players left the tournament. At the end of the tournament, it was found that the total number of games played is 100. How many players were there at the start of the tournament?
Answer: 105
Problem: Let $y_0$ be chosen randomly from $\{0, 50\}$, let $y_1$ be chosen randomly from $\{40, 60, 80\}$, let $y_2$ be chosen randomly from $\{10, 40, 70, 80\}$, and let $y_3$ be chosen randomly from $\{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $P$ be the unique polynomial of degree less than or equal to $3$ such that $P(0) = y_0$, $P(1) = y_1$, $P(2) = y_2$, and $P(3) = y_3$. What is the expected value of $P(4)$?
Answer: 107
Problem: Let $ABC$ be a triangle with sides $51$, $52$, and $53$. Let $\Omega$ denote the incircle of $\triangle ABC$. Draw tangents to $\Omega$ which are parallel to the sides of $ABC$. Let $r_1$, $r_2$, $r_3$ be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed $r_1 + r_2 + r_3$.
Answer: 15
Problem: Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. The area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Answer: 14800
Problem: Let $A, B, C, \ldots, Z$ be 26 nonzero real numbers. Suppose that $T = TNYWR$. Compute the smallest possible value of \[ \left\lceil A^2 + B^2 + \cdots + Z^2 \right\rceil. \] (The notation $\left\lceil x \right\rceil$ denotes the least integer $n$ such that $n \geq x$.)
Answer: 5
Problem: Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ 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 $k_2 + k_3.$
Answer: 2030612
Problem: Let $ABC$ be a triangle with $AB = 2$ and $AC = 3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2 = \frac{a + b\sqrt{c}}{d}$ for positive integers $a, b, c$, and $d$, where $\gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Compute $1000a + 100b + 10c + d$.
Answer: 270382
Problem: In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.
Answer: 24
Problem: Equilateral triangle $ABC$ has an area of $1$. Points $A'$, $B'$, and $C'$ are the midpoints of sides $BC$, $CA$, and $AB$, respectively. Points $A''$, $B''$, and $C''$ are the midpoints of segments $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m+n$.
Answer: 21
Problem: Solve the following equation in positive numbers:
$$ (2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1) = 828567056280801 $$
Answer: 100
Problem: A canister contains $2\frac{1}{2}$ cups of flour. Greg and Sally have a brownie recipe that calls for $1\frac{1}{3}$ cups of flour. They want to make $1\frac{1}{2}$ recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?
Answer: 80
Problem: In triangle $ABC$, where $AB = 28$, $AC = 36$, and $BC = 32$, let $D$ be the point on segment $BC$ such that $\angle BAD = \angle DAC$. Let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $\triangle ABC$. Find the length of segment $AE$.
Answer: 18
Problem: A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied:

1. All the squares are congruent.
2. If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares.
3. Each square touches exactly three other squares.

How many positive integers $n$ are there with $2018 \leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?
Answer: 501
Problem: For how many integers $n$ with $1 \leq n \leq 2012$ is the product
\[
  \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero?
Answer: 335
Problem: Find the least positive integer that cannot be represented as \( \frac{2^a - 2^b}{2^c - 2^d} \), where \( a, b, c, \) and \( d \) are positive integers.
Answer: 11
Problem: Given a square $ABCD$ with side length $10$. On sides $BC$ and $AD$ of this square, points $E$ and $F$ are selected such that they form a rectangle $ABEF$. Rectangle $KLMN$ is positioned such that its vertices $K, L, M$, and $N$ lie on segments $CD, DF, FE$, and $EC$, respectively. It is given that rectangles $ABEF$ and $KLMN$ are congruent with $AB = MN$. Find the length of segment $AL$.
Answer: 5
Problem: What is the number formed by the last three digits of $1201^{1202}$?
Answer: 401
Problem: A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members such that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be happy if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?
Answer: 1012
Problem: Convex quadrilateral $ABCD$ satisfies the following conditions: \( \angle{CAB} = \angle{ADB} = 30^{\circ} \), \( \angle{ABD} = 77^{\circ} \), \( BC = CD \), and \( \angle{BCD} = n^{\circ} \) for some positive integer \( n \). Compute \( n \).
Answer: 68
Problem: At a party, there were some couples attending. As they arrive, each person gets to talk with all the other persons who are already in the room. During the party, after all the guests arrive, groups of persons form such that:
1. No two persons forming a couple belong to the same group.
2. For each two persons that do not form a couple, there is one and only one group to which both belong.

Find the number of couples attending the party, knowing that there are fewer groups than persons at the party.
Answer: 4
Problem: Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and said:

"Today my daughter has a birthday. If in the polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter, then $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$."

How old is the daughter of the math teacher?
Answer: 1
Problem: Find the number of integers $n$ such that $1 \le n \le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.
Answer: 99
Problem: Let $p, q$ be two-digit numbers, neither of which are divisible by $10$. Let $r$ be the four-digit number formed by placing the digits of $p$ followed by the digits of $q$ in order. As $p$ and $q$ vary, a computer prints $r$ on the screen if $\gcd(p, q) = 1$ and $p + q$ divides $r$. Suppose that the largest number printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).
Answer: 13
Problem: Compute the integer $$\frac{2^{\left(2^5-2\right)/5-1}-2}{5}.$$
Answer: 6
Problem: Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that 
\[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \]
Answer: 504
Problem: Distinct prime numbers $p$, $q$, and $r$ satisfy the following equation:

$$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$

for some positive integer $A$. What is the value of $A$?
Answer: 1980
Problem: A sequence $a_n$ of real numbers satisfies the following conditions:

- $a_1 = 1$
- $a_2 = 0$
- $a_n = (S_{n-1} + 1) S_{n-2}$ for all integers $n \geq 3$

where $S_k = a_1 + a_2 + \dots + a_k$ for positive integers $k$. 

Determine the smallest integer $m > 2$ such that $127$ divides $a_m$.
Answer: 10
Problem: Define the *hotel elevator cubic* as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$?
Answer: 13
Problem: Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$. Point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $\overline{OE}$, respectively. Compute the area of quadrilateral $WXYZ$.
Answer: 12
Problem: Given that $9 \times 10 \times 11 \times \cdots \times 15 = 32432400$, what is $1 \times 3 \times 5 \times \cdots \times 15$?
Answer: 2027025
Problem: Jenny places 100 pennies on a table, with 30 showing heads and 70 showing tails. She randomly selects 40 pennies and flips them over. This means if a selected penny was showing heads, it will now show tails, and vice versa. What is the expected number of pennies showing heads at the end?
Answer: 46
Problem: Find the number of triples \((a, b, c)\) of positive integers such that \(a + ab + abc = 11\).
Answer: 3
Problem: Let \( \gamma = \alpha \times \beta \) where \[ \alpha = \underbrace{999 \cdots 9}_{2010 \text{ '9's}} \] and \[ \beta = \underbrace{444 \cdots 4}_{2010 \text{ '4's}} \] Find the sum of digits of \( \gamma \).
Answer: 18090
Problem: How many positive five-digit integers have the product of their five digits equal to $900$?
Answer: 210
Problem: For some positive integer $n$, consider the usual prime factorization \[ n = \prod_{i=1}^{k} p_{i}^{e_{i}} = p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}, \] where $k$ is the number of prime factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?
Answer: 56
Problem: Triangles $OAB$, $OBC$, and $OCD$ are isosceles triangles with $\angle OAB = \angle OBC = \angle OCD = 90^\circ$. Find the area of triangle $OAB$ if the area of triangle $OCD$ is 12.
Answer: 3
Problem: A **palindromic table** is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below.
\[
\begin{array}{ccc}
  O & M & O \\
  N & M & N \\
  O & M & O
\end{array}
\]

How many palindromic tables are there that use only the letters $O$ and $M$? (The table may contain only a single letter.)
Answer: 16
Problem: A point $(x, y)$ in the first quadrant lies on a line with intercepts $(a, 0)$ and $(0, b)$, where $a, b > 0$. Rectangle $M$ has vertices $(0, 0)$, $(x, 0)$, $(x, y)$, and $(0, y)$, while rectangle $N$ has vertices $(x, y)$, $(x, b)$, $(a, b)$, and $(a, y)$. What is the ratio of the area of $M$ to that of $N$?
Answer: 1
Problem: In the MTRP district, there are $10$ cities. Bob the builder wants to construct roads between these cities such that there is exactly one unique path between any two cities. The government has provided him with a budget of Rs. $20$, and each road requires a positive integer amount (in Rs.) to build. In how many ways can he build such a network of roads, ensuring the full budget is utilized? Determine the last two digits of your answer.
Answer: 0
Problem: Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.
Answer: 189
Problem: For the NEMO, Kevin needs to compute the product:
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
Answer: 870
Problem: Compute $\left\lceil\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.
Answer: 2019
Problem: A tailor met a tortoise sitting under a tree. When the tortoise was the tailor's age, the tailor was only a quarter of his current age. When the tree was the tortoise's age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise?
Answer: 77
Problem: What is the largest positive integer $n$ for which there are no positive integers $a, b$ such that $8a + 11b = n$?
Answer: 88
Problem: Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}\,dx.$
Answer: 0
Problem: (MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
Answer: 1944
Problem: Given that a duck found that $5 - 2\sqrt{3}i$ is one of the roots of the polynomial $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots?
Answer: 12
Problem: Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
Answer: 408
Problem: On a $10 \times 10$ chessboard, several knights are placed. In any $2 \times 2$ square, there is at least one knight. What is the smallest number of cells these knights can threaten? (Note: A knight does not threaten the square on which it stands, but it does threaten the squares on which other knights are standing.)
Answer: 50
Problem: For real numbers $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$.
Answer: 999
Problem: The function $f: \mathbb{N} \to \mathbb{N}_0$ is defined such that:
- $f(2) = 0$
- $f(3) > 0$
- $f(6042) = 2014$
- For all $m, n \in \mathbb{N}$, $f(m + n) - f(m) - f(n) \in \{0, 1\}$

Determine $f(2014)$.

Note: $\mathbb{N}_0 = \{0, 1, 2, \ldots\}$
Answer: 671
Problem: A square is inscribed in a circle with a radius of $6$. A quarter circle is inscribed in the square, as depicted in the diagram below. Given that the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$, determine the value of $n$.

[asy]
size(5 cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw(circle((1,1),1.41));
draw(arc((0,0),2,0,90));[/asy]
Answer: 18
Problem: Find the number of ordered triples of positive integers $(a, b, c)$ such that 
\[6a + 10b + 15c = 3000.\]
Answer: 4851
Problem: A tamer wants to line up five lions and four tigers. We know that a tiger cannot be placed after another tiger. How many ways can the animals be arranged if the tamer cannot distinguish between two animals of the same species?
Answer: 15
Problem: Let $a, b, c, d$ be four positive integers such that $a > b > c > d$. Given that $ab + bc + ca + d^2 \mid (a + b)(b + c)(c + a)$. Find the minimal value of $\Omega(ab + bc + ca + d^2)$. Here $\Omega(n)$ denotes the number of prime factors $n$ has. For example, $\Omega(12) = 3$.
Answer: 3
Problem: Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$.
Answer: 78
Problem: How many values of $x$, where $-19 < x < 98$, satisfy the equation $\cos^2 x + 2\sin^2 x = 1$?
Answer: 38
Problem: A four-digit number has the following properties:
1. The units digit equals the tens digit increased by $1$.
2. The hundreds digit equals twice the tens digit.
3. The thousands digit is at least twice the units digit.
4. The number is twice a prime number.

Determine this four-digit number.
Answer: 9634
Problem: Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many 5-digit numbers does Kelvin like?
Answer: 14034
Problem: At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61,629,875}$ and $n = \sqrt[7]{170,859,375}$. Given that $m$ and $n$ are both integers, compute $100m+n$.
Answer: 39515
Problem: We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday. On the first Monday, the number we record is $10.5$. On every Tuesday and every Thursday, the number we record is one third of what it was on the previous workday. On every Monday, Wednesday, and Friday, the number we record is double what it was on the previous workday. This pattern continues indefinitely. What is the sum of all the numbers we will record?
Answer: 252
Problem: Let $\alpha = \cos^{-1} \left( \frac{3}{5} \right)$ and $\beta = \sin^{-1} \left( \frac{3}{5} \right)$. 

\[\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta  m)}{2^n3^m}\]

can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.
Answer: 15013
Problem: How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum.
Answer: 41
Problem: In $\triangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\triangle ABC$ so that each $n$-gon shares a side with a distinct side of $\triangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?
Answer: 11
Problem: $A, B, C$ are points in the plane such that $\angle ABC = 90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA = 20$ and $BC = 21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Answer: 449
Problem: Let $P$ be a polynomial satisfying $P(x + 1) + P(x - 1) = x^3$ for all real numbers $x$. Find the value of $P(12)$.
Answer: 846
Problem: Parallelogram $ABCD$ is given such that $\angle ABC = 30^\circ$. Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ onto $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram.
Answer: 880
Problem: A $3 \times 3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.
Answer: 405
Problem: For each integer $1 \le j \le 2017$, let $S_j$ denote the set of integers $0 \le i \le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1 \le j \le 2017} \left(1 - \prod_{i \in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right) \in \{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by $2017$.
Answer: 1840
Problem: Find the 2000th positive integer that is not the difference between any two integer squares.
Answer: 7998
Problem: After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates. Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally. If there are $2000$ gold coins in the chest, what is the greatest number of gold coins Tony could take as his share? (Assume each gold coin is equally valuable.)
Answer: 982
Problem: Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?
Answer: 270
Problem: Compute the number of three-digit numbers such that all three digits are distinct, in descending order, and one of the digits is a $5$.
Answer: 36