Navvye/Olympiad-Problems-CSV huggingface dataset:
2017 USA Team Selection Test
In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is color-identifiable if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.



For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.
2017 USA Team Selection Test
Let $ABC$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $BC$ such that $\angle TAO = 90^{\circ}$. The circle with diameter $\overline{AT}$ intersects the circumcircle of $\triangle BOC$ at two points $A_1$ and $A_2$, where $OA_1 < OA_2$. Points $B_1$, $B_2$, $C_1$, $C_2$ are defined analogously.



Prove that $\overline{AA_1}$, $\overline{BB_1}$, $\overline{CC_1}$ are concurrent.

Prove that $\overline{AA_2}$, $\overline{BB_2}$, $\overline{CC_2}$ are concurrent on the Euler line of triangle $ABC$.



Evan Chen
2017 USA Team Selection Test
Let $P, Q \in \mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\lambda$ such that $P + \lambda Q$ is the square of a polynomial.



Alison Miller
2017 USA Team Selection Test
You are cheating at a trivia contest. For each question, you can peek at each of the $n > 1$ other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth $0$ points. An incorrect guess is worth $-2$ points for other contestants, but only $-1$ point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by $2^{n - 1}$ points at any time, then you can surely win first place.



Linus Hamilton
2017 USA Team Selection Test
Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus.



Danielle Wang and Evan Chen
2017 USA Team Selection Test
Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$, and $b < p$, such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$.



Noam Elkies
2016 USA Team Selection Test
Let $S = \{1, \dots, n\}$.  Given a bijection $f : S \to S$ an orbit of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$.  We denote by $c(f)$ the number of distinct orbits of $f$.  For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$.



Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that $$ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) $$where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$.



Proposed by Maria Monks Gillespie
2016 USA Team Selection Test
Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$.  The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$.  The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$.  Prove that line $AT$ passes through either $S_1$ or $S_2$.



Proposed by Evan Chen
2016 USA Team Selection Test
Let $p$ be a prime number.  Let $\mathbb F_p$ denote the integers modulo $p$, and let $\mathbb F_p[x]$ be the set of polynomials with coefficients in $\mathbb F_p$.  Define $\Psi : \mathbb F_p[x] \to \mathbb F_p[x]$ by $$ \Psi\left( \sum_{i=0}^n a_i x^i \right) = \sum_{i=0}^n a_i x^{p^i}. $$Prove that for nonzero polynomials $F,G \in \mathbb F_p[x]$, $$ \Psi(\gcd(F,G)) = \gcd(\Psi(F), \Psi(G)). $$Here, a polynomial $Q$ divides $P$ if there exists $R \in \mathbb F_p[x]$ such that $P(x) - Q(x) R(x)$ is the polynomial with all coefficients $0$ (with all addition and multiplication in the coefficients taken modulo $p$), and the gcd of two polynomials is the highest degree polynomial with leading coefficient $1$ which divides both of them.  A non-zero polynomial is a polynomial with not all coefficients $0$.  As an example of multiplication, $(x+1)(x+2)(x+3) = x^3+x^2+x+1$ in $\mathbb F_5[x]$.



Proposed by Mark Sellke
2016 USA Team Selection Test
Let $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$ be the binary representation of $\sqrt 3$. Prove that for any positive integer $n$, at least one of the digits $b_n$, $b_{n+1}$, $\dots$, $b_{2n}$ equals $1$.
2016 USA Team Selection Test
Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, $$ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. $$
2016 USA Team Selection Test
Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.
2015 USA Team Selection Test
Let $ABC$ be a non-isosceles triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Denote by $M$ the midpoint of $\overline{BC}$. Let $Q$ be a point on the incircle such that $\angle AQD = 90^{\circ}$.  Let $P$ be the point inside the triangle on line $AI$ for which $MD = MP$.  Prove that either $\angle PQE = 90^{\circ}$ or $\angle PQF = 90^{\circ}$.



Proposed by Evan Chen
2015 USA Team Selection Test
Prove that for every $n\in \mathbb N$, there exists a set $S$ of $n$ positive integers such that for any two distinct $a,b\in S$, $a-b$ divides $a$ and $b$ but none of the other elements of $S$.



Proposed by Iurie Boreico
2015 USA Team Selection Test
A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference.  The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step.  The physicist's goal is to isolate two usamons that she is  sure are currently in the same state. Is there any series of diode usage that makes this possible?



Proposed by Linus Hamilton
2015 USA Team Selection Test
Let $f : \mathbb Q \to \mathbb Q$ be a function such that for any $x,y \in \mathbb Q$, the number $f(x+y)-f(x)-f(y)$ is an integer. Decide whether it follows that there exists a constant $c$ such that $f(x) - cx$ is an integer for every rational number $x$.



Proposed by Victor Wang
2015 USA Team Selection Test
A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other.  Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors.  Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color.  The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring.  For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices.



Proposed by Po-Shen Loh
2015 USA Team Selection Test
Let $ABC$ be a non-equilateral triangle and let $M_a$, $M_b$, $M_c$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively.  Let $S$ be a point lying on the Euler line. Denote by $X$, $Y$, $Z$ the second intersections of $M_aS$, $M_bS$, $M_cS$ with the nine-point circle. Prove that $AX$, $BY$, $CZ$ are concurrent.
2014 USA Team Selection Test
Let $ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $BC$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$. Let $\ell$ be the line through $P$ parallel to $XR$. Prove that as $X$ varies along minor arc $BC$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $BC$, line $\ell$ passes through $F$.)



Robert Simson et al.
2014 USA Team Selection Test
Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity $$\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}$$ is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square).



Evan O'Dorney and Victor Wang
2014 USA Team Selection Test
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be amicable if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable.



Zoltán Füredi (suggested by Po-Shen Loh)
2014 USA Team Selection Test
Let $n$ be a positive even integer, and let $c_1, c_2, \dots, c_{n-1}$ be real numbers satisfying $$ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. $$ Prove that $$

	2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2

$$ has no real roots.
2014 USA Team Selection Test
Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.
2014 USA Team Selection Test
For a prime $p$, a subset $S$ of residues modulo $p$ is called a sum-free multiplicative subgroup of $\mathbb F_p$ if

$\bullet$ there is a nonzero residue $\alpha$ modulo $p$ such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$ (all considered mod $p$), and

$\bullet$ there are no $a,b,c \in S$ (not necessarily distinct) such that $a+b \equiv c \pmod p$.

Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb F_p$ such that $\left\lvert S \right\rvert \ge N$.



Proposed by Noga Alon and Jean Bourgain
2013 USA Team Selection Test
A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages.  Any pair of members always talk to each other in only one language.  Suppose that there were no three members such that they use only one language among them.  Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages.  Find the maximum possible value of $A$.
2013 USA Team Selection Test
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and

$$x^3(y^3+z^3)=2012(xyz+2).$$
2013 USA Team Selection Test
Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$, and let $D$ be the foot of the altitude from $C$.  Let $X$ be a point in the interior of the segment $CD$.  Let $K$ be the point on the segment $AX$ such that $BK = BC$.  Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$.  The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$).  Prove that $\angle ACT = \angle BCT$.
2013 USA Team Selection Test
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded.



Proposed by Palmer Mebane, United States
2013 USA Team Selection Test
Two incongruent triangles $ABC$ and $XYZ$ are called a pair of pals if they satisfy the following conditions:

(a) the two triangles have the same area;

(b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$.  The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers.



Determine if there are infinitely many pairs of triangles that are pals of each other.
2013 USA Team Selection Test
Let $ABC$ be an acute triangle.  Circle $\omega_1$, with diameter $AC$, intersects side $BC$ at $F$ (other than $C$).  Circle $\omega_2$, with diameter $BC$, intersects side $AC$ at $E$ (other than $C$).  Ray $AF$ intersects $\omega_2$ at $K$ and $M$ with $AK < AM$.  Ray $BE$ intersects $\omega_1$ at $L$ and $N$ with $BL < BN$.  Prove that lines $AB$, $ML$, $NK$ are concurrent.
2013 USA Team Selection Test
In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones.  For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column.  Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$.  We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row).



(a) Show that $f(C) \in \mathcal F$.

(b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.
2013 USA Team Selection Test
Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is not a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.
2012 USA Team Selection Test
In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.
2012 USA Team Selection Test
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for every pair of real numbers $x$ and $y$,

$$f(x+y^2)=f(x)+|yf(y)|.$$
2012 USA Team Selection Test
Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation

$$a^3+2b^3+4c^3=6abc+1.$$
2012 USA Team Selection Test
There are 2010 students and 100 classrooms in the Olympiad High School. At the beginning, each of the students is in one of the classrooms. Each minute, as long as not everyone is in the same classroom, somebody walks from one classroom into a different classroom with at least as many students in it (prior to his move). This process will terminate in $M$ minutes. Determine the maximum value of $M$.
2012 USA Team Selection Test
Consider (3-variable) polynomials

$$P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}$$

and

$$Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.$$

Determine all positive integers $n$ such that the quotient $Q_n(x,y,z)/P_n(x,y,z)$ is a (3-variable) polynomial with rational coefficients.
2012 USA Team Selection Test
In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.
2012 USA Team Selection Test
Determine all positive integers $n$, $n\ge2$, such that the following statement is true:

If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.
2012 USA Team Selection Test
Find all positive integers $a,n\ge1$ such that for all primes $p$ dividing $a^n-1$, there exists a positive integer $m<n$ such that $p\mid a^m-1$.
2011 USA Team Selection Test
In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$.



Proposed by Zuming Feng
2011 USA Team Selection Test
In the nation of Onewaynia, certain pairs of cities are connected by roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges). Some roads have a traffic capacity of 1 unit and other roads have a traffic capacity of 2 units. However, on every road, traffic is only allowed to travel in one direction. It is known that for every city, the sum of the capacities of the roads connected to it is always odd. The transportation minister needs to assign a direction to every road. Prove that he can do it in such a way that for every city, the difference between the sum of the capacities of roads entering the city and the sum of the capacities of roads leaving the city is always exactly one.



Proposed by Zuming Feng and Yufei Zhao
2011 USA Team Selection Test
Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a $p$-pod if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum

$$\sum_{k=0}^m (-1)^k {m \choose k} z_k.$$

Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.
2011 USA Team Selection Test
Find a real number $t$ such that for any set of 120 points $P_1, \ldots P_{120}$ on the boundary of a unit square, there exists a point $Q$ on this boundary with $|P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t$.
2011 USA Team Selection Test
Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that

$$\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.$$
2011 USA Team Selection Test
A polynomial $P(x)$ is called nice if $P(0) = 1$ and the nonzero coefficients of $P(x)$ alternate between $1$ and $-1$ when written in order. Suppose that $P(x)$ is nice, and let $m$ and $n$ be two relatively prime positive integers. Show that

$$Q(x) = P(x^n) \cdot \frac{(x^{mn} - 1)(x-1)}{(x^m-1)(x^n-1)}$$

is nice as well.
2011 USA Team Selection Test
Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.
2011 USA Team Selection Test
Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.
2011 USA Team Selection Test
Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality:

$$\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.$$
2010 USA Team Selection Test
Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and

$$\gcd(P(0), P(1), P(2), \ldots ) = 1.$$

Show there are infinitely many $n$ such that

$$\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.$$
2010 USA Team Selection Test
Let $a, b, c$ be positive reals such that $abc=1$. Show that $$\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.$$
2010 USA Team Selection Test
Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that

$$\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.$$
2010 USA Team Selection Test
Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.
2010 USA Team Selection Test
Define the sequence $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and, for $n > 1$,

$$a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.$$

Prove that there are infinitely many $n$ such that $a_n \equiv n \pmod{2^{2010}}$.
2010 USA Team Selection Test
Let $T$ be a finite set of positive integers greater than 1. A subset $S$ of $T$ is called good if for every $t \in T$ there exists some $s \in S$ with $\gcd(s, t) > 1$. Prove that the number of good subsets of $T$ is odd.
2010 USA Team Selection Test
In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.
2010 USA Team Selection Test
Let $m,n$ be positive integers with $m \geq n$, and let $S$ be the set of all $n$-term sequences of positive integers $(a_1, a_2, \ldots a_n)$ such that $a_1 + a_2 + \cdots + a_n = m$. Show that

$$\sum_S 1^{a_1} 2^{a_2} \cdots n^{a_n} =

{n \choose n} n^m - {n \choose n-1} (n-1)^m + \cdots +

(-1)^{n-2} {n \choose 2} 2^m + (-1)^{n-1} {n \choose 1}.$$
2010 USA Team Selection Test
Determine whether or not there exists a positive integer $k$ such that $p = 6k+1$ is a prime and

$$\binom{3k}{k} \equiv 1  \pmod{p}.$$
2009 USA Team Selection Test
Let $m$ and $n$ be positive integers.  Mr. Fat has a set $S$ containing every rectangular tile with integer side lengths and area of a power of $2$.  Mr. Fat also has a rectangle $R$ with dimensions $2^m \times 2^n$ and a $1 \times 1$ square removed from one of the corners.  Mr. Fat wants to choose $m + n$ rectangles from $S$, with respective areas $2^0, 2^1, \ldots, 2^{m + n - 1}$, and then tile $R$ with the chosen rectangles.  Prove that this can be done in at most $(m + n)!$ ways.



Palmer Mebane.
2009 USA Team Selection Test
Let $ ABC$ be an acute triangle.  Point $ D$ lies on side $ BC$.  Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively.  Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$.  Let $ H$ be the orthocenter of triangle $ ABC$.  Prove that $ \angle{DAX} = \angle{DAH}$.



Zuming Feng.
2009 USA Team Selection Test
For each positive integer $ n$, let $ c(n)$ be the largest real number such that

$$ c(n) \le \left| \frac {f(a) - f(b)}{a - b}\right|$$

for all triples $ (f, a, b)$ such that



--$ f$ is a polynomial of degree $ n$ taking integers to integers, and

--$ a, b$ are integers with $ f(a) \neq f(b)$.



Find $ c(n)$.



Shaunak Kishore.