Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Ukraine Team Selection Test
There is graph $ G_0$ on vertices $ A_1, A_2, \ldots, A_n$. Graph $ G_{n + 1}$ on vertices $ A_1, A_2, \ldots, A_n$ is constructed by the rule: $ A_i$ and $ A_j$ are joined only if in graph $ G_n$ there is a vertices $ A_k\neq A_i, A_j$ such that $ A_k$ is joined with both $ A_i$ and $ A_j$. Prove that the sequence $ \{G_n\}_{n\in\mathbb{N}}$ is periodic after some term with period $ T \le 2^n$.
2008 Ukraine Team Selection Test
Given $ \triangle ABC$ with point $ D$ inside. Let $ A_0=AD\cap BC$, $ B_0=BD\cap AC$, $ C_0 =CD\cap AB$ and $ A_1$, $ B_1$, $ C_1$, $ A_2$, $ B_2$, $ C_2$ are midpoints of $ BC$, $ AC$, $ AB$, $ AD$, $ BD$, $ CD$ respectively. Two lines parallel to $ A_1A_2$ and $ C_1C_2$ and passes through point $ B_0$ intersects $ B_1B_2$ in points $ A_3$ and $ C_3$respectively. Prove that $ \frac{A_3B_1}{A_3B_2}=\frac{C_3B_1}{C_3B_2}$.
2008 Ukraine Team Selection Test
Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.



Author: Dan Brown, Canada
2008 Ukraine Team Selection Test
Let $ ABCDE$ be convex pentagon such that $ S(ABC) = S(BCD) = S(CDE) = S(DEA) = S(EAB)$. Prove that there is a point $ M$ inside pentagon such that $ S(MAB) = S(MBC) = S(MCD) = S(MDE) = S(MEA)$.
2008 Ukraine Team Selection Test
Prove that for all natural $ m$, $ n$ polynomial $ \sum_{i = 0}^{m}\binom{n+i}{n}\cdot x^i$ has at most one real root.
2007 Ukraine Team Selection Test
$\{a,b,c\}\subset\left(\frac{1}{\sqrt6},+\infty\right)$ such that $a^{2}+b^{2}+c^{2}=1.$ Prove that

$\frac{1+a^{2}}{\sqrt{2a^{2}+3ab-c^{2}}}+\frac{1+b^{2}}{\sqrt{2b^{2}+3bc-a^{2}}}+\frac{1+c^{2}}{\sqrt{2c^{2}+3ca-b^{2}}}\ge2(a+b+c).$
2007 Ukraine Team Selection Test
$ ABCD$ is convex $ AD\parallel BC$, $ AC\perp BD$. $ M$ is interior point of $ ABCD$ which is not a intersection of diagonals $ AC$ and $ BD$ such that $ \angle AMB =\angle CMD =\frac{\pi}{2}$ .$ P$   is intersection of angel bisectors of $ \angle A$  and $ \angle C$. $ Q$ is intersection of angel bisectors of $ \angle B$ and $ \angle D$. Prove that $ \angle PMB =\angle QMC$.
2007 Ukraine Team Selection Test
It is known that $ k$ and $ n$ are positive integers and $$ k + 1\leq\sqrt {\frac {n + 1}{\ln(n + 1)}}.$$ Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, - 1\}$ such that $ (x - 1)^{k}$ divides $ P(x)$.
2007 Ukraine Team Selection Test
Find all functions $f: \mathbb Q \to \mathbb Q$  such that $ f(x^{2}+y+f(xy)) = 3+(x+f(y)-2)f(x)$ for all $x,y \in \mathbb Q$.
2007 Ukraine Team Selection Test
$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA = x$. Find $ \angle A_{3}MB_{3}$.
2007 Ukraine Team Selection Test
Find all primes $ p$ for that there is an integer $ n$ such that there are no integers $ x,y$ with $ x^3 + y^3 \equiv n \mod p$ (so not all residues are the sum of two cubes).



E.g. for $ p = 7$, one could set $ n = \pm 3$ since $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$, thus $ x^3 + y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7$ only.
2007 Ukraine Team Selection Test
There are 25 people. Every two of them are use some language to speak between. They use only one language even if they both know another one. Among every three of them there is one who speaking with two other on the same language. Prove that there exist one who speaking with 10 other on the same language.
2007 Ukraine Team Selection Test
$ F(x)$ is polynomial with real coefficients. $ F(x) = x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}x^{1}+a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}-\frac{2a_{2}}{3^{2}})^{2}< M-m < 3(\frac{a_{1}^{2}}{4}-\frac{2a_{2}}{3^{2}})^{2}$
2007 Ukraine Team Selection Test
Find all positive integers $ n$ such that acute-angled $ \triangle ABC$ with $ \angle BAC<\frac{\pi}{4}$ could be divided into $ n$ quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression.

Click to reveal hidden textbe carefull !  :lol:
2007 Ukraine Team Selection Test
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}+n+1$ are not more then $ \sqrt{n}$.



Click to reveal hidden textStronger one.

Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}-1$ are not more then $ \sqrt{n}$.
1999 Ukraine Team Selection Test
A triangle $ABC$ is given. Points $E,F,G$ are arbitrarily selected on the sides $AB,BC,CA$, respectively, such that $AF\perp EG$ and the quadrilateral $AEFG$ is cyclic. Find the locus of the intersection point of $AF$ and $EG$.
1999 Ukraine Team Selection Test
Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.
1999 Ukraine Team Selection Test
Let $m,n$ be positive integers with $m \le n$, and let $F$ be a family of $m$-element subsets of $\{1,2,...,n\}$ satisfying $A \cap B \ne \varnothing$ for all $A,B \in F$. Determine the maximum possible number of elements in $F$.
1999 Ukraine Team Selection Test
If $n \in N$ and $0 < x <\frac{\pi}{2n}$, prove the inequality $\frac{\sin 2x}{\sin x}+\frac{\sin 3x}{\sin 2x} +...+\frac{\sin (n+1)x}{\sin nx} < 2\frac{\cos x}{\sin^2 x}$.

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1999 Ukraine Team Selection Test
A convex pentagon $ABCDE$ with $DC = DE$ and $\angle DCB = \angle DEA = 90^o$ is given.

Let $F$ be a point on the segment $AB$ such that $AF : BF = AE : BC$.

Prove that $\angle FCE = \angle ADE$ and $\angle FEC = \angle BDC$.
1999 Ukraine Team Selection Test
Show that for any $n \in N$ the polynomial $f(x) = (x^2 +x)^{2^n}+1$ is irreducible over $Z[x]$.
1999 Ukraine Team Selection Test
Let $P_1P_2...P_n$ be an oriented closed polygonal line with no three segments passing through a single point. Each point $P_i$ is assinged the angle $180^o - \angle P_{i-1}P_iP_{i+1} \ge 0$ if $P_{i+1}$ lies on the left from the ray $P_{i-1}P_i$, and the angle $-(180^o -\angle P_{i-1}P_iP_{i+1}) < 0$ if $P_{i+1}$ lies on the right. Prove that if the sum of all the assigned angles is a multiple of $720^o$, then the number of self-intersections of the polygonal line is odd
1999 Ukraine Team Selection Test
Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.
1999 Ukraine Team Selection Test
Find all functions $u : R \to R$ for which there is a strictly increasing function $f : R \to R$ such that $f(x+y) = f(x)u(y)+ f(y)$ for all $x,y \in R$.
1999 Ukraine Team Selection Test
For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$.

Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.
1999 Ukraine Team Selection Test
In a group of $n \ge 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeated and that there are $m \ge 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $\left[n+3 -\frac{18m}{n}\right]$
2020 USA EGMO Team Selection Test
Vulcan and Neptune play a turn-based game on an infinite grid of unit squares. Before the game starts, Neptune chooses a finite number of cells to be flooded. Vulcan is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop*.



The game then begins with Vulcan moving first. On each of Vulcan’s turns, he may add up to three new walls to the levee (maintaining the conditions for the levee). On each of Neptune’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Prove that Vulcan can always, in a finite number of turns, build the levee into a closed loop such that all flooded cells are contained in the interior of the loop, regardless of which cells Neptune initially floods.



*More formally, there must exist lattice points $\mbox{\footnotesize \(A_0, A_1, \dotsc, A_k\)}$, pairwise distinct except possibly $\mbox{\footnotesize \(A_0 = A_k\)}$, such that the set of walls is exactly $\mbox{\footnotesize \(\{A_0A_1, A_1A_2, \dotsc , A_{k-1}A_k\}\)}$. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. $\mbox{\footnotesize \(A_0 = A_k\)}$) then Vulcan cannot add more walls. Since each wall has length $\mbox{\footnotesize \(1\)}$, the length of the levee is $\mbox{\footnotesize \(k\)}$.
2020 USA EGMO Team Selection Test
Let $ABC$ be a triangle and let $P$ be a point not lying on any of the three lines $AB$, $BC$, or $CA$. Distinct points $D$, $E$, and $F$ lie on lines $BC$, $AC$, and $AB$, respectively, such that $\overline{DE}\parallel \overline{CP}$ and $\overline{DF}\parallel \overline{BP}$. Show that there exists a point $Q$ on the circumcircle of $\triangle AEF$ such that $\triangle BAQ$ is similar to $\triangle PAC$.



Andrew Gu
2020 USA EGMO Team Selection Test
Choose positive integers $b_1, b_2, \dotsc$ satisfying

$$1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb$$and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?



Carl Schildkraut and Milan Haiman
2020 USA EGMO Team Selection Test
Let $ABC$ be a triangle.  Distinct points $D$, $E$, $F$ lie on sides $BC$, $AC$, and $AB$, respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic.  Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$.  Let $Q$ denote the reflection of $P$ across $BC$.  Show that $Q$ lies on the circumcircle of $\triangle AEF$.



Proposed by Ankan Bhattacharya
2020 USA EGMO Team Selection Test
Let $G = (V, E)$ be a finite simple graph on $n$ vertices.  An edge $e$ of $G$ is called a bottleneck if one can partition $V$ into two disjoint sets $A$ and $B$ such that



at most $100$ edges of $G$ have one endpoint in $A$ and one endpoint in $B$; and

the edge $e$ is one such edge (meaning the edge $e$ also has one endpoint in $A$ and one endpoint in $B$).



Prove that at most $100n$ edges of $G$ are bottlenecks.



Proposed by Yang Liu
2020 USA EGMO Team Selection Test
Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$.  Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.
2019 USA EGMO Team Selection Test
A $3 \times 3$ grid of unit cells is given. A snake of length $k$ is an animal which occupies an ordered $k$-tuple of cells in this grid, say $(s_1, \dots, s_k)$. These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$. After being placed in a finite $n \times n$ grid, if the snake is currently occupying $(s_1, \dots, s_k)$ and $s$ is an unoccupied cell sharing a side with $s_1$, the snake can move to occupy $(s, s_1, \dots, s_{k-1})$ instead. The snake has turned around if it occupied $(s_1, s_2, \dots, s_k)$ at the beginning, but after a finite number of moves occupies $(s_k, s_{k-1}, \dots, s_1)$ instead.



Find the largest integer $k$ such that one can place some snake of length $k$ in a $3 \times 3$ grid which can turn around.
2019 USA EGMO Team Selection Test
Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.



Merlijn Staps
2019 USA EGMO Team Selection Test
Let $n$ be a positive integer such that the number

$$\frac{1^k + 2^k + \dots + n^k}{n}$$is an integer for any $k \in \{1, 2, \dots, 99\}$. Prove that $n$ has no divisors between 2 and 100, inclusive.
2019 USA EGMO Team Selection Test
For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that

$$a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}$$for all positive integers $m$ and $n$. Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$. Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$.
2019 USA EGMO Team Selection Test
Let the excircle of a triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively. Denote by $\gamma$ the circumcircle of triangle $A_1B_1C_1$ and assume that $\gamma$ passes through vertex $A$.



Show that $\overline{AA_1}$ is a diameter of $\gamma$.

Show that the incenter of $\triangle ABC$ lies on line $B_1C_1$.
2019 USA EGMO Team Selection Test
Let $n$ be a positive integer. Tasty and Stacy are given a circular necklace with $3n$ sapphire beads and $3n$ turquoise beads, such that no three consecutive beads have the same color. They play a cooperative game where they alternate turns removing three consecutive beads, subject to the following conditions:



Tasty must remove three consecutive beads which are turquoise, sapphire, and turquoise, in that order, on each of his turns.

Stacy must remove three consecutive beads which are sapphire, turquoise, and sapphire, in that order, on each of her turns.



They win if all the beads are removed in $2n$ turns. Prove that if they can win with Tasty going first, they can also win with Stacy going first.



Yannick Yao
2021 USA IMO Team Selection Test
Determine all integers $s \ge 4$ for which there exist positive integers $a$, $b$, $c$, $d$ such that $s = a+b+c+d$ and $s$ divides $abc+abd+acd+bcd$.



Proposed by Ankan Bhattacharya and Michael Ren
2021 USA IMO Team Selection Test
Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$.



Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$.  Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$.  Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively.



Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$.



Proposed by Andrew Gu and Frank Han
2021 USA IMO Team Selection Test
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the inequality

$$ f(y) - \left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2} $$for all real numbers $x < y < z$.



Proposed by Gabriel Carroll
2020 USA IMO Team Selection Test
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.



Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.



Merlijn Staps
2020 USA IMO Team Selection Test
Let $\alpha \geq 1$ be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be flooded. Hephaestus is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop*.



The game then begins with Hephaestus moving first. On each of Hephaestus’s turns, he adds one or more walls to the levee, as long as the total length of the levee is at most $\alpha n$ after his $n$th turn. On each of Poseidon’s turns, every cell which is adjacent to an already flooded cell and with no wall between them becomes flooded as well. Hephaestus wins if the levee forms a closed loop such that all flooded cells are contained in the interior of the loop — hence stopping the flood and saving the world. For which $\alpha$ can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood?



*More formally, there must exist lattice points $\mbox{\footnotesize \(A_0, A_1, \dotsc, A_k\)}$, pairwise distinct except possibly $\mbox{\footnotesize \(A_0 = A_k\)}$, such that the set of walls is exactly $\mbox{\footnotesize \(\{A_0A_1, A_1A_2, \dotsc , A_{k-1}A_k\}\)}$. Once a wall is built it cannot be destroyed; in particular, if the levee is a closed loop (i.e. $\mbox{\footnotesize \(A_0 = A_k\)}$) then Hephaestus cannot add more walls. Since each wall has length $\mbox{\footnotesize \(1\)}$, the length of the levee is $\mbox{\footnotesize \(k\)}$.



Nikolai Beluhov
2020 USA IMO Team Selection Test
For a finite simple graph $G$, we define $G'$ to be the graph on the same vertex set as $G$, where for any two vertices $u \neq v$, the pair $\{u,v\}$ is an edge of $G'$ if and only if $u$ and $v$ have a common neighbor in $G$.



Prove that if $G$ is a finite simple graph which is isomorphic to $(G')'$, then $G$ is also isomorphic to $G'$.



Mehtaab Sawhney and Zack Chroman
2020 USA IMO Team Selection Test
Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions:



$m > 1$ and $\gcd(m,n) = 1$;

the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ 		are not divisible by $n$; and

$P^m(0)$ is divisible by $n$.



Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc.



Carl Schildkraut
2020 USA IMO Team Selection Test
Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$.



Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic.



Michael Ren
2019 USA IMO Team Selection Test
Let $\mathbb{Z}/n\mathbb{Z}$ denote the set of integers considered modulo $n$ (hence $\mathbb{Z}/n\mathbb{Z}$ has $n$ elements). Find all positive integers $n$ for which there exists a bijective function $g: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, such that the 101 functions

$$g(x), \quad g(x) + x, \quad g(x) + 2x, \quad \dots, \quad g(x) + 100x$$are all bijections on $\mathbb{Z}/n\mathbb{Z}$.



Ashwin Sah and Yang Liu
2019 USA IMO Team Selection Test
A snake of length $k$ is an animal which occupies an ordered $k$-tuple $(s_1, \dots, s_k)$ of cells in a $n \times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$. If the snake is currently occupying $(s_1, \dots, s_k)$ and $s$ is an unoccupied cell sharing a side with $s_1$, the snake can move to occupy $(s, s_1, \dots, s_{k-1})$ instead. The snake has turned around if it occupied $(s_1, s_2, \dots, s_k)$ at the beginning, but after a finite number of moves occupies $(s_k, s_{k-1}, \dots, s_1)$ instead.



Determine whether there exists an integer $n > 1$ such that: one can place some snake of length $0.9n^2$ in an $n \times n$ grid which can turn around.



Nikolai Beluhov
2019 USA IMO Team Selection Test
We say that a function $f: \mathbb{Z}_{\ge 0} \times \mathbb{Z}_{\ge 0} \to \mathbb{Z}$ is great if for any nonnegative integers $m$ and $n$,

$$f(m + 1, n + 1) f(m, n) - f(m + 1, n) f(m, n + 1) = 1.$$If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$ are two sequences of integers, we write $A \sim B$ if there exists a great function $f$ satisfying $f(n, 0) = a_n$ and $f(0, n) = b_n$ for every nonnegative integer $n$ (in particular, $a_0 = b_0$).



Prove that if $A$, $B$, $C$, and $D$ are four sequences of integers satisfying $A \sim B$, $B \sim C$, and $C \sim D$, then $D \sim A$.



Ankan Bhattacharya
2019 USA IMO Team Selection Test
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on line $BC$ satisfying $\angle AID=90^{\circ}$. Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to $\overline{BC}$ at $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively.



Prove that if quadrilateral $AB_1A_1C_1$ is cyclic, then $\overline{AD}$ is tangent to the circumcircle of $\triangle DB_1C_1$.



Ankan Bhattacharya
2018 USA Team Selection Test
Let $n \ge 2$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Prove that the $n^{\text{th}}$ smallest positive integer relatively prime to $n$ is at least $\sigma(n)$, and determine for which $n$ equality holds.



Proposed by Ashwin Sah
2018 USA Team Selection Test
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,

$$f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.$$

Proposed by Yang Liu and Michael Kural
2018 USA Team Selection Test
At a university dinner, there are 2017 mathematicians who each order two distinct entrées, with no two mathematicians ordering the same pair of entrées. The cost of each entrée is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entrée (ties broken arbitrarily). Over all possible sets of orders, what is the maximum total amount the university could have paid?



Proposed by Evan Chen
2018 USA Team Selection Test
Let $n$ be a positive integer and let $S \subseteq \{0, 1\}^n$ be a set of binary strings of length $n$. Given an odd number $x_1, \dots, x_{2k + 1} \in S$ of binary strings (not necessarily distinct), their majority is defined as the binary string $y \in \{0, 1\}^n$ for which the $i^{\text{th}}$ bit of $y$ is the most common bit among the $i^{\text{th}}$ bits of $x_1, \dots,x_{2k + 1}$. (For example, if $n = 4$ the majority of 0000, 0000, 1101, 1100, 0101 is 0100.)



Suppose that for some positive integer $k$, $S$ has the property $P_k$ that the majority of any $2k + 1$ binary strings in $S$ (possibly with repetition) is also in $S$. Prove that $S$ has the same property $P_k$ for all positive integers $k$.



Proposed by Joshua Brakensiek
2018 USA Team Selection Test
Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that



ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and

$\angle BEN = \angle MED$.





Proposed by Evan Chen
2018 USA Team Selection Test
Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope $+1$ or $-1$, she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can determine the set $S$.



Prove that if Alice picks $S$ to be of the form

$$S = \{(x, y) \in \mathbb{Z}^2 \mid m \le x^2 + y^2 \le n\}$$for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.)



Proposed by Mark Sellke