Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 ELMO Shortlist
Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$.



Evan O'Dorney.
2011 ELMO Shortlist
In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds:

$$\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.$$



Calvin Deng.
2011 ELMO Shortlist
Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that

$$\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,$$where $k=2+\sqrt{3}$.



Victor Wang.
2011 ELMO Shortlist
Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$,

$$\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),$$where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$.



Calvin Deng.
2011 ELMO Shortlist
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.



Evan O'Dorney.
2011 ELMO Shortlist
Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that



a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$;



b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$.



Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if

$$\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).$$



Evan O'Dorney.
2011 ELMO Shortlist
A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to.



David Yang.



Stronger VersionSee here.
2011 ELMO Shortlist
Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where



a) $f(n)=n\ln{n}$;



b) $f(n)=n$.



David Yang.
2011 ELMO Shortlist
Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$?



David Yang.
2011 ELMO Shortlist
Let $T$ be a tree. Prove that there is a constant $c>0$ (independent of $n$) such that every graph with $n$ vertices that does not contain a subgraph isomorphic to $T$ has at most $cn$ edges.



David Yang.
2011 ELMO Shortlist
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.



David Yang.
2011 ELMO Shortlist
Let $ABC$ be a triangle. Draw circles $\omega_A$, $\omega_B$, and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$, and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$. Define $P_B$ and $P_C$ similarly. Prove that $AP_A$, $BP_B$, and $CP_C$ are concurrent.



Tom Lu.
2011 ELMO Shortlist
Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$.



Calvin Deng.
2011 ELMO Shortlist
Prove that $n^3-n-3$ is not a perfect square for any integer $n$.



Calvin Deng.
2011 ELMO Shortlist
Let $p\ge5$ be a prime. Show that

$$\sum_{k=0}^{(p-1)/2}\binom{p}{k}3^k\equiv 2^p - 1\pmod{p^2}.$$



Victor Wang.
2011 ELMO Shortlist
Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ good if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples.



Mitchell Lee.
2010 ELMO Shortlist
Let $a,b,c$ be positive reals. Prove that

$$ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. $$



Calvin Deng.
2010 ELMO Shortlist
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.



George Xing.
2010 ELMO Shortlist
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$.



Evan O' Dorney.
2010 ELMO Shortlist
Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of $$\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}$$ over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers.



Brian Hamrick.
2010 ELMO Shortlist
For all positive real numbers $a,b,c$, prove that $$\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.$$



In-Sung Na.
2010 ELMO Shortlist
Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$.



Evan O' Dorney.
2010 ELMO Shortlist
For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$.



Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?

Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?





Brian Hamrick.
2010 ELMO Shortlist
For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$,

$$\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.$$



Alex Zhu.
2010 ELMO Shortlist
The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$.



Brian Hamrick.
2010 ELMO Shortlist
Hamster is playing a game on an $m \times n$ chessboard. He places a rook anywhere on the board and then moves it around with the restriction that every vertical move must be followed by a horizontal move and every horizontal move must be followed by a vertical move. For what values of $m,n$ is it possible for the rook to visit every square of the chessboard exactly once? A square is only considered visited if the rook was initially placed there or if it ended one of its moves on it.



Brian Hamrick.
2010 ELMO Shortlist
The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game?



Brian Hamrick.
2010 ELMO Shortlist
A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$?



David Yang.
2010 ELMO Shortlist
Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent.



Carl Lian.
2010 ELMO Shortlist
Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$.



Brian Hamrick.
2010 ELMO Shortlist
A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$.



Evan O' Dorney.
2010 ELMO Shortlist
Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic.



Carl Lian.
2010 ELMO Shortlist
For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have

$$\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, $$

and determine when equality holds.



Wenyu Cao.
2010 ELMO Shortlist
Given a prime $p$, show that $$\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.$$



Timothy Chu.
2010 ELMO Shortlist
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that

\begin{align*}

a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a

\end{align*}



Travis Hance.
2010 ELMO Shortlist
Find the set $S$ of primes such that $p \in S$ if and only if there exists an integer $x$ such that $x^{2010} + x^{2009} + \cdots + 1 \equiv p^{2010} \pmod{p^{2011}}$.



Brian Hamrick.
2019 Estonia Team Selection Test
Some positive integer $n$ is written on the board. Andrey and Petya is playing the following game. Andrey finds all the different representations of the number n as a product of powers of prime numbers (values degrees greater than 1), in which each factor is greater than all previous or equal to the previous one. Petya finds all different representations of the number $n$ as a product of integers greater than $1$, in which each factor is divisible by all the previous factors. The one who finds more performances wins, if everyone finds the same number of representations, the game ends in a draw. Find all positive integers $n$ for which the game will end in a draw.



Note.

The representation of the number $n$ as a product is also considered a representation consisting of a single factor $n$.
2019 Estonia Team Selection Test
In an acute-angled triangle $ABC$, the altitudes intersect at point $H$, and point $K$ is the foot of the altitude drawn from the vertex $A$. Circle $c$ passing through points $A$ and $K$ intersects sides $AB$ and $AC$ at points $M$ and $N$, respectively. The line passing through point $A$ and parallel to line $BC$ intersects the circumcircles of triangles $AHM$ and $AHN$ for  second time, respectively, at points $X$ and $Y$. Prove that $ | X Y | = | BC |$.
2019 Estonia Team Selection Test
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
2019 Estonia Team Selection Test
Let us call a real number $r$ interesting, if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.
2019 Estonia Team Selection Test
Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country
2019 Estonia Team Selection Test
It is allowed to perform the following transformations in the plane with any integers $a$:

(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,

(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.

Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:

a) Vertices of a square,

b) Vertices of a rectangle with unequal side lengths?
2019 Estonia Team Selection Test
An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.
2019 Estonia Team Selection Test
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
2019 Estonia Team Selection Test
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:

(a) the circumcircle of each triangle in the set $T$ is $\omega$;

(b) The interior of any two triangles in the set $T$ has no common point.

Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
2018 Estonia Team Selection Test
There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.
2018 Estonia Team Selection Test
Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected.
2018 Estonia Team Selection Test
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.
2018 Estonia Team Selection Test
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y  \in R$
2018 Estonia Team Selection Test
We call a positive integer $n$ whose all digits are distinct bright, if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)
2018 Estonia Team Selection Test
Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for  second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$  respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.
2018 Estonia Team Selection Test
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors

$1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.
2018 Estonia Team Selection Test
Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times  n$, and player $B$ has a $1  \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$.

Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.
2018 Estonia Team Selection Test
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations

$b_1 = a_1$ ,

$b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ ,

$b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$.

Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$
2018 Estonia Team Selection Test
Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied:

1) there is at least $1$ marked point on each side,

2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$
2018 Estonia Team Selection Test
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.

Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
2017 Estonia Team Selection Test
Do there exist two positive powers of $5$ such that the number obtained by writing one after the other is also a power of $5$?
2017 Estonia Team Selection Test
Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine:

a) which vertex of $BCF$ is its apex,

b) the size of $\angle BAC$
2017 Estonia Team Selection Test
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2017 Estonia Team Selection Test
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
2017 Estonia Team Selection Test
Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that

a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$

b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?
2017 Estonia Team Selection Test
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
2017 Estonia Team Selection Test
Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be nice if



$\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and



$\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar.



Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.
2017 Estonia Team Selection Test
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$  and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.
2017 Estonia Team Selection Test
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$

Proposed by Warut Suksompong, Thailand
2017 Estonia Team Selection Test
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
2016 Estonia Team Selection Test
There are $k$ heaps on the table, each containing a different positive number of stones. Juri and Mari make moves alternatingly, Juri starts. On each move, the player making the move has to pick a heap and remove one or more stones in it from the table; in addition, the player is allowed to distribute any number of remaining stones from that heap in any way between other non-empty heaps. The player to remove the last stone from the table wins. For which positive integers $k$ does Juri have a winning strategy for any initial state that satisfies the conditions?
2016 Estonia Team Selection Test
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.
2016 Estonia Team Selection Test
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y  \in  R$.