A real number is written next to each vertex of a regular pentagon. All five numbers are different. A triple of vertices is called successful if they form an isosceles triangle for which the number written on the top vertex is either larger than both numbers written on the base vertices, or smaller than both. Find the maximum possible number of successful triples.

Let $SABCDE$ be a pyramid whose base $ABCDE$ is a regular pentagon and whose other faces are acute triangles. The altitudes from $S$ to the base sides dissect them into ten triangles, colored red and blue alternatingly. Prove that the sum of the squared areas of the red triangles is equal to the sum of the squared areas of the blue triangles.

Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which \[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\]holds for any $x,y\in \mathbb{Z}$.

For positive integers $n$, let $f_2(n)$ denote the number of divisors of $n$ which are perfect squares, and $f_3(n)$ denotes the number of positive divisors which are perfect cubes. Prove that for each positive integer $k$ there exists a positive integer $n$ for which $\frac{f_2(n)}{f_3(n)}=k$.

In an $8 \times 8$ grid of squares, each square was colored black or white so that no $2\times 2$ square has all its squares in the same color. A sequence of distinct squares $x_1,\dots, x_m$ is called a snake of length $m$ if for each $1\leq i <m$ the squares $x_i, x_{i+1}$ are adjacent and are of different colors. What is the maximum $m$ for which there must exist a snake of length $m$?

In triangle $ABC$ the orthocenter is $H$ and the foot of the altitude from $A$ is $D$. Point $P$ satisfies $AP=HP$, and the line $PA$ is tangent to $(ABC)$. Line $PD$ intersects lines $AB, AC$ at points $X,Y$ respectively. Prove that $\angle YHX = \angle BAC$ or $\angle YHX+\angle BAC= 180^\circ$.

Toph wants to tile a rectangular $m\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?

For each positive integer $n$, define $A(n)$ to be the sum of its divisors, and $B(n)$ to be the sum of products of pairs of its divisors. For example, \[A(10)=1+2+5+10=18\]\[B(10)=1\cdot 2+1\cdot 5+1\cdot 10+2\cdot 5+2\cdot 10+5\cdot 10=97\]Find all positive integers $n$ for which $A(n)$ divides $B(n)$.

Let $ABC$ be a fixed triangle. Three similar (by point order) isosceles trapezoids are built on its sides: $ABXY, BCZW, CAUV$, such that the sides of the triangle are bases of the respective trapezoids. The circumcircles of triangles $XZU, YWV$ meet at two points $P, Q$. Prove that the line $PQ$ passes through a fixed point independent of the choice of trapezoids.

A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

Let $ABC$ be an isosceles triangle, $AB=AC$ inscribed in a circle $\omega$. The $B$-symmedian intersects $\omega$ again at $D$. The circle through $C,D$ and tangent to $BC$ and the circle through $A,D$ and tangent to $CD$ intersect at points $D,X$. The incenter of $ABC$ is denoted $I$. Prove that $B,C,I,X$ are concyclic.

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called perfect if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds: \[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]

Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums \[a_{k_1}\]\[a_{k_1}+a_{k_2}\]\[a_{k_1}+a_{k_2}+a_{k_3}\]\[\vdots\]\[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\]have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and incenter $I$. The midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$ is denoted $S$. Points $E, F$ were chosen on line $OI$ for which $BE$ and $CF$ are both perpendicular to $OI$. Point $X$ was chosen so that $XE\perp AC$ and $XF\perp AB$. Point $Y$ was chosen so that $YE\perp SC$ and $YF\perp SB$. $D$ was chosen on $BC$ so that $DI\perp BC$. Prove that $X$, $Y$, and $D$ are collinear.