Triangle $ABC$ with $\angle BAC=60^\circ$ is given. The circumcircle of $ABC$ is $\Omega$, and the orthocenter of $ABC$ is $H$. Let $S$ denote the midpoint of the arc $BC$ of $\Omega$ which doesn't contain $A$. Point $P$ was chosen on $\Omega$ so that $\angle HPS=90^\circ$. Prove that there exists a circle that goes through $P$ and $S$ and is tangent to lines $AB$, $AC$.

Let $n>1$ be an integer. Given a simple graph $G$ on $n$ vertices $v_1, v_2, \dots, v_n$ we let $k(G)$ be the minimal value of $k$ for which there exist $n$ $k$-dimensional rectangular boxes $R_1, R_2, \dots, R_n$ in a $k$-dimensional coordinate system with edges parallel to the axes, so that for each $1\leq i<j\leq n$, $R_i$ and $R_j$ intersect if and only if there is an edge between $v_i$ and $v_j$ in $G$. Define $M$ to be the maximal value of $k(G)$ over all graphs on $n$ vertices. Calculate $M$ as a function of $n$.

Let $n$ be a positive integer and $p$ be a prime number of the form $8k+5$. A polynomial $Q$ of degree at most $2023$ and nonnegative integer coefficients less than or equal to $n$ will be called "cool" if \[p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1.\]Prove that the number of cool polynomials is even.

Solve in positive integers: \[x^{y^2+1}+y^{x^2+1}=2^z\]

In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.

Let $0<c<1$ and $n$ a positive integer. Alice and Bob are playing a game. Bob writes $n$ integers on the board, not all equal. On a player's turn, they erase two numbers from the board and write their arithmetic mean instead. Alice starts and performs at most $cn$ moves. After her, Bob makes moves until there are only two numbers left on the board. Alice wins if these two numbers are different, and otherwise, Bob wins. For which values of $c$ does Alice win for all large enough $n$?

Let $ABC$ be a triangle and let $D$ be a point on $BC$ so that $AD$ bisects the angle $\angle BAC$. The common tangents of the circles $(BAD)$, $(CAD)$ meet at the point $A'$. The points $B'$, $C'$ are defined similarly. Show that $A'$, $B'$, $C'$ are collinear.

A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

Find all continuous functions $f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1}$ for which the following equation holds for all positive reals $x$, $y$: \[f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)\]

Let $G$ be a connected (simple) graph with $n$ vertices and at least $n$ edges. Prove that it is possible to color the vertices of $G$ red and blue, so that the following conditions hold: i. There is at least one vertex of each color, ii. There is an even number of edges connecting a red vertex to a blue vertex, and iii. If all such edges are deleted, one is left with two connected graphs.

Let $n$ be a positive integer. Find all polynomials $Q(x)$ with integer coefficients so that the degree of $Q(x)$ is less than $n$ and there exists an integer $m\geq 1$ for which \[x^n-1\mid Q(x)^m-1\]

Let $ABCD$ be a parallelogram. Let $\omega_1$ be the circle passing through $D$ tangent to $AB$ at $A$. Let $\omega_2$ be the circle passing through $A$ tangent to $CD$ at $D$. The tangents from $B$ to $\omega_1$ touch it at $A$ and $P$. The tangents from $C$ to $\omega_2$ touch it at $D$ and $Q$. Lines $AP$ and $DQ$ intersect at $X$. The perpendicular bisector of $BC$ intersects $AD$ at $R$. Show that the circumcircles of triangles $\triangle PQX$, $\triangle BCR$ are concentric.

For each positive integer $n$ let $a_n$ be the largest positive integer satisfying \[(a_n)!\left| \prod_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor\right.\]Show that there are infinitely many positive integers $m$ for which $a_{m+1}<a_m$.

Triangle $ABC$ is inscribed in circle $\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\Omega$ at $A$ with respect to $EF$ will be denoted $\ell_A$. We similarly define $\ell_B$, $\ell_C$. Show that the orthocenter of the triangle with sides $\ell_A$, $\ell_B$, $\ell_C$ lies on $OI$.

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-balanced if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\]Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.