The incircle of a triangle $ABC$ touches $BC$ at point $D$. Let $M$ be the midpoint of arc $\widehat{BAC}$ of the circumcircle, and $P$, $Q$ be the projections of $M$ to the external bisectors of angles $B$ and $C$ respectively. Prove that the line $PQ$ bisects $AD$.

Determine the lowest possible value of the expression \[ \frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y} \] where $a,b,x,$ and $y$ are positive real numbers satisfying the inequalities \[ \frac{1}{a+x} \ge \frac{1}{2} \] \[\frac{1}{a+y} \ge \frac{1}{2} \] \[ \frac{1}{b+x} \ge \frac{1}{2} \] \[ \frac{1}{b+y} \ge 1. \]

Alice and Bob play the following game on a square grid with $2024 \times 2024$ unit squares. They take turns covering unit squares with stickers including their names. Alice plays the odd-numbered turns, and Bob plays the even-numbered turns. \\ On the $k$-th turn, let $n_k$ be the least integer such that $n_k\geqslant\tfrac{k}{2024}$. If there is at least one square without a sticker, then the player taking the turn: [list = i] [*] selects at most $n_k$ unit squares on the grid such that at least one of the chosen unit squares does not have a sticker. [*] covers each of the selected unit squares with a sticker that has their name on it. If a selected square already has a sticker on it, then that sticker is removed first. [/list] At the end of their turn, a player wins if there exist $123$ unit squares containing stickers with that player's name that are placed on horizontally, vertically, or diagonally consecutive unit squares. We consider the game to be a draw if all of the unit squares are covered but no player has won yet. \\ Does Alice have a winning strategy? [i]Proposed by Erik Paemurru, Estonia[/i]

Solve the equation in the set of natural numbers $xyz+yzt+xzt+xyt = xyzt + 3$

Let $M$ be the midpoint of side $BC$ of triangle $ABC$, $Q$ the point of intersection of its angle bisectors. It is known that $MQ=QA$. Find the smallest possible value of angle $\angle MQA$.

$10$ people are sitting at a round table. There are some nuts in front of each of them, $100$ nuts altogether. After a certain signal each person passes some of his nuts to the person sitting to his right . If he has an even number of nuts, he passes half of them; otherwise he passes one nut plus half of the remaining nuts. This procedure is repeated over and over again. Prove that eventually everyone will have exactly $10$ nuts. (A Shapovalov)

Let $ ABCDA'B'C'D'$ be a cube. On the sides $ (A'D')$, $ (A'B')$ and $ (A'A)$ we consider the points $ M_1$, $ N_1$ and $ P_1$ respectively. On the sides $ (CB)$, $ (CD)$ and $ (CC')$ we consider the points $ M_2$, $ N_2$ and $ P_2$ respectively. Let $ d_1$ be the distance between the lines $ M_1N_1$ and $ M_2N_2$, $ d_2$ be the distance between the lines $ N_1P_1$ and $ N_2P_2$, and $ d_3$ be the distance between the lines $ P_1M_1$ and $ P_2M_2$. Suppose that the distances $ d_1$, $ d_2$ and $ d_3$ are pairwise distinct. Prove that the lines $ M_1M_2$, $ N_1N_2$ and $ P_1P_2$ are concurrent.

A whole number is written on each square of a board of $2019\times 2021$ squares. If the number written in each square is equal to the arithmetic mean of the numbers written in two of its neighboring squares, how many different numbers written on the blackboard can there be at most? Note: Two squares on the board are neighbors when they have a common side.

There are $n \ge 7$ points in the plane, no $3$ of which are collinear. At least $7$ pairs of points are joined by line segments. For every aforementioned line segment $s$, let $t(s)$ be the number of triangles for which the segment $s$ is a side. Prove that there exist different line segments $s_1, s_2, s_3,$ and $s_4$ such that \[t(s_1) = t(s_2) = t(s_3) = t(s_4)\] holds. Proposed by [i]Viktor Simjanoski[/i]

Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.

Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

Let $X =\{ X_1 , X_2 , ...\}$ be a countable set of points in space. Show that there is a positive sequence $\{a_k\}$ such that for any point $Z\not\in X$ the distance between the point Z and the set $\{X_1,X_2 , ...,X_k\}$ is at least $a_k$ for infinitely many k.

Let $ABC$ be an acute triangle with $AC > AB$ and circumcircle $\Gamma$. The tangent from $A$ to $\Gamma$ intersects $BC$ at $T$. Let $M$ be the midpoint of $BC$ and let $R$ be the reflection of $A$ in $B$. Let $S$ be a point so that $SABT$ is a parallelogram and finally let $P$ be a point on line $SB$ such that $MP$ is parallel to $AB$. Given that $P$ lies on $\Gamma$, prove that the circumcircle of $\triangle STR$ is tangent to line $AC$. [i]Proposed by Sam Bealing, United Kingdom[/i]

Show that there are only finitely many solutions to $1/a + 1/b + 1/c = 1/1983$ in positive integers.

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

A permutation $a_1, a_2, ..., a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of \[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \][i]Evan Chen[/i]