Let $n$ be a positive integer. Aslı and Zehra are playing a game on an $n\times n$ grid. Initially, $10n^2$ stones are placed on some of the unit squares of this grid. On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square. For which values of $n$, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?

$A, B, C, D$ are consecutive vertices of a regular $7$-gon. $AL$ and $AM$ are tangents to the circle center $C$ radius $CB$. $N$ is the intersection point of $AC$ and $BD$. Show that $L, M, N$ are collinear.

Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.

Given a regular tetrahedron $PQRS$ with side length $d$. Find the volume of the solid generated by a rotation around the line passing through $P$ and the midpoint $M$ of $QR$.

Given the vertex $A$ and the centroid $M$ of a triangle $ABC$, find the locus of vertices $B$ such that all the angles of the triangle lie in the interval $[40^\circ, 70^\circ].$

Consider two uniform spherical planets of equal density but unequal radius. Which of the following quantities is the same for both planets? (a) The escape velocity from the planet’s surface. (b) The acceleration due to gravity at the planet’s surface. (c) The orbital period of a satellite in a circular orbit just above the planet’s surface. (d) The orbital period of a satellite in a circular orbit at a given distance from the planet’s center. (e) None of the above.

For every non-constant polynomial $p$, let $H_p=\big\{z\in \mathbb{C} \, \big| \, |p(z)|=1\big\}$. Prove that if $H_p=H_q$ for some polynomials $p,q$, then there exists a polynomial $r$ such that $p=r^m$ and $q=\xi\cdot r^n$ for some positive integers $m,n$ and constant $|\xi|=1$.

$T$ is a set of all the positive integers of the form $2^k 3^l$, where $k, l$ are some non-negetive integers. Show that there exists $1998$ different elements of $T$ that satisfy the following condition. [b]Condition[/b] The sum of the $1998$ elements is again an element of $T$.

We have $ 100$ equal cubes. Player $ A$ has to paint the faces of the cubes, each white or black, such that every cube has at least one face of each colour, at least $ 50$ cubes have more than one black face and at least $ 50$ cubes have more than one white face . Player $ B$ has to place the coloured cubes in a table in a way that their bases form the frame that surrounds a $ 40*12$ rectangle. There are some faces that can not been seen because they are overlapped with other faces or based on the table, we call them invisible faces. On the other hand, the ones which can be seen are called visible faces. Prove that player $ B$ can always place the cubes in such a way that the number of visible faces is the the same as the number of invisible faces, despite the initial colouring of player $ A$ Note: It is easy to see that in the configuration, each cube has three visible faces and three invisible faces

Let $ z_1,z_2,z_3\in\mathbb{C} , $ different two by two, having the same modulus $ \rho . $ Show that: $$ \frac{1}{\left| z_1-z_2\right|\cdot \left| z_1-z_3\right|} +\frac{1}{\left| z_2-z_1\right|\cdot \left| z_2-z_3\right|} +\frac{1}{\left| z_3-z_1\right|\cdot \left| z_3-z_2\right|}\ge\frac{1}{\rho^2} . $$