We have two $20 \times 13$ rectangular grids with $260$ unit cells. each one. We insert in the boxes of each of the grids the numbers $1, 2, ..., 260$ as follows: $\bullet$ For the first grid, we start by inserting the numbers $1, 2, ..., 13$ in the boxes in the top row from left to right. We continue inserting numbers $14$, $ 15$, $...$, $26$ in the second row from left to right. We maintain the same procedure until in the last row, $20$, the numbers are placed $248$, $249$, $...$, $260$ from left to right. $\bullet$ For the second grid we start by inserting the numbers $1$, $2$,$ ..$., $20$ from top to bottom in the farthest column right. We continue inserting the numbers $21$, $22$,$ ...$, $40$ in the second column from the right also from top to bottom. We maintain that same procedure until we reach the column on the left where we place the numbers from top to bottom $241$, $242$, $ ...$, $260$. Determines the integers inserted in the boxes located in the same position in both grids.

The numbers $1, 2, ..., 2012$ are written on a blackboard, in some order, each of them exactly once. Between every two neighboring numbers the absolute value of their difference is written and the original numbers are deleted. This process is repeated until only a number remains on the board. What is the largest number that can stay on the board?

Ana and Carlos entertain themselves with the next game. At the beginning of game in each vertex of the square there is an empty box. In each step, the corresponding player has two possibilities: either he adds a stone to an arbitrary box, or move each box clockwise to the next vertex of the square. Carlos starts and they take 2012 steps in turn (each player 1006). So Carlos marks one of the vertices of the square and allows Ana to make a more play. Carlos wins if after this last step the number ofstones in some box is greater than the number of stones in the box which is at the vertex marked by Carlos; otherwise Ana wins. Which of the two players has a winning strategy?

Each unit square of a $5 \times 5$ board is colored blue or yellow. Prove that there is a rectangle with sides parallel to the sides. axes of the board, such that its four corners are the same color.

The number 2013 is written on a blackboard. Two players participate, alternating in turns, in the next game. A movement consists in changing the number that is on the board for the difference of this number and one of its divisors. The player who writes a zero loses. Which of the two players can guarantee victory?

Find all the integer solutions of the equation $ m^4 + 2n^2 = 9mn$.

Let $a$ and $b$ be real numbers with $0 \le a, b \le 1$. (a) Prove that $ \frac{a}{b+1} +\frac{b}{a+1} \le 1.$ (b) Find the case of equality.

Let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $AP$ and $AQ$. Let $T$ be another point of intersection of the circles $\Gamma_1$ and $\Gamma_2$. Let $Q_1$ be another point of intersection of the circle $\Gamma_1$ and the line $AQ$, and $P_1$ the other point of intersection of the circle $\Gamma_2$ and the line $AP$. The circle $\Gamma_3$ passes through the points $T$, $P$ and $P_1$ and the circle $\Gamma_4$ passes through the points $T$, $Q$ and $Q_1$. Prove that the line containing the common chord of the circles $\Gamma_3$ and $\Gamma_4$ passes through$A$.

Find all positive integers $a, b$ such that the numbers $\frac{a^2b + a}{a^2 + b}$ and $\frac{ab^2 + b}{b^2 - a}$ are integers.

Determine all real solutions to the system of equations: $$x^2 - y = z^2$$$$y^2 - z = x^2$$$$z^2 - x = y^2$$

Let $ABC$ be an isosceles triangle with $AB = AC$. Points $D$, $E$ and $F$ are on sides $BC$, $CA $ and $AB$ respectively, such that $\angle FDE =\angle ABC$ and $FE$ is not parallel to $BC$. Prove that $BC$ is tangent to the circumcircle of the triangle $DEF$, if and only if, $D$ is the midpoint of $BC$.

Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$

Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.

The triangle $ABC$ is inscribed in circle $\Gamma$. The points X, Y, Z are the midpoints of the arcs $BC$, $CA$ and $AB$ respectively in $\Gamma$ (those that do not contain the third vertex, in each case). The intersection points of the sides of the triangles $\vartriangle ABC$ and $\vartriangle XY Z$ form the hexagon $DEFGHK$. Prove that the diagonals $DG$, $EH$ and $FK$ are concurrent