The first $360$ natural numbers are separated into $9$ blocks in such a way that the numbers in each block are consecutive. Then, the numbers in each block are added, obtaining $9$ numbers. Is it possible to fill a $3 \times 3$ grid and form a magic square with these numbers? Note: In a magic square, the sum of the numbers written in any column, diagonal or row of the grid is the same.

In the triangle $ABC$, we have that $\angle BAC$ is acute. Let $\Gamma$ be the circle that passes through $A$ and is tangent to the side $BC$ at $C$. Let $M$ be the midpoint of $BC$ and let $D$ be the other point of intersection of $\Gamma$ with $AM$. If $BD$ cuts back to$ \Gamma$ at $E$, show that $AC$ is the bisector of $\angle BAE$.

A board of size $2015 \times 2015$ is covered with sub-boards of size $2 \times 2$, each of which is painted like chessboard. Each sub-board covers exactly $4$ squares of the board and each square of the board is covered with at least one square of a sub-board (the painted of the sub-boards can be of any shape). Prove that there is a way to cover the board in such a way that there are exactly $2015$ black squares visible. What is the maximum number of visible black squares?

Find all natural integers $m, n$ such that $m, 2+m, 2^n+m, 2+2^n+m$ are all prime numbers

In the triangle $ABC$, we have that $M$ and $N$ are points on $AB$ and $AC$, respectively, such that $BC$ is parallel to $MN$. A point $D$ is chosen inside the triangle $AMN$. Let $E$ and $F$ be the points of intersection of $MN$ with $BD$ and $CD$, respectively. Show that the line joining the centers of the circles circumscribed to the triangles $DEN$ and $DFM$ is perpendicular to $AD$.

We have $3$ circles such that any $2$ of them are externally tangent. Let $a$ be length of the outer tangent common to a pair of them. The lengths $b$ and $c$ are defined similarly. If $T$ is the sum of the areas of such circles, show that $\pi (a + b + c)^2 \le 12T $. Note: In In the case of externally tangent circles, the common external tangent is the segment tangent to them that touches them at different points.