Let $m$ be a positive integer and $n=m^2+1$. Determine all real numbers $x_1,x_2,\dotsc ,x_n$ satisfying $$x_i=1+\frac{2mx_i^2}{x_1^2+x_2^2+\cdots +x_n^2}\quad \text{for all }i=1,2,\dotsc ,n.$$

If $n\geqslant 3$ is an integer and $a_1,a_2,\dotsc ,a_n$ are non-zero integers such that $$a_1a_2\cdots a_n\left( \frac{1}{a_1^2}+\frac{1}{a_2^2} +\cdots +\frac{1}{a_n^2}\right)$$is an integer, does it follow that the product $a_1a_2\cdots a_n$ is divisible by each $a_i^2$?

Let $ABC$ be a triangle. Let $M$ be a variable point interior to the segment $AB$, and let $\gamma_B$ be the circle through $M$ and tangent at $B$ to $BC$. Let $P$ and $Q$ be the touch points of $\gamma_B$ and its tangents from $A$, and let $X$ be the midpoint of the segment $PQ$. Similarly, let $N$ be a variable point interior to the segment $AC$, and let $\gamma_C$ be the circle through $M$ and tangent at $C$ to $BC$. Let $R$ and $S$ be the touch points of $\gamma_C$ and its tangents from $A$, and let $Y$ be the midpoint of the segment $RS$. Prove that the line through the centers of the circles $AMN$ and $AXY$ passes through a fixed point.

Given a positive integer $n$. A triangular array $(a_{i,j})$ of zeros and ones, where $i$ and $j$ run through the positive integers such that $i+j\leqslant n+1$ is called a binary anti-Pascal $n$-triangle if $a_{i,j}+a_{i,j+1}+a_{i+1,j}\equiv 1\pmod{2}$ for all possible values $i$ and $j$ may take on. Determine the minimum number of ones a binary anti-Pascal $n$-triangle may contain.

Determine all positive integers $n$ such that for every positive devisor $ d $ of $n$, $d+1$ is devisor of $n+1$.

Let $A$ and $C$ be two points on a circle $X$ so that $AC$ is not diameter and $P$ a segment point on $AC$ different from its middle. The circles $c_1$ and $c_2$, inner tangents in $A$, respectively $C$, to circle $X$, pass through the point $P$ ¸ and intersect a second time at point $Q$. The line $PQ$ intersects the circle $X$ in points $B$ and $D$. The circle $c_1$ intersects the segments $AB$ and $AD$ in $K$, respectively $N$, and circle $c_2$ intersects segments $CB$ and ¸ $CD $ in $L$, respectively $M$. Show that: a) the $KLMN$ quadrilateral is isosceles trapezoid; b) $Q$ is the middle of the segment $BD$. Proposed by Thanos Kalogerakis

On a board the numbers $(n-1, n, n+1)$ are written where $n$ is positive integer. On a move choose 2 numbers $a$ and $b$, delete them and write $2a-b$ and $2b-a$. After a succession of moves, on the board there are 2 zeros. Find all possible values for $n$. Proposed by Andrei Eckstein

For positive real numbers $a_1, a_2, ..., a_n$ with product 1 prove: $$\left(\frac{a_1}{a_2}\right)^{n-1}+\left(\frac{a_2}{a_3}\right)^{n-1}+...+\left(\frac{a_{n-1}}{a_n}\right)^{n-1}+\left(\frac{a_n}{a_1}\right)^{n-1} \geq a_1^{2}+a_2^{2}+...+a_n^{2}$$Proposed by Andrei Eckstein