Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$m+f(n) \mid f(m)^2 - nf(n)$$for all positive integers $m$ and $n$. (Here, $f(m)^2$ denotes $\left(f(m)\right)^2$.)

Some cells of an infinite chessboard (infinite in all directions) are coloured blue so that at least one of the $100$ cells in any $10 \times 10$ rectangular grid is blue. Prove that, for any positive integer $n$, it is possible to select $n$ rows and $n$ columns so that all of the $n^2$ cells in their intersections are blue.

Let $\triangle ABC$ be a triangle such that $AB = AC$, and let its circumcircle be $\Gamma$. Let $\omega$ be a circle which is tangent to $AB$ and $AC$ at $B$ and $C$. Point $P$ belongs to $\omega$, and lines $PB$ and $PC$ intersect $\Gamma$ again at $Q$ and $R$. $X$ and $Y$ are points on lines $BR$ and $CQ$ such that $AX = XB$ and $AY = YC$. Show that as $P$ varies on $\omega$, the circumcircle of $\triangle AXY$ passes through a fixed point other than $A$.

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. Proposed by Angelo Di Pasquale, Australia

For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that \[ \left\lfloor{(n-1)!\over n(n+1)}\right\rfloor \] is even for every positive integer $n$.

Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.

Let $n \geq 2$ be a positive integer. A total of $2n$ balls are coloured with $n$ colours so that there are two balls of each colour. These balls are put inside $n$ cylindrical boxes with two balls in each box, one on top of the other. Phoe Wa Lone has an empty cylindrical box and his goal is to sort the balls so that balls of the same colour are grouped together in each box. In a move, Phoe Wa Lone can do one of the following: Select a box containing exactly two balls and reverse the order of the top and the bottom balls. Take a ball $b$ at the top of a non-empty box and either put it in an empty box, or put it in the box only containing the ball of the same colour as $b$. Find the smallest positive integer $N$ such that for any initial placement of the balls, Phoe Wa Lone can always achieve his goal using at most $N$ moves in total.

Find all real numbers $a, b, c$ that satisfy $$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$