CMO 2015

Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^{2}<f(n) f(f(n))<n^{2}+n$ for every positive integer $n$.

Let $A B C$ be an acute-angled triangle with altitudes $A D, B E$, and $C F$. Let $H$ be the orthocentre, that is, the point where the altitudes meet. Prove that

$$\frac{A B \cdot A C+B C \cdot B A+C A \cdot C B}{A H \cdot A D+B H \cdot B E+C H \cdot C F} \leq 2 .$$

On a $(4 n+2) \times(4 n+2)$ square grid, a turtle can move between squares sharing a side. The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?

Let $A B C$ be an acute-angled triangle with circumcenter $O$. Let $\Gamma$ be a circle with centre on the altitude from $A$ in $A B C$, passing through vertex $A$ and points $P$ and $Q$ on sides $A B$ and $A C$. Assume that $B P \cdot C Q=A P \cdot A Q$. Prove that $\Gamma$ is tangent to the circumcircle of triangle BOC.

Let $p$ be a prime number for which $\frac{p-1}{2}$ is also prime, and let a, $b, c$ be integers not divisible by $p$. Prove that there are at most $1+\sqrt{2 p}$ positive integers $n$ such that $n<p$ and $p$ divides $a^{n}+b^{n}+c^{n}$.