EGMO 2014

Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.

Proposed by S. Khan, UNK The answer is the interval $[2 / 3,2]$.

Let $D$ and $E$ be two points on the sides $A B$ and $A C$, respectively, of a triangle $A B C$, such that $D B=B C=C E$, and let $F$ be the point of intersection of the lines $C D$ and $B E$. Prove that the incenter $I$ of the triangle $A B C$, the orthocenter $H$ of the triangle $D E F$ and the midpoint $M$ of the $\operatorname{arc} B A C$ of the circumcircle of the triangle $A B C$ are collinear.

Proposed by Danylo Khilko, UKR

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n)=k$ and $d(n)$ does not divide $d\left(a^{2}+b^{2}\right)$ for any positive integers $a, b$ satisfying $a+b=n$.

Proposed by JPN

Determine all integers $n \geq 2$ for which there exist integers $x_{1}, x_{2}, \ldots, x_{n-1}$ satisfying the condition that if $0<i<n, 0<j<n, i \neq j$ and $n$ divides $2 i+j$, then $x_{i}<x_{j}$.

Proposed by Merlijn Staps, NLD The answer is that $n=2^{k}$ with $k \geq 1$ or $n=3 \cdot 2^{k}$ where $k \geq 0$.

Let $n$ be a positive integer. We have $n$ boxes where each box contains a nonnegative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.

Proposed by Dan Schwarz, ROU

The answer is any configuration with $2 n-2$ pebbles which has even numbers of pebbles in each box.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition

$$f\left(y^{2}+2 x f(y)+f(x)^{2}\right)=(y+f(x))(x+f(y))$$

for all real numbers $x$ and $y$. Proposed by Daniël Kroes, NLD The answer is the functions $f(x)=x, f(x)=-x$ and $f(x)=\frac{1}{2}-x$.