Let $ABC$ be a triangle with circumcenter $O$. Point $D, E, F$ are chosen on sides $AB, BC$ and $AC$, respectively, such that $ADEF$ is a rhombus. The circumcircles of $BDE$ and $CFE$ intersect $AE$ at $P$ and $Q$ respectively. Show that $OP=OQ$. Proposed by Ariel García

Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice can achieve this no matter how Bob plays. Proposed by Oriol Solé

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} Proposed by Isaac Jiménez

Show that the equation $$a^2b=2017(a+b)$$has no solutions for positive integers $a$ and $b$. Proposed by Oriol Solé

Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation $$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$. Proposed by Maximiliano Sánchez

In a certain country there are $n$ cities. Some pairs of cities are connected by highways in such a way that for each two cities there is at most one highway connecting them. Assume that for a certain positive integer $k$, the total number of highways is greater than $\frac{nk}{2}$. Show that there exist $k+2$ distinct cities $C_1, C_2, \dots, C_{k+2}$ such that $C_i$ and $C_{i+1}$ are connected by a highway for $i=1, 2, \dots, k+1$. Proposed by Oriol Solé