Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively. Prove that $AB = V W$ Proposed by Warut Suksompong, Thailand

Let $S = \{2, 3, 4, \ldots\}$ denote the set of integers that are greater than or equal to $2$. Does there exist a function $f : S \to S$ such that \[f (a)f (b) = f (a^2 b^2 )\text{ for all }a, b \in S\text{ with }a \ne b?\] Proposed by Angelo Di Pasquale, Australia

A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold. (i) The value of $a_0$ is a positive integer. (ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\frac{a_i}{a_i + 2} $ (iii) There exists a positive integer $k$ such that $a_k = 2014$. Find the smallest positive integer $n$ such that there exists a good sequence $a_0, a_1, . . .$ of real numbers with the property that $a_n = 2014$. Proposed by Wang Wei Hua, Hong Kong

Let $n$ be a positive integer. Consider $2n$ distinct lines on the plane, no two of which are parallel. Of the $2n$ lines, $n$ are colored blue, the other $n$ are colored red. Let $\mathcal{B}$ be the set of all points on the plane that lie on at least one blue line, and $\mathcal{R}$ the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects $\mathcal{B}$ in exactly $2n - 1$ points, and also intersects $\mathcal{R}$ in exactly $2n - 1$ points. Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand

Determine all sequences $a_0 , a_1 , a_2 , \ldots$ of positive integers with $a_0 \ge 2015$ such that for all integers $n\ge 1$: (i) $a_{n+2}$ is divisible by $a_n$ ; (ii) $|s_{n+1} - (n + 1)a_n | = 1$, where $s_{n+1} = a_{n+1} - a_n + a_{n-1} - \cdots + (-1)^{n+1} a_0$ . Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand