CAPS 2016

Let $P$ be a non-degenerate polygon with $n$ sides, where $n > 4$ . Prove that there exist three distinct vertices $A, B, C$ of $P$ with the following property:If $\ell_1,\ell_2,\ell_3$ are the lengths of the three polygonal chains into which $A, B, C$ break the perimeter of $P$ , then there is a triangle with side lengths $\ell_1,\ell_2$ and $\ell_3$ . Remark: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.(Poland)

Let $n$ be a positive integer. For a finite set $M$ of positive integers and each $i \in \{0,1,..., n-1\}$ , we denote $s_i$ the number of non-empty subsets of $M$ whose sum of elements gives remainder $i$ after division by $n$ . We say that $M$ is " $n$ -balanced" if $s_0 = s_1 =....= s_{n-1}$ . Prove that for every odd number $n$ there exists a non-empty $n$ -balanced subset of $\{0,1,..., n\}$ . For example if $n = 5$ and $M = \{1,3,4\}$ , we have $s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2$ so $M$ is not $5$ -balanced.(Czech Republic)

Find all quadruplets $(a, b, c, d)$ of real numbers satisfying the system $(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)$ $(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)$ $(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)$ (Slovakia)

Prove that for every non-negative integer $n$ there exist integers $x, y, z$ with $gcd(x, y, z) = 1$ , such that $x^2 + y^2 + z^2 = 3^{2^n}$ .(Poland)

Let $ABC$ be an acute-angled triangle with $AB < AC$ . Tangent to its circumcircle $\Omega$ at $A$ intersects the line $BC$ at $D$ . Let $G$ be the centroid of $\triangle ABC$ and let $AG$ meet $\Omega$ again at $H \neq A$ . Suppose the line $DG$ intersects the lines $AB$ and $AC$ at $E$ and $F$ , respectively. Prove that $\angle EHG = \angle GHF$ .(Slovakia)