AT-OEMO-FED1 2014

Determine all real numbers $x$ and $y$ such that $x^2 + x = y^3 - y$ , $y^2 + y = x^3 - x$

We call a set of squares with sides parallel to the coordinate axes and vertices with integer coordinates friendly if any two of them have exactly two points in common. We consider friendly sets in which each of the squares has sides of length $n$ . Determine the largest possible number of squares in such a friendly set.

Let $a_n$ be a sequence defined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$ . Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$ .

We are given a right-angled triangle $MNP$ with right angle in $P$ . Let $k_M$ be the circle with center $M$ and radius $MP$ , and let $k_N$ be the circle with center $N$ and radius $NP$ . Let $A$ and $B$ be the common points of $k_M$ and the line $MN$ , and let $C$ and $D$ be the common points of $k_N$ and the line $MN$ with with $C$ between $A$ and $B$ . Prove that the line $PC$ bisects the angle $\angle APB$ .