Let $ a_1=2, a_2=5$ and $ a_{n+2}=(2-n^2)a_{n+1}+ (2+n^2)a_n$ for $ n\geq 1$. Do there exist $ p,q,r$ so that $ a_pa_q =a_r$?

Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$ and such that $g(x) = g(y)$ only if $x = y$.

Consider all triangles $ABC$ in the cartesian plane whose vertices are at lattice points (i.e. with integer coordinates) and which contain exactly one lattice point (to be denoted $P$) in its interior. Let the line $AP$ meet $BC$ at $E$. Determine the maximum possible value of the ratio $\frac{AP}{PE}$.

For each real number $p > 1$, find the minimum possible value of the sum $x+y$, where the numbers $x$ and $y$ satisfy the equation $(x+\sqrt{1+x^2})(y+\sqrt{1+y^2}) = p$.

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.

Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $