ÖMO 1986

Show that a square can be inscribed in any regular polygon.

For $s,t \in \mathbb{N}$ , consider the set $M=\{ (x,y) \in \mathbb{N} ^2 | 1 \le x \le s, 1 \le y \le t \}$ . Find the number of rhombi with the vertices in $M$ and the diagonals parallel to the coordinate axes.

Find all possible values of $x_0$ and $x_1$ such that the sequence defined by:

$x_{n+1}=\frac{x_{n-1} x_n}{3x_{n-1}-2x_n}$ for $n \ge 1$

contains infinitely many natural numbers.

Find the largest $n$ for which there is a natural number $N$ with $n$ decimal digits which are all different such that $n!$ divides $N$ . Furthermore, for this largest $n$ find all possible numbers $N$ .

Show that for every convex $n$ -gon $( n \ge 4)$ , the arithmetic mean of the lengths of its sides is less than the arithmetic mean of the lengths of all its diagonals.

Given a positive integer $n$ , find all functions $F: \mathbb{N} \rightarrow \mathbb{R}$ such that $F(x+y)=F(xy-n)$ whenever $x,y \in \mathbb{N}$ satisfy $xy>n$ .