OME 1988

A sequence of integers $(x_n)_{n=1}^{\infty}$ satisfies $x_1 = 1$ and $x_n < x_{n+1} \le 2n$ for all $n$ . Show that for every positive integer $k$ there exist indices $r, s$ such that $x_r-x_s = k$ .

We choose $n > 3$ points on a circle and number them $1$ to $n$ in some order. We say that two non-adjacent points $A$ and $B$ are related if, in one of the arcs $AB$ , all the points are marked with numbers less than those at $A,B$ . Show that the number of pairs of related points is exactly $n-3$ .

Prove that if one of the numbers $25x+31y, 3x+7y$ (where $x,y \in Z$ ) is a multiple of $41$ , then so is the other.

The Fibonacci sequence is given by $a_1 = 1, a_2 = 2$ and $a_{n+1} = a_n +a_{n-1}$ for $n > 1$ . Express $a_{2n}$ in terms of only $a_{n-1},a_n,a_{n+1}$ .

A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One solution is exhibited on the picture. Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)

For all integral values of parameter $t$ , find all integral solutions $(x,y)$ of the equation $$y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .