AUSTRIAN-POLISH 1987

Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$ . Show that if $x,y$ are rational numbers with $P(x) = P(y)$ , then $x = y$ .

A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$ . Given $a \in R$ , define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$ . Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$ . Show that one of these sets contains, for each $d > 0$ , a pair of points at mutual distance $d$ .

Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max} \underset{i}{min}\delta (P_i, d)$ , where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$ . Let $F_n$ be the family of all such sets $A$ . Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$ .

For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$ , $s(n)=a_0+ a_1+ ... + a_k$ , $n^*= \overline{a_0a_1...a_k}$ . Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$ . (a) Show that $P$ is infinite. (b) Show that $Q$ is finite. (c) Write down all the elements of $Q$ .

A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$ . A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.

Let $M$ be the set of all points $(x,y)$ in the cartesian plane, with integer coordinates satisfying $1 \le x \le 12$ and $1 \le y \le 13$ . (a) Prove that every $49$ -element subset of $M$ contains four vertices of a rectangle with sides parallel to the coordinate axes. (b) Give an example of a $48$ -element subset of $M$ without this property.