AUSTRIAN-POLISH 1983

Nonnegative real numbers $a, b,x,y$ satisfy $a^5 + b^5 \le$ 1 and $x^5 + y^5 \le 1$ . Prove that $a^2x^3 + b^2y^3 \le 1$ .

Find all triples of positive integers $(p, q, n)$ with $p$ and $q$ prime, such that $p(p+1)+q(q+1) = n(n+1)$ .

A bounded planar region of area $S$ is covered by a finite family $F$ of closed discs. Prove that $F$ contains a subfamily consisting of pairwise disjoint discs, of joint area not less than $S/9$ .

The set $N$ has been partitioned into two sets A and $B$ . Show that for every $n \in N$ there exist distinct integers $a, b > n$ such that $a, b, a + b$ either all belong to $A$ or all belong to $B$ .

Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system $$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$ has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.

Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$ .

Let $P_1,P_2,P_3,P_4$ be four distinct points in the plane. Suppose $\ell_1,\ell_2, … , \ell_6$ are closed segments in that plane with the following property: Every straight line passing through at least one of the points $P_i$ meets the union $\ell_1 \cup \ell_2\cup … \cup\ell_6$ in exactly two points. Prove or disprove that the segments $\ell_i$ necessarily form a hexagon.

(a) Prove that $(2^{n+1}-1)!$ is divisible by $\prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$ , for every natural number n (b) Define the sequence ( $c_n$ ) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$ . Show that each $c_n$ is an integer.

To each side of the regular $p$ -gon of side length $1$ there is attached a $1 \times k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ( $kp$ unit cells and the $p$ -gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?