AUSTRIAN-POLISH 1978

Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy $$f(x+y)=f(x^2+y^2) \forall x,y\in (0;\infty)$$

A parallelogram is inscribed into a regular hexagon so that the centers of symmetry of both figures coincide. Prove that the area of the parallelogram does not exceed $2/3$ the area of the hexagon.

Prove that $$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$

Let $c\neq 1$ be a positive rational number. Show that it is possible to partition $\mathbb{N}$ , the set of positive integers, into two disjoint nonempty subsets $A,B$ so that $x/y\neq c$ holds whenever $x$ and $y$ lie both in $A$ or both in $B$ .

We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$ . Prove that all the sets have a common element.

We are given a family of discs in the plane, with pairwise disjoint interiors. Each disc is tangent to at least six other discs of the family. Show that the family is infinite.

Let $M$ be the set of all lattice points in the plane (i.e. points with integer coordinates, in a fixed Cartesian coordinate system). For any point $P=(x,y)\in M$ we call the points $(x-1,y)$ , $(x+1,y)$ , $(x,y-1)$ , $(x,y+1)$ neighbors of $P$ . Let $S$ be a finite subset of $M$ . A one-to-one mapping $f$ of $S$ onto $S$ is called perfect if $f(P)$ is a neighbor of $P$ , for any $P\in S$ . Prove that if such a mapping exists, then there exists also a perfect mapping $g:S\to S$ with the additional property $g(g(P))=P$ for $P\in S$ .

For any positive integer $k$ consider the sequence $$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$ where there are $n$ square-root signs on the right-hand side.

(a) Show that the sequence converges, for every fixed integer $k\ge 1$ . (b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.

In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$ . Show that there exist at least two vertices of $P$ , neither one of them being an endpoint of any one of those diagonals.