1st Austrian-Polish 1978


1. Find all real-valued functions f on the positive reals which satisfy f(x + y) = f(x² + y²) for all positive x, y.
2. A parallelogram has its vertices on the boundary of a regular hexagon and its center at the center of the hexagon. Show that its area is at most 2/3 the area of the hexagon.
3. Let x = 1^o. Show that (tan x tan 2x ... tan 44x)^1/44 < √2 - 1 < (tan x + tan 2x + ... + tan 44x)/44.
4. Given a positive rational k not equal to 1, show that we can partition the positive integers into sets A_k and B_k, so that if m and n are both in A_k or both in B_k then m/n does not equal k.
5. The sets A₁, A₂, ... , A₁₉₇₈ each have 40 elements and the intersection of any two distinct sets has just one element. Show that the intersection of all the sets has one element.
6. S is a set of disks in the plane. No point belongs to the interior of more than one disk. Each disk has a point in common with at least 6 other disks. Show that S is infinite.
7. S is a finite set of lattice points in the plane such that we can find a bijection f: S → S satisfying \|P - f(P)\| = 1 for all P in S. Show that we can find a bijection g: S → S such that \|P - g(P)\| = 1 for all P in S and g(g(P) ) = P for all P in S.
8. k is a positive integer. Define a₁ = √k, a_n+1 = √(k + a_n). Show that the sequence a_n converges. Find all k such that the limit is an integer. Show that if k is odd, then the limit is irrational.
9. P is a convex polygon. Some of the diagonals are drawn, so that no interior point of P lies on more than one diagonal. Show that at least two vertices of P do not lie on any diagonals.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.