# 1972 IMO Problems

Problems of the 14th IMO 1972 in Poland.

## Problem 1

Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.

## Problem 2

Prove that if , every quadrilateral that can be inscribed in a circle can be dissected into quadrilaterals each of which is inscribable in a circle.

## Problem 3

Let and be arbitrary non-negative integers. Prove that is an integer. (.)

## Problem 4

Find all solutions of the system of inequalities where are positive real numbers.

## Problem 5

Let and be real-valued functions defined for all real values of and , and satisfying the equation for all . Prove that if is not identically zero, and if for all , then for all .

## Problem 6

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.