# 1971 IMO Problems

Problems of the 13th IMO 1971 in Czechoslovakia.

## Problem 1

Prove that the following assertion is true for and , and that it is false for every other natural number :

If are arbitrary real numbers, then

## Problem 2

Consider a convex polyhedron with nine vertices ; let be the polyhedron obtained from by a translation that moves vertex to . Prove that at least two of the polyhedra have an interior point in common.

## Problem 3

Prove that the set of integers of the form contains an infinite subset in which every two members are relatively prime.

## Problem 4

All the faces of tetrahedron are acute-angled triangles. We consider all closed polygonal paths of the form defined as follows: is a point on edge distinct from and ; similarly, are interior points of edges , respectively. Prove:

(a) If , then among the polygonal paths, there is none of minimal length.

(b) If , then there are infinitely many shortest polygonal paths, their common length being , where .

## Problem 5

Prove that for every natural number , there exists a finite set of points in a plane with the following property: For every point in , there are exactly points in which are at unit distance from .

## Problem 6

Let be a square matrix whose elements are non-negative integers. Suppose that whenever an element , the sum of the elements in the th row and the th column is . Prove that the sum of all the elements of the matrix is .