# 1969 IMO Problems

Problems of the 11th IMO 1969 in Romania.

## Problem 1

Prove that there are infinitely many natural numbers with the following property: the number is not prime for any natural number .

## Problem 2

Let be real constants, a real variable, and Given that , prove that for some integer .

## Problem 3

For each value of , find necessary and sufficient conditions on the number so that there exists a tetrahedron with k edges of length , and the remaining edges of length 1.

## Problem 4

A semicircular arc is drawn on as diameter. is a point on other than and , and is the foot of the perpendicular from to . We consider three circles, , all tangent to the line . Of these, is inscribed in , while and are both tangent to and to , one on each side of . Prove that , and have a second tangent in common.

## Problem 5

Given points in the plane such that no three are collinear. Prove that there are at least convex quadrilaterals whose vertices are four of the given points.

## Problem 6

Prove that for all real numbers , with , the inequality is satisfied. Give necessary and sufficient conditions for equality.