# 1970 IMO Problems

Problems of the 12th IMO 1970 Hungary.

## Contents

## Day 1

### Problem 1

Let be a point on the side of . Let , and be the inscribed circles of triangles , and . Let , and be the radii of the exscribed circles of the same triangles that lie in the angle . Prove that

.

### Problem 2

Let , and be integers greater than 1, and let and be the bases of two number systems. and are numbers in the system with base and and are numbers in the system with base ; these are related as follows:

,

,

.

Prove:

if and only if .

### Problem 3

The real numbers satisfy the condition:

.

The numbers are defined by

(a) Prove that for all .

(b) given with , prove that there exist numbers with the above properties such that for large enough .

## Day 2

### Problem 4

Find the set of all positive integers with the property that the set can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

### Problem 5

In the tetrahedron , angle is a right angle. Suppose that the foot of the perpendicular from to the plane in the tetrahedron is the intersection of the altitudes of . Prove that

.

For what tetrahedra does equality hold?

### Problem 6

In a plane there are points, no three of which are collinear. Consider all possible triangles having these point as vertices. Prove that no more than of these triangles are acute-angled.