# 1975 IMO Problems

Problems of the 17th IMO 1975 in Bulgaria.

## Problem 1

Let be real numbers such that Prove that, if is any permutation of , then

## Problem 2

Let be an infinite increasing sequence of positive integers. Prove that for every there are infinitely many which can be written in the form with positive integers and .

## Problem 3

On the sides of an arbitrary triangle , triangles are constructed externally with . Prove that and .

## Problem 4

When is written in decimal notation, the sum of its digits is . Let be the sum of the digits of . Find the sum of the digits of . ( and are written in decimal notation.)

## Problem 5

Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.

## Problem 6

Find all polynomials , in two variables, with the following properties:

(i) for a positive integer and all real (that is, is homogeneous of degree ),

(ii) for all real

(iii)