# 1973 IMO Problems

Problems of the 15th IMO 1973 in USSR.

## Problem 1

Point lies on line ; are unit vectors such that points all lie in a plane containing and on one side of . Prove that if is odd, Here denotes the length of vector .

## Problem 2

Determine whether or not there exists a finite set of points in space not lying in the same plane such that, for any two points and of , one can select two other points and of so that lines and are parallel and not coincident.

## Problem 3

Let and be real numbers for which the equation has at least one real solution. For all such pairs , find the minimum value of .

## Problem 4

A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission?

## Problem 5

is a set of non-constant functions of the real variable of the form and has the following properties:

(a) If and are in , then is in ; here .

(b) If is in , then its inverse is in ; here the inverse of is .

(c) For every in , there exists a real number such that .

Prove that there exists a real number such that for all in .

## Problem 6

Let be positive numbers, and let be a given real number such that . Find numbers for which

(a) for ,

(b) for ,

(c) .