# 1965 IMO Problems

## Problem 1

Determine all values in the interval which satisfy the inequality

## Problem 2

Consider the system of equations with unknowns , , . The coefficients satisfy the conditions:

(a) , , are positive numbers;

(b) the remaining coefficients are negative numbers;

(c) in each equation, the sum of the coefficients is positive.

Prove that the given system has only the solution .

## Problem 3

Given the tetrahedron whose edges and have lengths and respectively. The distance between the skew lines and is , and the angle between them is . Tetrahedron is divided into two solids by plane , parallel to lines and . The ratio of the distances of from and is equal to . Compute the ratio of the volumes of the two solids obtained.

## Problem 4

Find all sets of four real numbers , , , such that the sum of any one and the product of the other three is equal to .

## Problem 5

Consider with acute angle . Through a point perpendiculars are drawn to and , the feet of which are and respectively. The point of intersection of the altitudes of is . What is the locus of if is permitted to range over (a) the side , (b) the interior of ?

## Problem 6

In a plane a set of points () is given. Each pair of points is connected by a segment. Let be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length . Prove that the number of diameters of the given set is at most .