# 1960 IMO Problems

Problems of the 2nd IMO 1960 Romania.

## Contents

## Day I

### Problem 1

Determine all three-digit numbers having the property that is divisible by 11, and is equal to the sum of the squares of the digits of .

### Problem 2

For what values of the variable does the following inequality hold:

### Problem 3

In a given right triangle , the hypotenuse , of length , is divided into equal parts ( and odd integer). Let be the acute angle subtending, from , that segment which contains the midpoint of the hypotenuse. Let be the length of the altitude to the hypotenuse of the triangle. Prove that:

## Day II

### Problem 4

Construct triangle , given , (the altitudes from and ), and , the median from vertex .

### Problem 5

Consider the cube (with face directly above face ).

a) Find the locus of the midpoints of the segments , where is any point of and is any point of ;

b) Find the locus of points which lie on the segment of part a) with .

### Problem 6

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let be the volume of the cone and be the volume of the cylinder.

a) Prove that ;

b) Find the smallest number for which ; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

### Problem 7

An isosceles trapezoid with bases and and altitude is given.

a) On the axis of symmetry of this trapezoid, find all points such that both legs of the trapezoid subtend right angles at ;

b) Calculate the distance of from either base;

c) Determine under what conditions such points actually exist. Discuss various cases that might arise.