# 1961 IMO Problems

## Contents

## Day I

### Problem 1

(*Hungary*)
Solve the system of equations:

where and are constants. Give the conditions that and must satisfy so that (the solutions of the system) are distinct positive numbers.

### Problem 2

Let *a*,*b*, and *c* be the lengths of a triangle whose area is *S*. Prove that

In what case does equality hold?

### Problem 3

Solve the equation

where *n* is a given positive integer.

## Day 2

### Problem 4

In the interior of triangle a point *P* is given. Let be the intersections of with the opposing edges of triangle . Prove that among the ratios there exists one not larger than 2 and one not smaller than 2.

### Problem 5

Construct a triangle *ABC* if the following elements are given: , and where *M* is the midpoint of *BC*. Prove that the construction has a solution if and only if

In what case does equality hold?

### Problem 6

Consider a plane and three non-collinear points on the same side of ; suppose the plane determined by these three points is not parallel to . In plane take three arbitrary points . Let be the midpoints of segments ; Let be the centroid of the triangle . (We will not consider positions of the points such that the points do not form a triangle.) What is the locus of point as range independently over the plane ?