# 1981 IMO Problems

Problems of the 22nd IMO 1981 U.S.A.

## Contents

## Day I

### Problem 1

is a point inside a given triangle . are the feet of the perpendiculars from to the lines , respectively. Find all for which

is least.

### Problem 2

Let and consider all subsets of elements of the set . Each of these subsets has a smallest member. Let denote the arithmetic mean of these smallest numbers; prove that

### Problem 3

Determine the maximum value of , where and are integers satisfying and .

## Day II

### Problem 4

(a) For which values of is there a set of consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining numbers?

(b) For which values of is there exactly one set having the stated property?

### Problem 5

Three congruent circles have a common point and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point are collinear.

### Problem 6

The function satisfies

(1)

(2)

(3)

for all non-negative integers . Determine .