# 1987 USAMO Problems

## Problem 1

Find all solutions to , where m and n are non-zero integers.

## Problem 2

The feet of the angle bisectors of form a right-angled triangle. If the right-angle is at , where is the bisector of , find all possible values for .

## Problem 3

X is the smallest set of polynomials such that:

1. belongs to X 2. If belongs to X, then and both belong to X.

Show that if and are distinct elements of X, then for any .

## Problem 4

M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that and . For what value of is a minimum?

## Problem 5

is a sequence of 0's and 1's. T is the number of triples with which are not equal to (0, 1, 0) or (1, 0, 1). For , is the number of with plus the number of with . Show that . If n is odd, what is the smallest value of T?