# 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 the triangle ABC form a right-angled triangle. If the right-angle is at X, where AX is the bisector of angle A, find all possible values for angle A.

## 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 i<j<k1\le i\le nf(i)j<ia_j = a_ij>ia_j\neq a_i\displaystyle T=\sum_{i=1}^n f(i)\cdot\frac{f(i)-1}2$. If n is odd, what is the smallest value of T?