# 1976 USAMO Problems/Problem 5

## Problem

If , , , and are all polynomials such that prove that is a factor of .

## Solutions

### Solution 1

In general we will show that if is an integer less than and and are polynomials satisfying then , for all integers . For the problem, we may set , , and then note that since , is a factor of .

Indeed, let be the th roots of unity other than 1. Then for all integers , for all integers . This means that the th degree polynomial has distinct roots. Therefore all its coefficients must be zero, so for all integers , as desired.

### Solution 2

Let be three distinct primitive fifth roots of unity. Setting , we have These equations imply that or But by symmetry, Since , it follows that . Then, as noted above, so is a factor of , as desired.

*Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.*