# 2004 IMO Problems/Problem 2

## Problem

Find all polynomials with real coefficients such that for all reals such that we have the following relations

## Solution

### Solution 1

From , we have , so is even, and all the degrees all of its terms are even. Let

Let *; then we have . Comparing lead coefficients, we have , which cannot be true for . Hence, we have . We can easily verify by expanding that all such polynomials work.

- The substitution arises from writing .

### Solution 2

Let , , and . Then it is easy to check that , so

for all . Hence, for the coefficient of to be nonzero, we must have .

This does not hold for , and if is odd and , then the LHS is irrational and the RHS is a positive integer, so must be even.

Let . Then , so . This holds for and , and , so for . Therefore, must be of the form .

## See also

- <url>viewtopic.php?p=99448#99448 AoPS/MathLinks discussion</url>