2004 IMO Problems/Problem 2
Find all polynomials with real coefficients such that for all reals such that we have the following relations
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 .
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 .
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