Writeup: IMO 2004 Shortlist A4

21 Jun, 2025

Problem Statement:

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca=0$ we have the following relation:

My Solution

First, note that the the set $S$ of all polynomials satisfying the problem statement is closed under addition.

Now, note that $S$ must be even. This comes from the fact that if we set $b=c$, then $a=-\frac{1}{2}b$ so we must have

Now, note that if we set $c=\frac{1}{2}b$, then we must have $a=-\mathrm{\backslash fracb}3$ and we have

Let the degree of $f$ be $d$. Then note that we must have

A quick proof of this fact involves using limits. We know that if

and

are the same polynomial $P(b)$, then ${lim}_{x\to \infty }P(b)/x^d$ will be the same for both polynomials (and this limit exists and is definite).

But if we don't have

then the limit will not hold true so we have a contradiction.

Finally, note that $d=2$ and $d=4$ both satisfy this equation. Note, also, that $\frac{4}{3}\ge \frac{7}{6}$. So if we have

then we will have

Note that

so the only possible polynomials we can consider are linear combinations of $x^2$ and $x^4$, because we can't have odd polynomials and we can't have even polynomials above degree $4$.

Note that

and

so the polynomials $f(x)=x^2$ and $f(x)=x^4$ do work.

Therefore the only polynomials satisfying the problem statement are of the form