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Problem 1318

AIME late

Algebra Difficulty 5.1 Prove it Singapore Mathematical Olympiad (SMO) · Singapore

Find all functions $f: \mathbb{R} \to \mathbb{R}$ so that
$(x+y)(f(x)-f(y)) = (x-y)f(x+y)$
for all $x, y \in \mathbb{R}$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Suppose that $f$ is a solution. Let
$a = \frac{1}{2}(f(1) - f(-1)), \quad b = \frac{1}{2}(f(1) + f(-1))$
and $g(x) = f(x) - a x - b x^2$ . Then
$(x+y)(g(x)-g(y)) = (x-y)g(x+y)$
and $g(1) = g(-1) = 0$ . Letting $y = 1$ and $y = -1$ above give
$\begin{aligned} (x+1)g(x) &= (x-1)g(x+1) \\ xg(x+1) &= (x+2)g(x). \end{aligned}$
Thus
$x(x+1)g(x) = x(x-1)g(x+1) = (x-1)(x+2)g(x)$
for all $x$ . So $g(x) = 0$ for all $x$ . Hence $f(x) = a x + b x^2$ . We can check directly that any function of this form (for some $a, b \in \mathbb{R}$ ) satisfies the given equation.

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