IMO 1986 SL 1

Origin: GBR

Problem

Find, with proof, all functions f defined on the nonnegative real numbers and taking nonnegative real values such that (i) f[xf(y)]f(y) = f(x + y), (ii) f(2) = 0 but f(x) ̸= 0 for 0 \leqx < 2.

Solution

If w > 2, then setting in (i) x = w −2, y = 2, we get f(w) = f((w − 2)f(w))f(2) = 0. Thus f(x) = 0 if and only if x \geq2. Now let 0 \leqy < 2 and x \geq0. The LHS in (i) is zero if and only if xf(y) \geq2, while the RHS is zero if and only if x + y \geq2. It follows that x \geq2/f(y) if and only if x \geq2 −y. Therefore f(y) = . 2−y for 0 \leqy < 2; for y \geq2. The confirmation that f satisfies the given conditions is straightforward.