IMO 2001 SL A4

Origin: LIT | Category: Algebra

Problem

Find all functions f : R \toR satisfying f(xy)(f(x) −f(y)) = (x −y)f(x)f(y) for all x, y.

Solution

Let (∗) denote the given functional equation. Substituting y = 1 we get f(x)2 = xf(x)f(1). If f(1) = 0, then f(x) = 0 for all x, which is the trivial solution. Suppose f(1) = C ̸= 0. Let G = {y \inR | f(y) ̸= 0}. Then f(x) = . Cx if x \inG, otherwise. (1) We must determine the structure of G so that the function defined by (1) satisfies (∗). (1) Clearly 1 \inG, because f(1) ̸= 0. (2) If x \inG, y ̸\inG, then by (∗) it holds f(xy)f(x) = 0, so xy ̸\inG. (3) If x, y \inG, then x/y \inG (otherwise by 2◦, y(x/y) = x ̸\inG).

(4) If x, y \inG, then by 2◦we have x−1 \inG, so xy = y/x−1 \inG. Hence G is a set that contains 1, does not contain 0, and is closed under multiplication and division. Conversely, it is easy to verify that every such G in (1) gives a function satisfying (∗).