IMO 1989 SL 10

Origin: GRE

Problem

Let g : C \toC, w \inC, a \inC, w3 = 1 (w ̸= 1). Show that there is one and only one function f : C \toC such that f(z) + f(wz + a) = g(z), z \inC. Find the function f.

Solution

Plugging in wz + a instead of z into the functional equation, we obtain f(wz + a) + f(w2z + wa + a) = g(wz + a). (1) By repeating this process, this time in (1), we get f(w2z + wa + a) + f(z) = g(w2z + wa + a). (2) Solving the system of linear equations (1), (2) and the original functional equation, we easily get f(z) = g(z) + g(w2z + wa + a) −g(wz + a) . This function thus uniquely satisfies the original functional equation.