IMO 1967 LL SWE50

Origin: SWE

Problem

The function ϕ(x, y, z), defined for all triples (x, y, z) of real numbers, is such that there are two functions f and g defined for all pairs of real numbers such that ϕ(x, y, z) = f(x + y, z) = g(x, y + z) for all real x, y, and z. Show that there is a function h of one real variable such that ϕ(x, y, z) = h(x + y + z) for all real x, y, and z.

Solution

Since ϕ(x, y, z) = f(x+y, z) = ϕ(0, x+y, z) = g(0, x+y +z), it is enough to put h(t) = g(0, t).