6th Chinese 1991 Problem A2 X is the closed interval [0, 1]. Find all functions f: X x X → X such that: (1) f(x, 1) = f(1, x) = x for all x; (2) f( f(x, y), z) = f(x, f(y, z) ) for all x, y, z; (3) for some fixed positive real k, f(xy, xz) = x k f(y, z) for all x, y, z. Solution
: f(x, y) = xy or f(x, y) = min(x, y). Suppose y ≤ x, so that y/x is in X. Then, from (3), f(x, y) = x k f(1, y/x) = (from (1)) y x k-1 . Similarly, f(y, x) = x k f(y/x, 1) = y x k-1 . Now take x ≤ y ≤ z, but y z k-1  y z k-1 , then f(x, f(y, z) ) = x k-1 y z k-1 , so x k-2 = y k-2 . If x and y are unequal, then that implies k = 2. But care is needed here, because if k = 1, then we cannot have x > y z k-1 , since z k-1 = 1 and we have already assumed x ≤ y. So all we can conclude is that k cannot have any values except 1 and 2. If k = 2, then f(x, y) = xy. It is easy to check that this does indeed satisfy (3): f( (f(x, y), z) = f(xy, z) = xyz = f(x, yz) = f(x, f(y, z) ). If k = 1, then f(x, y) = min(x, y). Again it is easy to check that this satisfies (3) - both sides are min(x, y, z). 6th Chinese 1991 © John Scholes jscholes@kalva.demon.co.uk 1 May 2002