Ibero 93/A3 solution

Problem A3

Find all functions f on the positive integers with positive integer values such that (1) if x < y, then f(x) < f(y), and (2) f(y f(x)) = x²f(xy).

Solution

Answer: f(x) = x².

Note that (1) implies f is (1, 1).

Put y = 1. Then f( f(x) ) = x² f(x).

Put y = f(z), then f( f(z) f(x) ) = x² f(x f(z) ) = x²z² f(xz) = f( f(xz) ). But f is (1, 1) so f(xz) = f(x) f(z).

Now suppose f(m) > m² for some m. Then by (1), f( f(m) ) > f(m² = f(m.m) = f(m)². But f( f(m) ) = m² f(m), so m² > f(m). Contradiction.

Similarly, suppose f(m) < m². Then m² f(m) = f( f(m) ) < f(m²) = f(m)², so m² < f(m). Contradiction. So we must have f(m) = m².