Construct a function from the set of positive rational numbers into itself such that f(x f(y)) = f(x)/y for all x, y.
Solution
We show first that f(1) = 1. Taking x = y = 1, we have f(f(1)) = f(1). Hence f(1) = f(f(1)) = f(1 f(f(1)) ) = f(1)/f(1) = 1.
Next we show that f(xy) = f(x)f(y). For any y we have 1 = f(1) = f(1/f(y) f(y)) = f(1/f(y))/y, so if z = 1/f(y) then f(z) = y. Hence f(xy) = f(xf(z)) = f(x)/z = f(x) f(y).
Finally, f(f(x)) = f(1 f(x)) = f(1)/x = 1/x.
We are not required to find all functions, just one. So divide the primes into two infinite sets S = {p1, p2, ... } and T= {q1, q2, ... }. Define f(pn) = qn, and f(qn) = 1/pn. We extend this definition to all rationals using f(xy) = f(x) f(y): f(pi1pi2...qj1qj2.../(pk1...qm1...)) = pm1...qi1.../(pj1...qk1...). It is now trivial to verify that f(x f(y)) = f(x)/y.
Solutions are also available in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
14 Nov 1998