22nd USAMO 1993 Problem 3 Let S be the set of functions f defined on reals in the closed interval [0, 1] with non-negative real values such that f(1) = 1 and f(x) + f(y) ≤ f(x + y) for all x, y such that x + y ≤ 1. What is the smallest k such that f(x) ≤ kx for all f in S and all x? Solution
: k = 2. Consider the function f(x) = 0 for 0 ≤ x ≤ 1/2, 1 for 1/2  1/2, then 2x > 1, so f(x) < 2x. If x ≤ 1/2, choose n ≥ 1, so that 1/2 n+1 < x ≤ 1/2 n . Then 2 n f(x) ≤ f(2 n x) ≤ 1, so f(x) ≤ 1/2 n ≤ 2x. 22nd USAMO 1993 © John Scholes jscholes@kalva.demon.co.uk 23 Aug 2002 Last corrected/updated 25 Nov 03