25th Eötvös 1918 Problem 2 Find three distinct positive integers a, b, c such that 1/a + 1/b + 1/c is an integer.
Unfortunately, it is well-known that 1/2 + 1/3 + 1/6 = 1. Suppose we did not know it! wlog a < b < c, so 1/a + 1/b + 1/c ≤ 1/1 + 1/2 + 1/3 < 2. Hence we must have 1/a + 1/b + 1/c = 1. Since 1/3 + 1/4 + 1/5 < 1, we must have a = 1 or 2. We cannot have a = 1 for then 1/b + 1/c = 0. So a = 2. Hence 1/b + 1/c = 1/2. We have 1/4 + 1/5 < 1/2, so b = 3. Then c = 6. 25th Eötvös 1918 © John Scholes jscholes@kalva.demon.co.uk 1 Nov 2003 Last corrected/updated 1 Nov 03