IMO 1968 Problem A2 Find all natural numbers n the product of whose decimal digits is n 2 - 10n - 22.
Suppose n has m > 1 digits. Let the first digit be d. Then the product of the digits is at most d.9 m-1  0 for n >= 13. So there are no solutions for n ≥ 13. But n 2 - 10n - 22 < 0 for n ≤ 11, so the only possible solution is n = 12 and indeed that is a solution. s are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48- X. 10th IMO 1968 © John Scholes jscholes@kalva.demon.co.uk 27 Sep 1998 Last corrected/updated 27 Sep 1998