16th VMO 1978 Problem A1 Find all three digit numbers abc such that 2(abc) = bca + cab.
111, 222, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 777, 888, 999. Solution We have 200a + 20b + 2c = 100b + 10c + a + 100c + 10a + b, so 7a = 3b + 4c (*). There are the obvious solutions a = b = c. If any two of a, b, c are equal, then (*) implies that they are all equal. So we can assume they are all distinct. Note that we must have a = b mod 4. It is now a question of looking in turn at a = 1, 2, ... , 9. For example a = 1, so b = 5 or 9. But in both cases 3b > 7a. Thanks to Suat Namli 16th VMO 1978 © John Scholes jscholes@kalva.demon.co.uk 6 March 2004 Last corrected/updated 6 Mar 04