8th USAMO 1979 Problem 1 Find all sets of 14 or less fourth powers which sum to 1599. Solution
: none. The only 4th powers less than 1599 are 1, 16, 81, 256, 625, 1296 (7 4 = 2401). Note that 1, 81, 625 = 1 mod 16 and 16, 256, 1296 = 0 mod 16. But 1599 = 15 mod 16, so it cannot be expressed as a sum of 14 or less fourth powers. 8th USAMO 1979 © John Scholes jscholes@kalva.demon.co.uk 11 May 2002