Russian 2001 Problem 1 Are there more positive integers under a million for which the nearest square is odd or for which it is even?
odd Solution There are 2n integers for which the closest integer is n 2 , namely n 2 -n+1, n 2 -n+2, ... , n 2 +n. So there are 2(1 + 3 + 5 + ... + 999) = 500·1000 integers under a million for which the nearest square is odd. Hence there are 499999 integers for which the nearest square is even. Russian 2001 © John Scholes jscholes@kalva.demon.co.uk 31 December 2003 Last corrected/updated 31 Dec 03