Russian 01/1 solution

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?

Answer

odd

Solution

There are 2n integers for which the closest integer is n², namely n²-n+1, n²-n+2, ... , n²+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