Problem

Source: VI Caucasus Mathematical Olympiad

Tags: number theory



Let us call a set of positive integers nice, if its number of elements is equal to the average of all its elements. Call a number $n$ amazing, if one can partition the set $\{1,2,\ldots,n\}$ into nice subsets. a) Prove that any perfect square is amazing. b) Prove that there exist infinitely many positive integers which are not amazing.