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Number of elements is equal to the average of all its elements

Source: VI Caucasus Mathematical Olympiad

March 14, 2021

number theory

Problem Statement

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.