MathDB

8

Part of 2021 Caucasus Mathematical Olympiad

Problems(1)

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

Source: VI Caucasus Mathematical Olympiad

3/14/2021
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 nn amazing, if one can partition the set {1,2,…,n}\{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.
number theory