We call a positive integer $n$ delightful if there exists an integer $k$, $1 < k < n$, such that \[1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n\] Does there exist a delightful number $N$ satisfying the inequalities \[2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?\]
Problem
Source: 2013 Baltic Way, Problem 16
Tags: inequalities, number theory unsolved, number theory