Problem

Source: Finland 2012, Problem 4

Tags: inequalities, induction, combinatorics unsolved, combinatorics



Let $k,n\in\mathbb{N},0<k<n.$ Prove that \[\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.\]