Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that \[\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 \]
Source: Bulgaria MO 2011
Tags: number theory proposed, number theory
Prove whether or not there exist natural numbers $n,k$ where $1\le k\le n-2$ such that \[\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 \]