KURSCHAK 1999 Problem 3

We are given more than $2^k$ integers, where $k\in\mathbb{N}$ . Prove that we can choose $k+2$ of them such that if some of our selected numbers satisfy $$x_1+x_2+\dots+x_m=y_1+y_2+\dots+y_m$$ where $x_1<\dots<x_m$ and $y_1<\dots<y_m$ , then $x_i=y_i$ for any $1\le i\le m$ .