I'm reading through a combinatorics book and I got stuck on this problem.
Let $v_1=(x_1,y_1),..., v_n=(x_n,y_n)$ be $n$ two dimensional vectors such that each $x_i$ and $y_i$ is an integer whose absolute value does not exceed $\frac{2^{n/2}}{100\sqrt{n}}$. Prove there are two disjoint sets $I,J$ in $\{1,2,...,n\}$ such that $$\sum_{i\in I}v_i=\sum_{j\in J}v_j.$$
I am not sure how to prove this. I was thinking that it has something to do with distinct sums: A set $w_i,,,w_k$ of positive integers has distinct sums if all sums $$\sum_{i\in S}w_i,$$ $S\subset \{1,...,k\}$ are distinct, but I am not sure if this is even the right direction since the thing we want to prove would mean we don't have distinct sums. Any help would be greatly appreciated.