Nemotron-IMO-Bench 2026 C45

Let $v_1,v_2,...,v_n$ be $n$ vectors in the plane, whose coordinates are integers of absolute value less that $\frac{1}{100}\sqrt{\frac{2^n}{n}}$. Prove that there are disjoint subsets $I,J$ of $\{1,2,...,n\}$ such that $\sum_{i\in I} v_i=\sum_{j\in J} v_j$.