I came across this question:
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 $$\sum_{i\in I}v_i=\sum_{j\in J}v_j.$$
Im thinking about calculating the L2 norm variance and expectation, and using Chebyshev inequality afterwards, but im not sure if this is correct. Does anyone know how to solve this?