Question 3
Let $F$ be the set of all sequences $(a_1, a_2, \dots, a_{2020})$ with $a_i \in \{-1, 1\}$ for all $i = 1, 2, \dots, 2020$. Prove that there exists a set $S$, such that $S \subset F$, $|S| = 2020$ and for any $(a_1, a_2, \dots, a_{2020}) \in F$ there exists $(b_1, b_2, \dots, b_{2020}) \in S$, such that $\sum_{i=1}^{2020} a_i b_i = 0$.