Problem

Source: 2021 Saudi Arabia JBMO TST 1.4

Tags: combinatorics



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