MathDB
Sum a_ib_i =0

Source: 2021 Saudi Arabia JBMO TST 1.4

September 4, 2021
combinatorics

Problem Statement

Let FF is the set of all sequences {(a1,a2,...,a2020)}\{(a_1, a_2, . . . , a_{2020})\} with ai∈{−1,1}a_i \in \{-1, 1\} for all i=1,2,...,2020i = 1,2,...,2020. Prove that there exists a set SS, such that S⊂FS \subset F, ∣S∣=2020|S| = 2020 and for any (a1,a2,...,a2020)∈F(a_1,a_2,...,a_{2020}) \in F there exists (b1,b2,...,b2020)∈S(b_1,b_2,...,b_{2020}) \in S, such that ∑i=12020aibi=0\sum_{i=1}^{2020} a_ib_i = 0.
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