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Sigma_{i=1} ^{2006}a_ib_i = 0 where a_i, b_i in {-1,1}

Source: KJMO 2006 p8

May 1, 2019
SetscombinatoricsSumpermutations

Problem Statement

De ne the set FF as the following: F={(a1,a2,...,a2006):∀i=1,2,...,2006,ai∈{−1,1}}F = \{(a_1,a_2,... , a_{2006}) : \forall i = 1, 2,..., 2006, a_i \in \{-1,1\}\} Prove that there exists a subset of FF, called SS which satis es the following: ∣S∣=2006|S| = 2006 and for all (a1,a2,...,a2006)∈F(a_1,a_2,... , a_{2006})\in F there exists (b1,b2,...,b2006)∈S(b_1,b_2,... , b_{2006}) \in S, such that Σi=12006aibi=0\Sigma_{i=1} ^{2006}a_ib_i = 0.
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