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zero sum

Source: miklos schweitzer 1996 q3

October 12, 2021
linear algebra

Problem Statement

Let 1≤a1<a2<...<a2n≤4n−21\leq a_1 < a_2 <... < a_{2n} \leq 4n-2 be integers, such that their sum is even. Prove that for all sufficiently large n, there exist ε1,...,ε2n=±1\varepsilon_1 , ..., \varepsilon_{2n} = \pm1 such that ∑εi=∑εiai=0\sum\varepsilon_i = \sum\varepsilon_i a_i = 0
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