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4 divides n if x_1x_2 +x_2x_3+... +x_{n-1}x_n +x_nx_1=0 with x_i \in {-1,1}

Source: 2021 Czech-Polish-Slovak Match Junior, team p2 CPSJ

September 12, 2021
number theorydivisibledivides

Problem Statement

Let the numbers xi∈{−1,1}x_i \in \{-1, 1\} be given for i=1,2,...,ni = 1, 2,..., n, satisfying x1x2+x2x3+...+xn−1xn+xnx1=0.x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0. Prove that nn is divisible by 44.
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