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TOT 296 1991 Spring A S3 sum x_i=0, sum x_i^2=1

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June 9, 2024
algebraSumcombinatorics

Problem Statement

The numbers x1,x2,x3,...,xnx_1,x_2,x_3, ..., x_n satisfy the two conditions ∑i=1nxi=0  ,    ∑i=1nxi2=1\sum^n_{i=1}x_i=0 \,\, , \,\,\,\,\sum^n_{i=1}x_i^2=1 Prove that there are two numbers among them whose product is no greater than −1/n- 1/n.
(Stolov, Kharkov)
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