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Not quite Cauchy-Schwarz [ILL 1977]

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January 11, 2011
inequalitiesalgebra proposedalgebra

Problem Statement

Prove the following assertion: If c1,c2,…,cn (n≥2)c_1,c_2,\ldots ,c_n\ (n\ge 2) are real numbers such that (n−1)(c12+c22+⋯+cn2)=(c1+c2+⋯+cn)2, (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2, then either all these numbers are nonnegative or all these numbers are nonpositive.
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