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88 Citations•2017•
Mathematics Magazine
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Abstract
2025. Proposed by Valerian Nita, Sterling Heights, MI. Let n be a positive integer and let x1, x2, . . . , xn and a1, a2, . . . , an be real numbers such that ∑n k=1 xk = 0 and 0 < a1 < a2 < · · · < an . Define s1, s2, . . . , sn by sk = ∑k j=1 a j x j for k = 1, 2, 3, . . . , n. If there is at least one nonzero number among x1, x2, . . . , xn , prove that there is at least one positive and at least one negative number among s1, s2, . . . , sn .