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Hungary-Israel Binational 1996\4

Source: Prove that existence of a k

October 29, 2008
algebra unsolvedalgebra

Problem Statement

a1,a2,⋯ ,an a_1, a_2, \cdots, a_n is a sequence of real numbers, and b1,b2,⋯ ,bn b_1, b_2, \cdots, b_n are real numbers that satisfy the condition 1≥b1≥b2≥⋯≥bn≥0 1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0. Prove that there exists a natural number k≤n k \le n that satisifes |a_1b_1 \plus{} a_2b_2 \plus{} \cdots \plus{} a_nb_n| \le |a_1 \plus{} a_2 \plus{} \cdots \plus{} a_k|
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