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Obvious Cauchy Schwarz

Source: Czech and Slovak Olympiad 1978, National Round, Problem 1

October 11, 2024
algebrainequalities

Problem Statement

Let a1,…,an,b1,…,bna_1,\ldots,a_n,b_1,\ldots,b_n be positive numbers. Show that (a1+⋯+an)(b1+⋯+bn)≥a1b1+⋯+anbn\sqrt{\left(a_1+\cdots+a_n\right)\left(b_1+\cdots+b_n\right)}\ge\sqrt{a_1b_1}+\cdots+\sqrt{a_nb_n} and prove that equality holds if and only if a1b1=⋯=anbn.\frac{a_1}{b_1}=\cdots=\frac{a_n}{b_n}.
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