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sum x_iy_i >= 1/n (sum x_i\)( sum y_1)

Source: Mathcenter Contest / Oly - Thai Forum 2011 (R1) p3 sl-3 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 14, 2022
inequalitiesalgebraSum

Problem Statement

We will call the sequence of positive real numbers. a1,a2,…,ana_1,a_2,\dots ,a_n of length nn when a1≥a1+a22≥⋯≥a1+a2+⋯+ann.a_1\geq \frac{a_1+a_2}{2}\geq \dots \geq \frac{a_1+a_2+\cdots +a_n}{n}. Let x1,x2,…,xnx_1,x_2,\dots ,x_n and y1,y2,…,yny_1,y_2,\dots ,y_n be sequences of length n.n. Prove that ∑i=1nxiyi≥1n(∑i=1nxi)(∑i=1nyi).\sum_{i = 1}^{n}x_iy_i\geq\frac{1}{n}\left(\sum_{i = 1}^{n}x_i\right)\left(\sum_{i = 1}^{n}y_i\right).
(tatari/nightmare)
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