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(sum a_ip_i)(sum b_ip_ii)<=sum a_1b_1p_1 if pi>0 iwth sum p_i=0,

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2014 3.2

October 1, 2021
algebrainequalities

Problem Statement

Let p1,p2,p3p_1,p_2, p_3 be positive numbers such that p1+p2+p3=1p_1 + p_2 + p_3 = 1. If a1<a2<a3a_1 <a_2 <a_3 and b1<b2<b3b_1 <b_2 <b_3 prove that (a1p1+a2p2+a3p3)(b1p1+b2p2+b3p3)≤(a1b1p1+a2b2p2+a3b3p3)(a_1p_1 + a_2p_2 + a_3p_3) (b_1p_1 + b_2p_2 + b_3p_3)\le (a_1b_1p_1 + a_2b_2p_2 + a_3b_3p_3)
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