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China Mathematical Olympiad 1986 problem1

Source: China Mathematical Olympiad 1986 problem1

January 16, 2014
inequalities

Problem Statement

We are given nn reals a1,a2,⋯ ,ana_1,a_2,\cdots , a_n such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if nn non-negative reals x1,x2,⋯ ,xnx_1,x_2,\cdots ,x_n satisfy x1+x2+⋯+xn=1x_1+x_2+\cdots +x_n=1, then the inequality a1x1+a2x2+⋯+anxn≥a1x12+a2x22+⋯+anxn2a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n holds.
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