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China Mathematical Olympiad 1988 problem1

Source: China Mathematical Olympiad 1988 problem1

November 4, 2013
inequalitiesfunctioninequalities unsolved

Problem Statement

Let r1,r2,…,rnr_1,r_2,\dots ,r_n be real numbers. Given nn reals a1,a2,…,ana_1,a_2,\dots ,a_n that are not all equal to 00, suppose that inequality r1(x1−a1)+r2(x2−a2)+⋯+rn(xn−an)≤x12+x22+⋯+xn2−a12+a22+⋯+an2r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2} holds for arbitrary reals x1,x2,…,xnx_1,x_2,\dots ,x_n. Find the values of r1,r2,…,rnr_1,r_2,\dots ,r_n.
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