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The largest M for which there exists an arrangement

Source: China TST 2011 - Quiz 3 - D1 - P1

May 20, 2011
inequalities unsolvedinequalities

Problem Statement

Let n≥3n\geq 3 be an integer. Find the largest real number MM such that for any positive real numbers x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n, there exists an arrangement y1,y2,⋯ ,yny_1,y_2,\cdots,y_n of real numbers satisfying ∑i=1nyi2yi+12−yi+1yi+2+yi+22≥M,\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M, where yn+1=y1,yn+2=y2y_{n+1}=y_1,y_{n+2}=y_2.
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