MathDB
mT iNeQuAlItY ErUpTeD OmG

Source: USOMO #6

June 21, 2020
inequalitiesHi

Problem Statement

Let n≥2n \ge 2 be an integer. Let x1≥x2≥...≥xnx_1 \ge x_2 \ge ... \ge x_n and y1≥y2≥...≥yny_1 \ge y_2 \ge ... \ge y_n be 2n2n real numbers such that 0=x1+x2+...+xn=y1+y2+...+yn0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n and1=x12+x22+...+xn2=y12+y22+...+yn2.\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2. Prove that ∑i=1n(xiyi−xiyn+1−i)≥2n−1.\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}. Proposed by David Speyer and Kiran Kedlaya
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