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sum (-1)^{k+1} x_{k}^n >= (sum (-1)^{i} x_{k})^n for k=1 to 2003

Source: Austrian Polish 2003 APMC

April 25, 2020
inequalitiesSumalgebra

Problem Statement

Given reals x1≥x2≥...≥x2003≥0x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0, show that x1n−x2n+x2n−...−x2002n+x2003n≥(x1−x2+x3−x4+...−x2002+x2003)nx_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n for any positive integer nn.
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