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sum 1/x^k >= sum x^k - All-Russian MO 1999 Regional (R4) 9.3

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September 25, 2024
algebrainequalities

Problem Statement

The product of positive numbers x,yx, y and zz is equal to 11. Prove that if it holds that 1x+1y+1z≥x+y+z,\frac1x +\frac1y + \frac1z \ge x + y + z, then for any natural kk, holds the inequality 1xk+1yk+1zk≥xk+yk+zk.\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.
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