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Smallest number

Author: mathforces
Problem has been solved: 240 times

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Find the smallest positive integer $k$ for which there exist positive real numbers $x_1, \dots, x_k$ satisfying $$x_1^2 + \dots + x_k^2 < \frac{x_1 + \dots + x_k}{2} \quad \text{and} \quad x_1 + \dots + x_k < \frac{x_1^3 + \dots + x_k^3}{2}.$$





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