Hard Inequality

Algebra Level 4

Find the smallest positive integer kk for which there exist positive reals x1,x2,x3,,xkx_1, x_2, x_3, \ldots, x_k satisfying both x12+x22+x32++xk2<x1+x2+x3++xk2  andx1+x2+x3++xk<x13+x23+x33++xk32.\begin{aligned} x_1^2+x_2^2+ x_3^2 + \cdots+x_k^2 &< \dfrac{x_1+x_2+x_3+\cdots+x_k}{2} \ \ \text{and}\\\\ x_1+x_2+ x_3 + \cdots+x_k &< \dfrac{x_1^3+x_2^3+x_3^3+\cdots+x_k^3}{2}. \end{aligned}

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