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sum x_i^2 +n^3<=(2n-1) sum x_i +n^2 - 1998 Romania NMO VII p1

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August 14, 2024
inequalitiesalgebraPerfect Squarenumber theory

Problem Statement

Let nn be a positive integer and x1,x2,...,xnx_1,x_2,...,x_n be integer numbers such that x12+x22+...+xn2+n3≤(2n−1)(x1+x2+...+xn)+n2x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2 . Show that : a) x1,x2,...,xnx_1,x_2,...,x_n are non-negative integers b) the number x1+x2+...+xn+n+1x_1+x_2+...+x_n+n+1 is not a perfect square.
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