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MMO 281 Moscow MO 1954 sum x_i^2 >10000, sum x_1<300

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August 13, 2019
Sumsystem of inequalitiesinequalitiesalgebra

Problem Statement

*. Positive numbers x1,x2,...,x100x_1, x_2, ..., x_{100} satisfy the system {x12+x22+...+x1002>10000x1+x2+...+x100<300\begin{cases} x^2_1+ x^2_2+ ... + x^2_{100} > 10 000 \\ x_1 + x_2 + ...+ x_{100} < 300 \end{cases} Prove that among these numbers there are three whose sum is greater than 100100.
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