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There exist 3 with sum at least 5 [Austria Regional 2017, P1]

Source: Austrian Mathematics Olympiad Regional Competition (Qualifying Round) 2017, Problem 1

April 2, 2017
inequalities

Problem Statement

Let x1,x2,…,xnx_1, x_2, \dots, x_n be non-negative real numbers such that x12+x22+…x92≥25.x_1^2+x_2^2 + \dots x_9^2 \ge 25. Prove that one can choose three of these numbers such that their sum is at least 55.
Proposed by Karl Czakler
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