2011 OIM Problems/Problem 5

Problem

Let $x_1, \cdots , x_n$ be positive real numbers. Prove that there exist $a_1, \cdots , a_n \in \left\{ -1, 1 \right\}$ such that

\[a_1x_1^2+\cdots+a_nx_n^2\ge (a_1x_1+\cdots+a_nx_n)^2\]

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions