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n-variable inequality with a weird condition

Source: Romanian TST 2022, Test 1, P1

May 14, 2022
inequalities

Problem Statement

Given are positive reals x1,x2,...,xnx_1, x_2,..., x_n such that ∑11+xi2=1\sum\frac {1}{1+x_i^2}=1. Find the minimal value of the expression ∑xi∑1xi\frac{\sum x_i}{\sum \frac{1}{x_i}} and find when it is achieved.
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