MEMO 2016 pojedinačno problem 1


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29. kolovoza 2018.
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Let n \geqslant 2 be an integer and x_1, x_2, \ldots, x_n be real numbers satisfying

(a) x_j > -1 for j = 1, 2, \ldots, n and

(b) x_1 + x_2 + \ldots + x_n = n

Prove the inequality \sum_{j = 1}^{n} \frac{1}{1+x_j} \geqslant \sum_{j = 1}^{n} \frac{x_j}{1+x_j^2} and determine when equality holds.

Izvor: Srednjoeuropska matematička olimpijada 2016, pojedinačno natjecanje, problem 1