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inequality with sums of inverses of positive reals

Source: Nordic Mathematical Contest 1999 #4

October 3, 2017
inequalitiespositive

Problem Statement

Let a1,a2,...,ana_1, a_2, . . . , a_n be positive real numbers and n≥1n \ge 1. Show that n(1a1+...+1an)≥(11+a1+...+11+an)(n+1a1+...+1an)n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n}) When does equality hold?
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