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China Northern Mathematical Olympiad 2014 , Problem 2

Source: China Sijiazhuang , Aug 2014

August 13, 2014
inequalities proposedinequalitiesalgebraChina

Problem Statement

Define a positive number sequence sequence {an}\{a_n\} by a1=1,(n2+1)an−12=(n−1)2an2.a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.Prove that1a12+1a22+⋯+1an2≤1+1−1an2.\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .
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