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China Northern Mathematical Olympiad 2017, Problem 1

Source: China Northern Mathematical Olympiad 2017

July 29, 2017

inequalitiesSequence

Problem Statement

A sequence

\{a_n\}

is defined as follows:

a_1 = 1

a_2 = \frac{1}{3}

, and for all

n \geq 1,

\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}

.
Prove that, for all

n \geq 1

a_1 + a_2 + ... + a_n < \frac{34}{21}