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problem about equation

Source: China south east mathematical Olympiad 2006 problem4

July 4, 2013
algebra unsolvedalgebra

Problem Statement

Given any positive integer nn, let ana_n be the real root of equation x3+xn=1x^3+\dfrac{x}{n}=1. Prove that (1) an+1>ana_{n+1}>a_n; (2) ∑i=1n1(i+1)2ai<an\sum_{i=1}^{n}\frac{1}{(i+1)^2a_i} <a_n.
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