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2010 smo (6)

Source: 2010 China South East Mathematical Olympiad

July 18, 2011
algebra unsolvedalgebra

Problem Statement

Let N∗\mathbb{N}^* be the set of positive integers. Define a1=2a_1=2, and for n=1,2,…,n=1, 2, \ldots,an+1=min⁡{λ∣1a1+1a2+⋯1an+1λ<1,λ∈N∗} a_{n+1}=\min\{\lambda|\frac{1}{a_1}+\frac{1}{a_2}+\cdots\frac{1}{a_n}+\frac{1}{\lambda}<1,\lambda\in \mathbb{N}^*\} Prove that an+1=an2−an+1a_{n+1}=a_n^2-a_n+1 for n=1,2,…n=1,2,\ldots.
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