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Inequality on sequence of integers

Source: IMO Shortlist 2017 A7

July 10, 2018

IMO Shortlist

Problem Statement

Let

a_0,a_1,a_2,\ldots

be a sequence of integers and

b_0,b_1,b_2,\ldots

be a sequence of positive integers such that

a_0=0,a_1=1

, and

a_{n+1} = \begin{cases} a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\ a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$} \end{cases}\qquad\text{for }n=1,2,\ldots.

for

n=1,2,\ldots.

Prove that at least one of the two numbers

a_{2017}

and

a_{2018}

must be greater than or equal to

2017

.

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