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IMO ShortList 2001, algebra problem 5

Source: IMO ShortList 2001, algebra problem 5

September 30, 2004

inequalitiesalgebrarecurrence relationequationInteger sequenceIMO Shortlistimo shortlist 2001

Problem Statement

Find all positive integers

a_1, a_2, \ldots, a_n

such that

\frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n},

where

a_0 = 1

and

(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)

for

k = 1,2,\ldots,n-1