IMO 2017 Shortlist A7

Let a0,a1,a2,... be a sequen e of integers and b0,b1,b2,... be a sequen e of positive integers su h that a0 “ 0,a1 “ 1, ...

imomathematicsolympiadshortlistalgebra

IMO 2017 Shortlist A7

Category: Algebra

Problem

Let a0,a1,a2,... be a sequen e of integers and b0,b1,b2,... be a sequen e of positive integers su h that a0 “ 0,a1 “ 1, and an`1 “

anbn ` an´1, if bn´1 “ 1 anbn ´ an´1, if bn´1 ą 1 for n “ 1,2,.... Prove that at least one of the two numbers a2017 and a2018 must be greater than or equal to 2017. (Australia)