IMO 2013 Shortlist A1

Let n be a positive integer and let a1,...,an´1 be arbitrary real numbers. Define the sequences u0,...,un and v0,...,vn ...

imomathematicsolympiadshortlistalgebra

IMO 2013 Shortlist A1

Category: Algebra

Problem

Let n be a positive integer and let a1,...,an´1 be arbitrary real numbers. Define the sequences u0,...,un and v0,...,vn inductively by u0 “ u1 “ v0 “ v1 “ 1, and uk1 “ uk akuk´1, vk1 “ vk an´kvk´1 for k “ 1,...,n ´ 1. Prove that un “ vn. (France)