IMO 2014 Shortlist A2
Define the function f : p0,1q Ñ p0,1q by fpxq “ x 1 if x ă 1 , x2 if x ě 1 . Let a and b be two real numbers such that 0...
Category: Algebra
Problem
Define the function f : p0,1q Ñ p0,1q by fpxq “
x ` 1 if x ă 1 , x2 if x ě 1 . Let a and b be two real numbers such that 0 ă a ă b ă 1. We define the sequences an and bn by a0 “ a, b0 “ b, and an “ fpan´1q, bn “ fpbn´1q for n ą 0. Show that there exists a positive integer n such that pan ´ an´1qpbn ´ bn´1q ă 0. (Denmark)