Splitting numbers on the board with 2r^2 = ab
Source: APMO 2009 Q.1
March 13, 2009
inductioninvariantcombinatorics
Problem Statement
Consider the following operation on positive real numbers written on a blackboard:
Choose a number written on the blackboard, erase that number, and then write a pair of positive real numbers and satisfying the condition 2 r^2 \equal{} ab on the board.
Assume that you start out with just one positive real number on the blackboard, and apply this operation k^2 \minus{} 1 times to end up with positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.