MathDB

Consider the following operation on positive real numbers written on a blackboard: Choose a number

r

written on the blackboard, erase that number, and then write a pair of positive real numbers

a

and

b

satisfying the condition 2 r^2 \equal{} ab on the board. Assume that you start out with just one positive real number

r

on the blackboard, and apply this operation k^2 \minus{} 1 times to end up with

k^2

positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.

1