Asian Pacific Olympiad

Well This problem is from APMO 2009. Just Try This and post your Solutions.

Problem.  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 2r2 = ab on the board.
Assume that you start out with just one positive real number r on the blackboard, and
apply this operation k2 ¡ 1 times to end up with k2 positive real numbers, not necessarily
distinct. Show that there exists a number on the board which does not exceed kr.


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