Question 1
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 $2r^2=ab$ on the board.
Assume that you start out with just one positive real number $r$ on the blackboard, and apply this operation $k^2-1$ 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 $kr$.
Assume that you start out with just one positive real number $r$ on the blackboard, and apply this operation $k^2-1$ 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 $kr$.