2009 APMO 第 1 题

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 $2 r^{2}=a b$ 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 $k r$.

考虑对写在黑板上的正实数进行以下运算：

选择黑板上写的一个数字$r$，擦掉该数字，然后在黑板上写下一对满足条件$2 r^{2}=a b$ 的正实数$a$ 和$b$。

假设您一开始在黑板上只有一个正实数 $r$，然后应用此运算 $k^{2}-1$ 次，最终得到 $k^{2}$ 正实数，但不一定不同。证明棋盘上存在一个不超过$k r$的数字。