HMMT 二月 2017 · 冲刺赛 · 第 9 题

英文解析

9. [ 7 ] Jeffrey writes the numbers 1 and 100000000 = 10 on the blackboard. Every minute, if x, y are onthe board, Jeffrey replaces them with
   ( ) − 1
   x + y 1 1
   and 2 + .
   2 x y
   After 2017 minutes the two numbers are a and b . Find min( a, b ) to the nearest integer.
   Proposed by: Yang Liu
   Answer: 10000
   Note that the product of the integers on the board is a constant. Indeed, we have that
   ( ) − 1
   x + y 1 1
   · 2 + = xy.
   2 x y
   √
   Therefore, we expect that the answer to the problem is approximately 1 · 10 = 10 .84
   To be more rigorous, we have to show that the process indeed converges quickly enough. To show this,
   we bound the difference between the integers on the board at time i . Say that at time i , the integerson the board are a < b . Note thati i
   ( ) − 1
   a + b 1 1 ( a − b )2
   i i i id = b − a = − 2 + =
   i +1 i +1 i +1
   2 a b 2( a + b )
   i i i ib − a di i i < = .
   2 2
   The inequality at the end follows from that obvious fact that b − a < b + a . Therefore, d ≤ , soidi i i i i +1
   1082
   d < , which is extremely small. So the difference is essentially 0 at time 2017, which completes
   2017
   2017
   the argument.2