PUMaC 2016 · 团队赛 · 第 12 题
PUMaC 2016 — Team Round — Problem 12
题目详情
- (10) King Tin writes the first n perfect squares on the royal chalkboard, but he omits the first (so for n = 3, he writes 4 and 9). His son, Prince Tin, comes along and repeats the following process until only one number remains: He erases the two greatest numbers still on the board, calls them a and b , and writes the ab − 1 value of on the board. a + b − 2 Let S ( n ) be the last number that Prince Tin writes on the board. Let lim S ( n ) = r, n →∞ meaning that r is the unique number such that for every > 0 there exists a positive integer m N so that | S ( n ) − r | < for all n > N . If r can be written in simplest form as , find m + n . n
解析
- Define the operation o by a o b = . Then, noticing the similarity between and a + b − 2 a + b − 2 ab 1 = , we find that 1 1 a + b
a b 1 a o b = 1 + , 1 1 + a − 1 b − 1 which is both commutative and associative. Therefore, ( ) − 1 n ∑ 1 2 S ( n ) = 4 o 9 o · · · o n = 1 + , 2 k − 1 k =2 so ( ) ( ) − 1 − 1 ∞ ∞ ∑ ∑ 1 1 / 2 1 / 2 7 lim S ( n ) = 1 + = 1 + − = , 2 n →∞ k − 1 k − 1 k + 1 3 k =2 k =2 and our answer is 7 + 3 = 10 . Problem written by Matt Tyler.