PUMaC 2012 · 团队赛 · 第 10 题

PUMaC 2012 — Team Round — Problem 10

(3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at ( − 1 , 0) , (1 , 0) , (1 , 100) , ( − 1 , 100). You remove a section of the bottom of the paper by cutting along the function y = f ( x ), where f satisfies f (1) = f ( − 1) = 0. (In other words, you keep the bottom two vertices.) You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an L -shaped elbow tube. √ 1 1 a + b We can write f ( )+ f ( ) = , where a, b, c are integers and b is square-free. Find a + b + c . 3 6 πc

解析

Problem: (3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at ( − 1 , 0) , (1 , 0) , (1 , 100) , ( − 1 , 100). You remove a section of the bottom of the paper by cutting along the function y = f ( x ), where f satisfies f (1) = f ( − 1) = 0. (In other words, you keep the bottom two vertices.) You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an L -shaped elbow tube. √ 1 1 a + b We can write f ( )+ f ( ) = , where a, b, c are integers and b is square-free. Find a + b + c . 3 6 πc Answer: 532 1 Solution: The function is f ( x ) = (1 + cos( πx )), so π √ 5 + 3 1 1 f ( ) + f ( ) = 3 6 π 2 and the answer is 532 . Author: Henry