PUMaC 2021 · 组合（B 组） · 第 2 题

PUMaC 2021 — Combinatorics (Division B) — Problem 2

Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for n minutes once he arrives at the airport before going through security. What is the smallest n they can select such that they meet with at least 50% probability? The answer will be of √ the form a + b c for integers a , b , and c , where c has no perfect square factor other than 1. Report a + b + c .

解析

Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for n minutes once he arrives at the airport before going through security. What is the smallest n they can select such that they meet with at least 50% probability? The answer will be of √ the form a + b c for integers a , b , and c , where c has no perfect square factor other than 1. Report a + b + c . Proposed by: Rishi Dange Answer: 67 2 2 (50 − n ) (70 − n ) Use geometric probability to see that the desired n will occur where + = 2 2 0 . 5 × 3600. The larger solution obviously is not the correct one, leaving the smaller solution √ as the answer (60 − 10 17).