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HMMT 十一月 2022 · 冲刺赛 · 第 12 题

HMMT November 2022 — Guts Round — Problem 12

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [8] Candice starts driving home from work at 5:00 PM. Starting at exactly 5:01 PM, and every minute after that, Candice encounters a new speed limit sign and slows down by 1 mph. Candice’s speed, in miles per hour, is always a positive integer. Candice drives for 2 / 3 of a mile in total. She drives for a whole number of minutes, and arrives at her house driving slower than when she left. What time is it when she gets home? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2022, November 12, 2022 — GUTS ROUND Organization Team Team ID#
解析
  1. [8] Candice starts driving home from work at 5:00 PM. Starting at exactly 5:01 PM, and every minute after that, Candice encounters a new speed limit sign and slows down by 1 mph. Candice’s speed, in miles per hour, is always a positive integer. Candice drives for 2 / 3 of a mile in total. She drives for a whole number of minutes, and arrives at her house driving slower than when she left. What time is it when she gets home? Proposed by: Preston Bushnell Answer: 5:05 (PM) Solution: Suppose that Candice starts driving at n miles per hour. Then she slows down and drives ( n − 1) mph, ( n − 2) mph, and so on, with her last speed being ( m + 1) mph. Then the total distance traveled is n n − 1 m + 1 1 n ( n + 1) m ( m + 1)
    • · · · + = − 60 60 60 60 2 2 2 2 n + n − m − m = 120 ( n + m + 1)( n − m ) = . 120 Since the total distance travelled is 2 / 3, we have ( n + m + 1)( n − m ) = 120 · 2 / 3 = 80. We know m is nonnegative since Candice’s speed is always positive, so n + m + 1 > n − m . Thus, n + m + 1 and n − m are a factor pair of 80, with n + m + 1 greater and n − m smaller. Since one is even and one is odd, this means we either have ( n + m + 1 , n − m ) = (80 , 1) or (16 , 5). The first case is impossible since it gives n − m = 1, which would imply that Candice drives at n mph the whole way home. Therefore, ( n + m − 1 , n − m ) = (16 , 5). Since n − m = 5, she gets home at 5:05 pm.