HMMT 二月 2002 · 冲刺赛 · 第 27 题

HMMT February 2002 — Guts Round — Problem 27

[7] Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand exactly at the 5 and the minute hand exactly at the 6). How many different positions are there that would remain possible if the hour and minute hands were switched?

解析

Consider the two hands of an analog clock, each of which moves with constant angular velocity. Certain positions of these hands are possible (e.g. the hour hand halfway between the 5 and 6 and the minute hand exactly at the 6), while others are impossible (e.g. the hour hand exactly at the 5 and the minute hand exactly at the 6). How many different positions are there that would remain possible if the hour and minute hands were switched? Solution: 143 We can look at the twelve-hour cycle beginning at midnight and ending just before noon, since during this time, the clock goes through each possible position exactly once. The minute hand has twelve times the angular velocity of the hour hand, so if the hour hand has made t revolutions from its initial position (0 ≤ t < 1), the minute hand has made 12 t revolutions. If the hour hand were to have made 12 t revolutions, the minute hand would have made 144 t . So we get a valid configuration by reversing the hands precisely when 144 t revolutions land the hour hand in the same place as t revolutions — i.e. when 143 t = 144 t − t is an integer, which clearly occurs for exactly 143 values of t corresponding to distinct positions on the clock (144 − 1 = 143).