HMMT 二月 2002 · 冲刺赛 · 第 27 题

英文解析

27. Consider the two hands of an analog clock, each of which moves with constantangular velocity. Certain positions of these hands are possible (e.g. the hour hand halfwaybetween 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 differentpositions 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 endingjust before noon, since during this time, the clock goes through each possible position exactlyonce. The minute hand has twelve times the angular velocity of the hour hand, so if thehour hand has made t revolutions from its initial position (0 ≤ t < 1), the minute handhas made 12 t revolutions. If the hour hand were to have made 12 t revolutions, the minutehand would have made 144 t . So we get a valid configuration by reversing the hands preciselywhen 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 correspondingto distinct positions on the clock (144 − 1 = 143).