HMMT 二月 2000 · 团队赛 · 第 10 题

HMMT February 2000 — Team Round — Problem 10

Ho w man y times p er da y do at least t w o of the three hands on a lo k oin ide?

解析

The se ond hand rosses the min ute hand 59 times an hour. The se ond hand rosses the hour hand 60 times an hour, ex ept for 2 of the hours, due to the mo v emen t of the hour hand. The min ute hand and the hour hand ross 22 times total, b e ause the hour hand ompletes 2 rotations in a da y , and the min ute hand ompletes 24. The se ond, hour, and min ute hand all oin ide only at no on and midnigh t, but w e'v e oun ted ea h of these 12:00's 3 times instead of on e. Therefore, the answ er is 59 60 + 60 24 2 + 22 2 2, giving us 2872 rossings. m