HMMT 十一月 2017 · 冲刺赛 · 第 9 题

HMMT November 2017 — Guts Round — Problem 9

[ 3 ] A malfunctioning digital clock shows the time 9 : 57 AM; however, the correct time is 10 : 10 AM. There are two buttons on the clock, one of which increases the time displayed by 9 minutes, and another which decreases the time by 20 minutes. What is the minimum number of button presses necessary to correctly set the clock to the correct time? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2017, November 11, 2017 — GUTS ROUND Organization Team Team ID# x x + 6 y − 3 z − 2 y + z 2

解析

[ 3 ] A malfunctioning digital clock shows the time 9 : 57 AM; however, the correct time is 10 : 10 AM. There are two buttons on the clock, one of which increases the time displayed by 9 minutes, and another which decreases the time by 20 minutes. What is the minimum number of button presses necessary to correctly set the clock to the correct time? Proposed by: Kevin Sun Answer: 24 We need to increase the time by 13 minutes. If we click the 9 minute button a times and the 20 minute button b times, then we must have 9 a − 20 b = 13. Note that if this equation is satisfied, then b increases as a increases, so it suffices to minimize a . This means that a must end in a 7. However, since 63 − 20 b = 13 has no integer solution, the next smallest possible value of a is 17, which gives the solution ( a, b ) = (17 , 7), or 24 button presses. x x + 6 y − 3 z − 2 y + z 2