HMMT 二月 2006 · 代数 · 第 5 题
HMMT February 2006 — Algebra — Problem 5
题目详情
- Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given ◦ ◦ instant, the hands on each clock form an angle between 0 and 180 inclusive. At how many times during that day are the angles on the two clocks equal? √ √ √ 3 2 4 2
解析
- Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact ◦ time. At any given instant, the hands on each clock form an angle between 0 and ◦ 180 inclusive. At how many times during that day are the angles on the two clocks equal? Answer: 33 Solution: A tricky thing about this problem may be that the angles on the two clocks might be reversed and would still count as being the same (for example, both ◦ angles could be 90 , but the hour hand may be ahead of the minute hand on one clock and behind on the other). ◦ Let x , − 12 ≤ x < 12, denote the number of hours since noon. If we take 0 to mean upwards to the “XII” and count angles clockwise, then the hour and minute hands of ◦ ◦ ◦ the correct clock are at 30 x and 360 x , and those of the slow clock are at 15 x and ◦ ◦ ◦ 180 x . The two angles are thus 330 x and 165 x , of course after removing multiples of ◦ 360 and possibly flipping sign; we are looking for solutions to ◦ ◦ ◦ ◦ ◦ ◦ 330 x ≡ 165 x (mod 360 ) or 330 x ≡ − 165 x (mod 360 ) . In other words, 360 | 165 x or 360 | 495 x. Or, better yet, 165 11 495 11 x = x and/or x = x 360 24 360 8 2 must be an integer. Now x is any real number in the range [ − 12 , 12), so 11 x/ 8 ranges in [ − 16 . 5 , 16 . 5), an interval that contains 33 integers. For any value of x such that 11 x/ 24 is an integer, of course 11 x/ 8 = 3 × (11 x/ 24) is also an integer, so the answer is just 33. √ √ √ 3 2