HMMT 十一月 2019 · 团队赛 · 第 9 题
HMMT November 2019 — Team Round — Problem 9
题目详情
- [65] Will stands at a point P on the edge of a circular room with perfectly reflective walls. He shines ◦ ◦ two laser pointers into the room, forming angles of n and ( n + 1) with the tangent at P , where n is a positive integer less than 90. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach P again. ( P is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room?
解析
- [65] Will stands at a point P on the edge of a circular room with perfectly reflective walls. He shines ◦ ◦ two laser pointers into the room, forming angles of n and ( n + 1) with the tangent at P , where n is a positive integer less than 90. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach P again. ( P is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room? Proposed by: Handong Wang Answer: 28 Note that we want the path drawn out by the lasers to come back to P in as few steps as possible. Observe that if a laser is fired with an angle of n degrees from the tangent, then the number of points it creates 180 on the circle is . (Consider the regular polygon created by linking all the points that show up on gcd(180 ,n ) the circle–if the center of the circle is O, and the vertices are numbered V , V , . . . , V , the angle ∠ V OV 1 2 k 1 2 360 is equal to 2 gcd(180 , n ), so there are a total of sides). 2 gcd(180 ,n ) 180 Now, we consider the case with both n and n + 1. Note that we wish to minimize the value + gcd(180 ,n ) 180 , or maximize both gcd(180 , n ) and gcd(180 , n + 1). Note that since n and n + 1 are relatively gcd(180 ,n +1) prime and 180 = (4)(9)(5), the expression is maximized when gcd(180 , n ) = 20 and gcd(180 , n + 1) = 9 (or vice versa). This occurs when n = 80. Plugging this into our expression, we have that the number of 180 points that show up from the laser fired at 80 degrees is = 9 and the number of points that appear 20 180 from the laser fired at 81 degrees is = 20. However, since both have a point that shows up at P (and 9 no other overlapping points since gcd(9 , 20) = 1), we see that the answer is 20 + 9 − 1 = 28. 3