PUMaC 2017 · 团队赛 · 第 13 题
PUMaC 2017 — Team Round — Problem 13
题目详情
- (10) A point-sized cue ball is fired in a straight path from the center of a regular hexagonal billiards table of side length 1. If it is not launched directly into a pocket but travels an integer distance before falling into one of the pockets (located in the corners), find the minimum distance that it could have traveled.
解析
- Consider the tiling of the plane with hexagons. This represents the possible reflections of the cue ball as it hits a wall – i.e. instead of reflecting the ball’s path about the wall, we reflect the table across the wall and continue the path straight. Thus the following argument will get us our answer. √ √ ( ) ( ) 1 1 The cue ball starts at the origin; let C = 0 , n 3 and v = , 3 . We restrict our n 2 2 consideration to a sixth of the plane: the sector swept out by the positive y -axis rotated ◦ clockwise by 60 . The rest of the plane is just 5 other copies of this sector, rotated by ◦ some multiple of 60 . We wish to find points here that are an integer distance from the origin. These occur at H = C + kv for 3 - k but otherwise n, k ∈ N . Thus, H = n,k n n,k ( ( ) √ ) √ 1 1 2 2 k, n + k 3 and its magnitude is 3 n + 3 nk + k . We now search for the minimal 2 2 √ 2 integer value that this can take. For k = 1 we need to find integer values of 3 n + 3 n + 1 = √ 1 2 3(2 n + 1) + 1; computation gives that this first occurs at n = 7. For k = 2 we need to 2 4 √ √ 2 2 find integer values of 3 n + 6 k + 4 = 3( n + 1) + 1; computation gives that this occurs √ 2 at n = 3, an improvement. For k = 4 we need to find integer values of 3 n + 12 k + 16 = √ 2 3( n + 2) + 4; computation gives that this occurs at n = 6, worst than H . After seeing 3 , 2 that nothing close to H works for k = 5 we can rule out anything larger because that will 3 , 2 2 2 2 have k > 3 · 3 + 2 . Thus, ‖ H ‖ = 7 is our final answer. 3 , 2 Problem written by Zack Stier 5