HMMT 十一月 2010 · GEN2 赛 · 第 3 题

HMMT November 2010 — GEN2 Round — Problem 3

[ 5 ] Dragoons take up 1 × 1 squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the x -coordinates of their centers and the difference in the y -coordinates of their centers are both at most 6, regardless of any dragoons in between. For example, a dragoon centered at (4 , 5) can fire at a dragoon centered at the origin, but a dragoon centered at (7 , 0) can not. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?

解析

[ 5 ] Dragoons take up 1 × 1 squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the x -coordinates of their centers and the difference in the y -coordinates of their centers are both at most 6, regardless of any dragoons in between. For example, a dragoon centered at (4 , 5) can fire at a dragoon centered at the origin, but a dragoon centered at (7 , 0) can not. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously? Answer: 168 Assign coordinates in such a way that the dragoon being fired on is centered at (0 , 0). Any dragoon firing at it must have a center with x -coordinates and y -coordinates that are no smaller than − 6 and no greater than 6. That means that every dragoon firing at it must lie entirely in the region bounded by the lines x = − 6 . 5, x = 6 . 5, y = − 6 . 5, and y = 6 . 5. This is a square with sides of length 13, so there is room for exactly 169 dragoons in it. One of them is the dragoon being fired on, so there are at most 168 dragoons firing at it.