HMMT 十一月 2014 · 团队赛 · 第 8 题

英文解析

8. [ 3 ] Let H be a regular hexagon with side length one. Peter picks a point P uniformly and at randomwithin H , then draws the largest circle with center P that is contained in H . What is this probabilitythat the radius of this circle is less than ?1
   √2
   2 3 − 1
   Answer: We first cut the regular hexagon H by segments connecting its center to eachvertex into six different equilateral triangles with side lengths 1. Therefore, each point inside H is 3
   contained in some equilateral triangle. We first see that for each point inside an equilateral triangle,
   the radius of the largest circle with center P which is contained in H equals the shortest distance from
   P to the nearest side of the hexagon, which is also a side of the triangle in which it is contained.
   √
   Consider that the height of each triangle is . Therefore, the region inside the triangle containing all 3
   points with distance more than to the side of the hexagon is an equilateral triangle with a height of 12
   √2
   3 − 1 1
   . Consequently, the area inside the triangle containing all points with distance less than to the
   2 2
   ( )
   ( ) ( )
   √ √ √ √ √
   3 3 − 1 3 2 3 − 1 2 3 − 12
   √
   side of the hexagon has area 1 − = · . This is of the ratio to the
   4 4 3 3
   √3
   area of the triangle, which is . Since all triangles are identical and the point P is picked uniformly 3
   within H , the probability that the radius of the largest circle with center P which is contained in H is 4
   √
   1 2 3 − 1
   less than is , as desired.
   2 3