HMMT 十一月 2015 · THM 赛 · 第 3 题

英文解析

3. Consider a 3 × 3 grid of squares. A circle is inscribed in the lower left corner, the middle square ofthe top row, and the rightmost square of the middle row, and a circle O with radius r is drawn suchthat O is externally tangent to each of the three inscribed circles. If the side length of each square is
   1, compute r .
   Proposed by: Sam Korsky
   √
   5 2 − 3
   Answer:
   Let A be the center of the square in the lower left corner, let B be the center of the square in the 6
   middle of the top row, and let C be the center of the rightmost square in the middle row. It’s clearthat O is the circumcenter of triangle ABC - hence, the desired radius is merely the circumradius of
   √ √
   triangle ABC minus . Now note that by the Pythagorean theorem, BC = 2 and AB = AC = 5 so 1
   √2
   3 2
   we easily find that the altitude from A in triangle ABC has length . Therefore the area of triangle
   ABC is . Hence the circumradius of triangle ABC is given by 32
   √2
   BC · CA · AB 5 2
   3=
   4 ·6
   √2
   √
   5 2 − 3
   5 2 1
   and so the answer is − = .
   6 2 6