HMMT 二月 2009 · GEN2 赛 · 第 8 题

英文解析

8. [ 5 ] The incircle ω of equilateral triangle ABC has radius 1. Three smaller circles are inscribed tangentto ω and the sides of ABC , as shown. Three smaller circles are then inscribed tangent to the previouscircles and to each of two sides of ABC . This process is repeated an infinite number of times. Whatis the total length of the circumferences of all the circles?
   Answer: 5 π
   Solution: One can find using the Pythagorean Theorem that, in each iteration, the new circles haveradius 1 / 3 of that of the previously drawn circles. Thus the total circumference is 2 π +3 · 2 π ( − 1) =1
   1 − 1 / 3
   5 π.