HMMT 二月 2009 · GEN2 赛 · 第 8 题
HMMT February 2009 — GEN2 Round — Problem 8
题目详情
- [ 5 ] The incircle ω of equilateral triangle ABC has radius 1. Three smaller circles are inscribed tangent to ω and the sides of ABC , as shown. Three smaller circles are then inscribed tangent to the previous circles and to each of two sides of ABC . This process is repeated an infinite number of times. What is the total length of the circumferences of all the circles?
解析
- [ 5 ] The incircle ω of equilateral triangle ABC has radius 1. Three smaller circles are inscribed tangent to ω and the sides of ABC , as shown. Three smaller circles are then inscribed tangent to the previous circles and to each of two sides of ABC . This process is repeated an infinite number of times. What is 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 have 1 radius 1 / 3 of that of the previously drawn circles. Thus the total circumference is 2 π +3 · 2 π ( − 1) = 1 − 1 / 3 5 π.