HMMT 十一月 2009 · GEN1 赛 · 第 8 题

英文解析

8. [ 7 ] Let 4 ABC be an equilateral triangle with height 13, and let O be its center. Point X is chosenat random from all points inside 4 ABC . Given that the circle of radius 1 centered at X lies entirelyinside 4 ABC , what is the probability that this circle contains O ?
   √
   3 π
   Answer: The set of points X such that the circle of radius 1 centered at X lies entirely inside
   ′ ′ ′ ′ ′ ′ ′100
   4 ABC is itself a triangle, A B C , such that AB is parallel to A B , BC is parallel to B C , and CA is
   ′ ′ ′ ′ ′ ′ ′ ′
   parallel to C A , and furthermore AB and A B , BC and B C , and CA and C A are all 1 unit apart.
   ′ ′ ′
   We can use this to calculate that A B C is an equilateral triangle with height 10, and hence has area
   √100
   . On the other hand, the set of points X such that the circle of radius 1 centered at X contains Ois a circle of radius 1, centered at O , and hence has area π . The probability that the circle centered at 3
   √
   π 3 π
   X contains O given that it also lies in ABC is then the ratio of the two areas, that is, = .
   √ 100100 3