HMMT 十一月 2009 · GEN1 赛 · 第 8 题

HMMT November 2009 — GEN1 Round — Problem 8

[ 7 ] Let 4 ABC be an equilateral triangle with height 13, and let O be its center. Point X is chosen at random from all points inside 4 ABC . Given that the circle of radius 1 centered at X lies entirely inside 4 ABC , what is the probability that this circle contains O ?

解析

[ 7 ] Let 4 ABC be an equilateral triangle with height 13, and let O be its center. Point X is chosen at random from all points inside 4 ABC . Given that the circle of radius 1 centered at X lies entirely inside 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 O 3 is a circle of radius 1, centered at O , and hence has area π . The probability that the circle centered at √ π 3 π X contains O given that it also lies in ABC is then the ratio of the two areas, that is, = . 100 √ 100 3