HMMT 十一月 2016 · 团队赛 · 第 1 题

HMMT November 2016 — Team Round — Problem 1

[ 3 ] Two circles centered at O and O have radii 2 and 3 and are externally tangent at P . The common 1 2 external tangent of the two circles intersects the line O O at Q . What is the length of P Q ? 1 2

解析

[ 3 ] Two circles centered at O and O have radii 2 and 3 and are externally tangent at P . The common 1 2 external tangent of the two circles intersects the line O O at Q . What is the length of P Q ? 1 2 Proposed by: Eshaan Nichani Answer: 12 Let the common external tangent intersect the circles centered at O , O at X , Y respectively. Then 1 2 O Q O Q − O Q OY 3 O O 1 2 2 1 1 2 = = , so = = . Since O O = 2 + 3 = 5, O Q = 10 and hence 1 2 1 O Q OX 2 O Q O Q 2 1 1 1 P Q = O Q + O P = 12. 1 1