HMMT 十一月 2016 · 团队赛 · 第 1 题
HMMT November 2016 — Team Round — Problem 1
题目详情
英文原题
- [ 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 ?
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解析
英文解析
- [ 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
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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