HMMT 十一月 2015 · 冲刺赛 · 第 6 题
HMMT November 2015 — Guts Round — Problem 6
题目详情
- [ 6 ] Let AB be a segment of length 2 with midpoint M . Consider the circle with center O and radiusr that is externally tangent to the circles with diameters AM and BM and internally tangent to the circle with diameter AB . Determine the value of r .
HMMT November 2015, November 14, 2015 — GUTS ROUND
Organization Team Team ID#
英文原题
[ 6 ] Let AB be a segment of length 2 with midpoint M . Consider the circle with center O and radius
r that is externally tangent to the circles with diameters AM and BM and internally tangent to the
circle with diameter AB . Determine the value of r .
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HMMT November 2015, November 14, 2015 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 6 ] Let AB be a segment of length 2 with midpoint M . Consider the circle with center O and radiusr that is externally tangent to the circles with diameters AM and BM and internally tangent to the circle with diameter AB . Determine the value of r .
Proposed by: Sam Korsky
Answer:1
1 13
Let X be the midpoint of segment AM . Note that OM ⊥ M X and that M X = and OX = + r
2 2
and OM = 1 − r . Therefore by the Pythagorean theorem, we have
( )
1 12
2 2 2 2
OM + M X = OX = ⇒ (1 − r ) + = + r
2 22
which we can easily solve to find that r = .1 3