Difference between revisions of "1997 AIME Problems/Problem 4"

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[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]
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Revision as of 18:35, 4 July 2013

Problem

Circles of radii $5, 5, 8,$ and $\frac mn$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

1997 AIME-4.png

If (in the diagram above) we draw the line going through the centers of the circles with radii $8$ and $\frac mn = r$, that line is the perpendicular bisector of the segment connecting the centers of the two circles with radii $5$. Then we form two right triangles, of lengths $5, x, 5+r$ and $5, 8+r+x, 13$, wher $x$ is the distance between the center of the circle in question and the segment connecting the centers of the two circles of radii $5$. By the Pythagorean Theorem, we now have two equations with two unknowns:

$52+x2=(5+r)2x=10r+r2(8+r+10r+r2)2+52=1328+r+10r+r2=1210r+r2=4r10r+r2=168r+r2r=89$ (Error compiling LaTeX. Unknown error_msg)

So $m+n = \boxed{17}$.

See also

1997 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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