1997 AIME Problems/Problem 4

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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

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:

$\begin{eqnarray*}5^2 + x^2 &=& (5+r)^2 \\x &=& \sqrt{10r + r^2} \\&& \\(8 + r + \sqrt{10r+r^2})^2 + 5^2 &=& 13^2\\8 + r + \sqrt{10r+r^2} &=& 12\\\sqrt{10r+r^2}&=& 4-r\\10r+r^2 &=& 16 - 8r + r^2\\r &=& \frac{8}{9}\end{eqnarray*}$

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

1997 AIME (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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1997 AIME Problems/Problem 4

From AoPSWiki

Problem

Solution

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