2007 UNCO Math Contest II Problems/Problem 9

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Problem

A circle is inscribed in an equilateral triangle whose side length is $2$. Then another circle is inscribed externally tangent to the first circle but inside the triangle as shown. And then another, and another. If this process continues forever what is the total area of all the circles? Express your answer as an exact multiple of $\pi$ (and not as a decimal approximation).

[asy] path T=polygon(3); draw(unitcircle,black); draw(scale(2)*T,black); draw(shift(2/sqrt(3),-2/3)*scale(1/3)*unitcircle,black); draw(shift(2/sqrt(3)/3,-2/9)*shift(2/sqrt(3),-2/3)*scale(1/9)*unitcircle,black); [/asy]


Solution

See Also

2007 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
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