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Geometry Level 3

The above diagram shows $8$ circles. The yellow and green circles are concentric, with the radius of the green circle being $1$ unit and the radius of the yellow circle being $4$ units. The red circles are internally tangent to the yellow circle and externally tangent to the orange circles, and the orange circles are externally tangent to both the green circle and the red circles. Furthermore, the red and orange circles have the same radii.

If the radius of the red and orange circles can be written as $\frac{\sqrt{a}-b}{c}$ units, where $a$ , $b$ and $c$ are integers with $c>0$ , $a \geq 0$ , and the fraction is reduced to the lowest terms, find the value of $a+b+c$ .

The answer is 144.

1 solution

Marta Reece
Dec 15, 2017

Triangle $ABC$ has sides $1+R$ , $4-R$ , and $2R$

From symmetry the angle $\angle ACB=60^\circ$

So $cos (ACB)=\frac12$

Writing the law of cosines for this triangle

results in a quadratic equation for $R$

$(1+R)^2+(4-R)^2-2\times\frac12(1+R)(4-R)=(2R)^2$

This equation has only one positive solution

$\frac{\sqrt{133}-9}2$

So the answer is

$133+9+2=\boxed{144}$

Looks like pizza

The answer is 144.

1 solution

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