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# Geometry Warmups

Mathematics is filled with shapes that are kaleidoscopic in variety. Wielded since ancient times, the power of geometry helps us examine and measure these shapes.

A parabola passes through the following points

\[ \left( -1,1 \right) ,\left( 0,0 \right) ,\left( \dfrac { 1 }{ 2 } ,\dfrac { 1 }{ 4 } \right) ,\left( 1,1 \right) ,\left( -\dfrac { 7 }{ 5 } ,-\dfrac { 3 }{ 5 } \right). \]

It also passes through the point: \((\dfrac { 289 }{ 240 } ,\dfrac { a }{ b } ),\) where \(a,b\) are positive coprime integers. Find \(a+b\)

The two large gray circles are congruent, and each is half the diameter of the largest circle. All circles that appear to be tangent to each other *are* indeed tangent to each other.

\[ \frac{\text{radius of blue circle}}{\text{radius of red circle}} = \frac{a}{b}, \]

where \(a\) and \(b\) are coprime positive integers. Find \(a+b\).

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