Geometry
# Geometry Warmups

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 $a+b$

$\frac{m}{n}$, for relatively prime integers $m$ and $n$. Compute $m+n$.

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction

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 $a+b$.