What is the measure of $\angle D?$ Express your answer in degrees rounded to the nearest whole degree.

If your answer is $31.4^{\circ},$ you would enter: $31$

Sides $AB$ and $CD$ are parallel. The area of $\bigtriangleup BCD$ is $20.$ What is the area of $\bigtriangleup ABD?$ (The diagram is not drawn to scale.)

Suppose $B$ and $C$ are endpoints of a diameter of a circle. If $A$ is another point on this circle, and $AB = 12$ and $AC = 16,$ what is the radius of the circle?

If the four corners of this quadrilateral are all on the circle, and $\angle A = 125^\circ$ and $\angle B = 60^\circ,$ how many degrees are in $\angle C?$

Let $a$ be the area of this parallelogram with diagonals $8$ and $4.$ What is $a^2?$

In a circle with radius $12$, two chords intersect, creating segments of length $6,$ $8,$ and $9,$ as shown. What is the length of the missing segment?

How many different triangles with integer side lengths have a perimeter of $8?$

An isosceles triangle has sides of $AB = 300,$ $AC = 300,$ and $BC = 168.$ There are two inscribed circles with the same radius, both internally tangent to the triangle along $\overline{BC}.$ What is the length of the radius of the circles?

What is the square of the height of this triangle?

Two balls fit snugly between two walls that are $22$ inches apart. The sum of the radii of the balls is $12.$ What is the product of the two radii?