If there are 12 people in a room and every person shakes hands exactly once with all of the other people, how many handshakes will there be? Generalize this idea for a room with n people.
How many diagonals are in a regular polygon with n sides?
Consider a 12-sided polygon. If you join any 4 of the 12 vertices of the polygon, you get a quadrilateral. How many quadrilaterals can be formed without including the sides of the 12-sided polygon? How about in a 20-gon?
What is the area of a square inscribed in a circle with radius 1? What is the area of a hexagon inscribed in a circle with radius 1? What is the area of an equilateral triangle inscribed in a circle with radius 1? What patterns do you notice? How can we describe the areas regular polygons in terms of the number of sides?
A lune is a plane figure bounded by the arcs of two circles. We are given a right isosceles triangle ABC. A lune is constructed with vertices at A and B with the center of the inner arc being at C and the center of the outer arc being at the midpoint of segment AB. Prove that the area of the lune so constructed is the same as the area of the triangle ABC.
Two circles, each of which passes through the center of the other, intersect at points M and N. A line from M intersects the circles at K and L. If KL=6, compute the area of triangle KLN. If r is the measure of the radius of each circle, find the least value and the greatest value of the area of triangle KLN.
Given a semicircle of radius r. What is the maximum area of any trapezoid inscribed in the semicircle? How can you prove your result?
Suppose you're on a 5×5 grid, and want to go from the bottom left to the top right. How many different paths can you take? Avoid backtracking -- you can only move right or up. Generalize this idea to an m×n grid.