Q1.

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by  on every pair of shoes. Carlos also knew that he had to pay a  sales tax on the discounted price. He had  dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

Q2.

Let  be a convex pentagon with      and  For each point  in the plane, define  The least possible value of  can be expressed as  where  and  are positive integers and  is not divisible by the square of any prime. Find 

Q3.

Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws  more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these  line segments divide the disk.

Q4.

The set of points in -dimensional coordinate space that lie in the plane  whose coordinates satisfy the inequalitiesforms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form  where  and  are positive integers and  is not divisible by the square of any prime. Find