1 A point in the first quadrant lies on a line with intercepts and , with 0 a,b > 0″ />. Rectangle has vertices , , , and , while rectangle has vertices , , , and . What is the ratio of the area of to that of ?
3 Billy and Bobby are located at points and , respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point ; the second time they meet, they are located 10 units from point . Find all possible values for the distance between and .
5 We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?
6 Circle with diameter has chord drawn such that is perpendicular to at . Another circle is drawn, sharing chord . A point on minor arc of is chosen so that . Line is tangent to through and a point on is chosen such that . If and , find .
8 Define to be the nearest integer to , with the greater integer chosen if two integers are tied for being the nearest. For example, , , and . Define to be the area of region . Define region , for each positive integer , to be the region on the Cartesian plane which satisfies the inequality <img title=”f(|x|) + f(|y|) < n” src=”http://data.artofproblemsolving.com/images/latex/d/c/3/dc30d44d8951d826bd03abacc3ea16d5410bdb23.gif” alt=”f(|x|) + f(|y|) . We pick an arbitrary point on the perimeter of , and mark every two units around the perimeter with another point. Region is defined by connecting these points in order.