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
?
2 Two sequences
and
are defined as follows:
and
. If both sequences are defined with
ranging across the natural numbers, how many numbers belong to both sequences?
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
.
4 In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for
?

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
.
7 The number
has
trailing zeroes when expressed in base
.
a) Find the minimum possible
.
b) Find the maximum possible
.
c) Find the total number of possible
.
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.
a) Prove that the perimeter of
is always congruent to
.
b) Prove that
is constant for any
.
c) Prove that
.