1. a) (15 points) Find three positive integers a,b,c whose
sum is 407, and whose product when written in base 10 ends in six 0's.

b) (15 points) Prove that there do NOT exist positive
integers a,b,c whose sum is 407 and whose product ends in seven 0's.

2. (30 points) Three circles, each of radius r, are placed
on a plane so that the center of each circle lies on a point of intersection of
the other two circles. The region R consists of all points inside or on at
least one of these three circles. Find the area of R.

3. (30 points) Let f

_{ 1}(x)= a_{1}x^{2}+b_{1}x+c_{1}, f_{ 2}(x)=a_{2}x^{2}+b_{2}x+c_{2}, and f_{ 1}(x)=a_{3}x^{2}+b_{3}x+c_{3}be the equations of three parabolas such that a_{1}> a_{2}> a_{3}. Prove that if each pair of parabolas intersects in exactly one point, then all three parabolas intersect in a common point.
4. Gigafirm is a large corporation with many employees.

a) (10 points) Show that the number of employees with an odd number of
acquaintances is even.

b) (20 points) Suppose that each employee with an even
number of acquaintances sends a letter to each of these acquaintances. Each
employee with an odd number of acquaintances sends a letter to each
non-acquaintance. So far, Leslie has received 99 letters. Prove that Leslie
will receive at least one more letter.

(Notes: ``acquaintance'' and ``non-acquaintance'' refer to employees of
Gigafirm. If A is acquainted with B, then B is acquainted with A. However, no
one is acquainted with himself.)

5. a) (5 points) Prove that for every positive integer N,
if A is a subset of the numbers {1,2,...,N} and A has size at least 2N/3+1,
then A contains a three-term arithmetic progression (i.e., there are positive
integers a and b so that all three of the numbers a,a+b, and a+2b are elements
of A).

b) (25 points) Show that if A is a subset of {1,2,...,3500}
and A has size at least 2003, then A contains a three-term arithmetic
progression.

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