2. Let P P2 Pn be distinct primes. Suppose a1, a2.....am are distinct positive integers that are divisible only by these primes. Show that if m>2", then aja, is a square for some i < j. 3. Suppose each point in Zx Z is colored with one of n given colors. Find the smallest k and I such that in any kx I grid one is guaranteed to find four monochromatic points that are vertices of a rectangle. 4. A set of lines in the Euclidean plane is in general position if no two lines are parallel and no three lines are concurrent (contain a common point). Let , be the number of regions determined by n lines in general position in the Euclidean plane. (a) Find a recurrence relation that r, satisfies and give a proof of your statement. (b) Use your recurrence relation to find a simple formula for -