2. Let P P2 Pn be distinct primes. Suppose a1, a2.....am are distinct positive integers that...

 

Transcribed Image Text:

2. Let P₁ 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 ₁- 2. Let P₁ 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 ₁-

Answer rating: 100% (QA)

Solution Step 1 P P 1 2 and Are Now 3 5 3 91 92 integers So the It means and lower S... View the full answer