Putnam Preparation Sessions
in the Math Common Room
3-4:30 Thursdays
Putnam Exam is Saturday 10 am !
Problem of the week
If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart?.
| date, time | what happened |
|---|---|
| Oct 25, 2:15-3:45 | The first session. 10 of us worked on these problems (and we solved the first one) 1) If you pick four points on the surface of a sphere what is the probability that the tetrahedron with those vertices will contain the center of the sphere? 2) If you pick N points in the plane how many ways can you divide the set into two parts with a straight line? 3) The geometry problem below. |
| Nov 1, 2:45-4:15 | The second session. There were 5 of us. But we solved two problems: 1) You have one set X and 1066 subsets A1, A2, ... , A1066 each of which has more than half the elements of X. Show that there exist ten elements x1, x2, ... , x10 in X so that every set Ai contains at least one of the points xj. 2) n people sit around a table and play a game. They each start with 1 penny. The first person passes one penny to the second, the second person passes 2 pennies to the third, the third person passes 1 penny to the fourth and so on, alternating between 1 and 2. When someone runs out of pennies he is out of the game. To win, you have to get all the pennies. Find an infinite number of numbers n so that, starting with n players, someone wins. Then we worked on two other problems without success. One was: 3) Suppose that f, f', f'', f''' are continuous and f'''(x) ≤ f(x) for all x. Then prove that f'(x) < 2f(x) for all x. |
| Nov 8, 3-4:30 | The third session. There were again 10 of us and we solved 2.5 problems 1) How many prime numbers have decimal form 101010...01 (alternating 1 and 0)? 2) If P(x) is a polynomial of degree n and P(x)=Q(x)P''(x) where Q(x) is quadratic and P(x) has two distinct roots then show that it has n distinct roots. 3) (This is the problem that we half solved.) Find all real numbers c so that (exp(x)+exp(-x))/2 ≤ exp(cx2) for all real x. We were in the middle of a fourth problem when we ran out of time: 4) Does there exist a number L so that any n x m rectangle where n,m are integer larger than L is a union of 4 x 6 and 5 x 7 rectangles which meet only along their boundaries? |
| Nov 15, 3-4:30 | The fourth session. There were 10 of us who solved 6 problems! My favorite was the problem: Show that there exists a unique function from the positive reals to the positive reals so that f(f(x)) = 6x - f(x) for all x > 0. |
| Nov 22 | Thanksgiving-No meeting. |
| Nov 29, 3-4:30 | The last session. There was a good turnout. About 12 of us solved several problems. My favorite was the last problem: Suppose that M,N are 3-by-2 and 2-by-3 matrices with product MN= 8 2 -2 2 5 4 -2 4 5 Show that NM= 9 0 0 9 |
| Saturday, Dec 1, 10am | The Putnam Competition (Let me know if you can make it!) |
Updated: 11/30/07, 2pm.