Information about 2009 Putnam Competition

Fall-09. Math 191 (ccn 54662): Putnam Workshop

Instructor: Alexander Givental
Meetings: ThTh 3:30--5 p.m. in 81 Evans
Office hours: TuTh 5--6 p.m. in 701 Evans

This class will function as a problem-solving seminar intended, in the first place, for those who plan to take the Putnam Exam, the US and Canada math Olympiad for college students, in December 2009. Our objectives will be to build participant's confidence in solving challenging math problems and, more importantly, to learn interesting mathematics, using materials of past Putnam exams for motivation. Expect 2009 hours of work per week.

Useful links:
Past Putnam Exams 1985--2008
Introduction to LaTeX -- Harvard Math Dept
LaTeX help at Art of Problem Solving website
Website of Olga Holtz (previous instructor of this workshop)
Handouts:
Soluitions to Putnam-1985 (pdf)
Elementary Number Theory (pdf)
Permutations and Determinants (pdf)
Hints to Putnam-1986 (pdf)
Exercises on Determinants (pdf)
Solutions to Putnam-1986 (pdf)
Hints to Putnam-1988 (pdf)
Linear Algebra. Part I (pdf)
Homework due Tue, Sept. 22:
  • Be prepared to explain your solutions to Linear Algebra-I exercises with numbers congruent to 0 mod 9.
  • Write or type 2 pages with solutions to the following problems of past Putnam exams: 1987(B3,A4,A5), 1989(B1,A1,A4)
    By Th, Sept. 24, you are required to learn (or invent) a proof of the theorem:
    Every qudratic form in \C^n by a suitable linear changes of coordinates can be transformed to one of the n+1 normal forms: z_1^2+...+z_r^2, r=0, 1,..., n.
    Homework due Th, Oct. 1: 1990 (A3, A4, B1, B5) 1991 (A2, B2, B4, B5)
    Homework due Th, Oct. 8: 1989 (A5,B6), 1990 (A2), 1992(A3,B4,B5)
    By Tue, Oct 13, prove Pick's formula: The area of a parallelogram with integer vertices is equal to I+B/2+1, where I is the number of integer points in the interior of the parallelogram, and B in the interior of its sides.
    Homework due Th, Oct. 15: Do exercises on Linear Algebra. Part 0 (pdf)
    Homewoerk due Th, Oct. 22: Solve Putnam Problems 1993 (B1, B3, B6)
    Do the following linear algebra exercises:
    1. Prove that if a field has q elements, then q is a power of a prime.
    2. Classify up to linear transformations of F^n: (a) subspaces, (b) pairs of subspaces, (c) complete flags of subspaces F^1 \subset F^2 \subset ... \subset F^{n-1} (i.e. arrays of nested subspaces of dimensions 1,2,...,n-1).
    3. In space F^n over a field F of q elements, find the number of: (a) bases, (b) complete flags, (c) k-dimensional subspaces.
    Homework due Th, Oct. 29:
    I. Exercises 1,2,3 of the previous week - please complete solutions of those exercises that you have not yet submitted (in the absense of The Mathematical Mastermind in this region of the Universe, everyone should do each exercise).
    II. Solve this cute middle-school problem: A long log is divided by 6 blue marks into 7 equal parts, by 12 red marks into 13 equal parts, and then cut into 20 equal pieces; prove that each piece except the two extreme ones has exatly one mark (blue or red).
    III. Putnam problems: 1994(A1,A4,B2), 1995 (B4)
    Homework (due by Th, Nov. 5): 1995 (A5,B6) 1996(A2,B1,B3,B4)
    Homework (due by Th, Nov. 12): 1997(A2,A4,A5,B5) 1998(A3,B5)
    Homework (due by Th, Nov. 19): 1999(A2,A3,B4,B5)
    Is there an nxn-matrix A such that A^3=0 but A^2\neq 0, if (a) n=2, (b) n=3?
    Homework (due by Tue, Dec. 1): 2000(A1, A2, A4, A5, B2, B3)