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Math 101: General Course Outline

Catalog Description

Lecture, three hours. Prerequisite: Math 100 or significant experience with mathematical competitions. Advanced problem solving techniques and mathematical topics useful as preparation for Putnam Competition. Problems in abstract algebra, linear algebra, number theory, combinatorics, probability, real and complex analysis, differential equations, Fourier analysis. Regular practice tests given, similar in difficulty to the Putnam Competition. Enrollment is by permission of the instructor, based on a selection test or past Putnam results. May be repeated for maximum of 12 units. P/NP or letter grading.

Textbook

R. Gelca & T. Andreescu. Putnam and Beynd, Springer Verlag

Updated 10/14: C. Manolescu

Schedule of Lectures

Lecture Section Topics

1

Introduction to the Putnam Mathematical Competition. Selected test problems from previous years.

2

Methods of proof: contradiction, induction, the pigeonhole principle, invariants.

3

Algebra. Inequalities and identities. Real and complex polynomials.

4

Linear Algebra. Eigenvalues, the Cayley-Hamilton Theorem. Abstract algebra (groups, rings).

5

Geometry and trigonometry. Using vectors and complex numbers to solve gemetry problems.

6

Number theory. Euler's theorem. Diophantine equations.

7

Combinatorics and combinatorial geometry. Generating functions. Probability.

8

Real analysis problems. Sequences, series, continuity, derivatives and integrals. Convexity.

9

Multivariable differential and integral calculus. Solving integrals using complex analysis.

10

Differential equations and Fourier analysis.