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Math 100: General Course Outline

Catalog Description

100. Problem Solving. (Formerly numbered 192.)Lecture, three hours. Requisite: course 31B with grade of C- or better. Problem-solving techniques and mathematical topics useful as preparation for Putnam Examination and similar competitions. Continued fractions, inequalities, modular arithmetic, closed form evaluation of sums and products, problems in geometry, rational functions and polynomials, other nonroutine problems. Participants expected to take Putnam Examination. P/NP grading.

Math 100 is a course in problem solving. The problems are more varied and unexpected than in a typical undergraduate mathematics course. Often an original or imaginative step is required. Some variations of topics from year to year are expected. Topics may include: explicit summations of series, spherical trigonometry, advanced Euclidean geometry, elementary number theory, combinatorial problems, inequalities, continued fractions. There is a lot of classroom discussion. Homework is assigned regularly and makes a large contribution to the course grade. One three-hour final is given.

Textbook

Problem-Solving Through Problems by Loren C. Larson

Updated: 10/14 C. Manolescu

Schedule of Lectures

Lecture Section Topics

Week 1

Induction. Generalized induction. The pigeonhole principle.

Week 2

Inequalities (AM-GM, weighted AM-GM, Cauchy-Schwartz, Jensen, Holder, Minkowski).

Week 3

Number theory. Modular arithmetic. Fermat's little theorem, Euler's theorem. The Chinese remainder theorem.

Week 4

Algebra. Polynomials (factorization over different fields, Viete's relations). Some abstract algebra (groups, rings).

Week 5

Summation of series. Geometric progressions. Telescoping series and products. Taylor series.

Week 6

Combinatorics. Binomial coefficients and combinatorial identities.

Week 7

Recurent sequences (linear recurrences, generating functions). Discrete and continuous probability.

Week 8

Geometry problems. Elementary methods. Analytic geometry, conics. Vectors and complex numbers.

Week 9

Differential calculus. The extreme value theorem and the mean value theorem. Functional equations.

Week 10

Integral calculus. Approximating integrals by Riemann sums. Integral functional equations.