Math 100: Problem Solving

Fall 2013

Time and Place: MWF 11-11:50pm in Geology 6704


  • Instructor: Ciprian Manolescu
  • E-mail: cm@math.ucla.edu
  • Office: MS 6921
  • Office Hours: Wed 2-3 and Thu 11-12

Web page: http://www.math.ucla.edu/~cm/100.1.13f/index.html

Prerequisite: The official prerequisite is completion of Math 31A. However, Math 100 is aimed at the most talented students at UCLA, and you should only take it if you got A's in your lower division math classes. Freshmen and transfer students who have some experience in math contests are welcome to enroll. To help you decide if the course is right for you, we will have a 25 minute long quiz in class on Friday, October 4. Based on the results of the quiz, I will give you a non-binding recommendation about whether you should continue in the class. The quiz does not count toward your course grade.

Course description: 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. The course can serve preparation for the Putnam Mathematical Contest. See the departmental Putnam web site for more information.

Topics to be covered are: Induction, the pigeonhole principle, inequalities, modular arithmetic, closed form evaluation of sums and products, problems in combinatorics, probability, geometry, rational functions and polynomials, and calculus.

Textbook: Loren C. Larson, Problem Solving Through Problems, Springer, 2006. Also recommended is (AG) Titu Andreescu and Razvan Gelca, Putnam and Beyond, Springer, 2007, available online. Other resources can be found at http://www.math.ucla.edu/~cm/putnam.html

Grading: 50% final, 30% midterm, 20% homework. The final grades will be Pass or No Pass.

Exams: There will be one in-class midterm and one final. The midterm will take place on Friday, November 1, during class. The final will be on Wednesday, December 11, 8am-11am. No make-up exams will be given, except in case of a documented emergency. No books, notes or calculators will be allowed on the exams.

Office hours during the last week of classes are: Thursday (Dec.5) 11-12 and 2-3 and Friday (Dec.6) 10-11. I will be out of town during final week, but you can email me questions.

Homework: Homework will be assigned every week and will be due in class on Friday. The lowest homework score will be dropped.

The homework assignments will be posted on the web page. You are encouraged to talk about the problems with other students, but you should write up the solutions individually. You should acknowledge the assistance of any book, student or professor. No late homework will be accepted.

Syllabus (subject to change):

Week Topic Read   Homework Due
Sep.27-Oct.3 Induction and Pigeonhole Larson Ch.2, AG Ch.1 Larson 2.1.6, 2.1.9, 2.1.10, 2.2.4, 2.3.3 (a), 2.6.6, 2.6.7, 2.6.9, 2.6.10, 2.6.11 (a). Solutions Oct. 4
Oct.4-9 Inequalities Larson Ch.7.1-7.4, AG Ch.2.1 Larson 7.1.7, 7.1.11, 7.2.6, 7.2.7, 7.2.9(b)-(c), 7.2.11, 7.3.7(a)-(b), 7.4.10, 7.4.11, 7.4.17. Solutions Oct. 11
Oct.11-16 Number theory Larson Ch.3, AG Ch.5 Larson 3.1.10, 3.1.12, 3.1.13, 3.1.14, 3.2.11, 3.2.14(c), 3.2.15 (b)-(c), 3.2.23, 3.3.26, 3.4.7. Solutions Oct.18
Oct.18-23 Algebra Larson Ch.4, AG Ch.2.2, 2.4 Problem Set 4. Solutions Oct.25
Oct.25-30 Summation of series Larson Ch.5.2, 5.3, AG Ch.3.1 Larson 4.4.15, 4.4.26, 4.4.30 (a)-(b), 5.2.7, 5.2.8, 5.2.10, 5.2.11 (a) [Typo: Set k=i], 5.3.6, 5.3.7 (a) [Typo: Sum should start at n=2], 5.3.9 (b). Solutions Nov.1
Nov.1-6 Combinatorics Larson Ch.1.3, 5.1, AG Ch.6 Larson 1.3.9, 1.3.10, 1.3.12, 2.5.13(a), 5.1.9(a)-(d), 5.1.12 [Typo in 5.1.12(a): The sum should end with the term (r choose s-n)(s choose s)]. Solutions Nov.8
Nov.8-13 Recurrent sequences, probability Larson Ch.5.4, AG Ch.6.2, 6.3 Problem Set 7. Solutions Nov.15
Nov.15-20 Geometry and trigonometry Larson Ch.8, AG Ch.4 Larson 8.2.6, 8.2.7, 8.2.10, 8.3.10, 8.3.11, 8.3.13, 8.3.18, 8.4.6, 8.4.7, 8.4.8. Solutions Nov.22
Nov.22-Dec.6 Calculus Larson Ch.6, AG Ch.3 Larson 6.1.4, 6.1.7, 6.2.4, 6.2.5, 6.3.5(b), 6.5.5, 6.7.3, 6.7.6, 6.8.4(c)-(d), 6.9.8. Dec.6