MATH 189: PROBLEM SOLVING (WINTER 2007).
Course materials
All files are in the pdf format, which require the
free
Adobe Acrobat reader.
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Two sample problems and the
LaTeX file.
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Problem Set 1: Introductory Problems.
For class on Jan. 11: Turn in your work on problems 6, 17, 42 of Problem Set 1.
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Problem Set 2: Logic and Information.
For class on Jan. 16: Work on problems 4 and 6 from Problem Set 2, but do not
turn anything in. For class on Jan. 18: Turn in your work on problems 9 and 10 from Problem
Set 2.
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Problem Set 3: Pigeonhole Principle.
For class on Jan. 23: Read Chapter 22, Sections 1, 2, 3, and 5 in THE BOOK. Also work on
problems 6 and 7 from Problem Set 3, but do not turn anything in. For class on Jan. 25:
Turn in your work on problems 10 and 14 from Problem Set 3.
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Problem Set 4: Variational Method.
For class on Jan. 28: Read Chapter 9, Theorems 1 and 2, in THE BOOK, and
the introductory example to Problem set 4.
Also work on Problem 1 from Problem Set 4.
For class on Jan. 30: work on problems 2, 3, and 8 from Problem Set 4.
For class on Feb. 1: Turn in your work on problems 9 and 10 on Problem Set 4.
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Problem Set 5: Invariants.
For class on Feb. 4: Read Chapter 11 in THE BOOK, the parts on Euler's Formula and Pick's Theorem.
Work on problems 26 and 27 from Problem Set 1 and problems 1, 2, 3, 4 from Problem Set 5.
We will continue discussion on these on Feb. 6.
For class on Feb. 8:
Turn in your work on problems 9 and 10 from Problem Set 5.
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Problem Set 6: Inequalities.
For class on Feb. 11:
Read enough of Chapter 17 in THE BOOK and introduction to Problem Set 6 to make
yourself familiar with basic inequalities. We will not discuss this in class.
Work on problems 8, 19, and 21 from Problem Set 6 but do not hand anything in.
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Problem Set 7: Monotonicity.
For class on Feb. 13: Read the introductory example given in Problem Set 6.
Work on problems 4 and 5 from Problem Set 6, but do not turn anything in.
For class on Feb. 15: Turn in your work on problems 6 and 8 from Problem Set 7.
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Problem Set 8: Mathematical Induction.
For class on Feb. 20: Read Chapter 31 in THE BOOK.
Work on problems 5 and 9 from Problem Set 8, but do not
turn anything in.
For class on Feb. 22: turn in your work on problems 10, 11, and 12
from Problem Set 8.
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Problem Set 9: Geometry.
For classes on Feb. 25 and Feb. 27: Read Chapter 14 (Theorem 1 only) in THE BOOK.
Also, work on problems 5, 9, 10, 11, 12 from Problem Set 9, but do not turn anything in.
In problem 10, the first turn is the right turn.
For class on Feb. 29:
Turn in your work on problems 13 and 14 from Problem Set 9.
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Problem Set 10: Number Theory.
Read the following chapters from THE BOOK:
Chapter 2. A slightly more
condensed version of the proof is in
this Wikipedia article.
Chapter 3. At least skim a proof of the Sylvester-Schur theorem, e.g.,
this one.
Chapter 4. Look
here
for examples. Perhaps you will notice that the number of different representations
of a positive integer n as a sum of two squares equals to
4 (number of divisors of n which are 1 mod 3) -
4 (number of divisors of n which are 3 mod 3); this is the
Jacobi's Two-Square Theorem.
For classes on Mar. 3 and Mar. 5: work on problems 1, 5, 9, and 12 from Problem Set 10,
but do not turn anything in.
For class on March 7: turn in your work on problems 7 and 11 from Problem Set 10.
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Problem Set 11: Probability.
For classes on March 10 and 12: Work on problems 1, 5, 9, and 12 from Problem Set 11,
but do not turn anything in.
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Final Exam.
Final exam solutions.