Course
Syllabus
Ohio
Northern University
College of Arts
and Sciences
Department of
Mathematics
Date: Fall
2005
Course Math-
285 Name: Mathematical Problem Solving
Credit hours: 1
Lecture
hours/week: 1 Lab hours/week 0
Instructor: Mihai
Caragiu
Usual student level Sophomore
Course required of students in Mathematics
Course frequency per year Fall
Quarter
Average enrollment per year 10
This course has a prerequisite none
This course is a prerequisite for none
Catalogue
description
A seminar on mathematical problem
solving. Intended for students who
enjoy solving challenging
mathematical problems. Various techniques and
strategies are used in the problem solving.
Course
Objectives
To
expose the student to fundamental set of mathematical problems, problem
types and techniques of
problem-solving from various mathematical areas, including identities
and inequalities, combinatorics,
elementary number theory, probability and geometry.
Textbook: Variable. Some possible choices:
Geometry Revisited, by H.S.M. Coxeter and S.L.Greitzer, MAA, 1967
Challenging Mathematical Problems with Elementary Solutions by
A.M.Yaglom and I.M.Yaglom, vols. I and II, Dover 1987
Mathematical Quckies, by Charles W. Trigg, Dover 1985
Mathematical Olympiad Treasures, by T.
Andreescu and B. Enescu, Birkhauser (2003)
The Green Book of Mathematical Problems, by K. Hardy and K.S. Williams,
Dover 1997
The Red Book of Mathematical Problems, by K. Hardy and K.S. Williams,
Dover1996
Course Outline
An
inventory of basic identities
An inventory of basic techniques of proof.
Classical counting problems.
Combinatorial identities.
Path counting and binomial coefficients.
Problems involving inclusion-exclusion principle.
Problems involving the pigeonhole principle.
Probability problems
involving combinatorial analysis.
Representations of
integers as sums and products.
Problems involving divisibility of integers.
Problems involving
partitions and compositions of integers.
Problems involving congruences.
Problems involving sequences.
Problems involving Fibonacci and Lucas numbers.
Problems involving calculus.
Classical analytic inequalities
Using calculus to prove inequalities.
Problems in triangle geometry, special lines and special points in a
triangle.
Ceva and Menelaus
theorems
Concurrence and collinearity
problems.
Problems involving
geometrical transformations.
Inversions.
The nine-point circle.
Problems involving circles and the power of a point with respect to a
circle.
Discrete, combinatorial and convex geometry problems.