Syllabus

Syllabus for the Mobile Mathematics Circle

I. Graphs.

1. What Is a Graph?
2. The Degree of a Vertex. Isomorphism.
3. Connectivity and Cycles. Trees.
4. Eulerian Graphs.
5. Euler's Theorem.
6. Oriented Graphs.

II. Invariants.

1. Parity.
2. Colorings.
3. Remainders as Invariants.

III. Number Theory.

1. Prime and Composite Numbers.
2. Remainders and Euclid's Algorithm.
3. Decimal Representation and Divisibility Tests.
4. Diophantine Equations.
5. Congruence.
6. Fermat's Little Theorem.

IV. The Pigeonhole Principle.

1. Basic Pigeonhole.
2. Intermediate Pigeonhole
3. The Pigeonhole Principle in Geometry.
4. The Pigeonhole Principle in Number Theory.

V. Combinatorics.

1. Permutations and Combinations.
2. Pascal's Triangle and Newton's Binomial Theorem.
3. Strategies and Tactics of Counting.

VI. Inequalities.

1. Classical Inequalities
2. The AM-GM Inequality.
3. Cauchy and Chebyshev Inequalities.
4. The Triangle Inequality.
5. Transformations.
6. Induction and Inequalities.

VII. Games.

1. Fundamental Ideas.
2. Symmetry. Winning Positions.
3. A Strategy of Finding Winning Positions.

VIII. Induction

1. Process and Method of Induction.
2. MMI and Guessing by Analogy.
3. Classical Elementary Problems.

IX. Logical Problems.

X. Constructions and Weightings.

XI. Optimization Problems.

1. The Extreme principle.
2. Semi-invariant.