Classes

Intermediate Classes Syllabi:

Intermediate B Geometry Syllabus:

1. Introduction, Proofs of Basic Triangle centers. Review Basic geo theorem proofs
2. Quadrilaterals, Cyclic Quadrilaterals
3. Polygons, The Bigger Picture, Spiral Similarity
4. Circles
5. Trig in Geo
6. Locus
7. 3D
8. Transformations
9. Coordinates
10. Complex Numbers

Intermediate B Counting/Probability Syllabus

0. Casework Pset
1. Probability as Geometry
2. Probability as Counting
3. Probability as Algebra (States)
4. PIE Correcting for Overcounting
5. Symmetry, Burnside's Lemma
6. Sets, Distributions, Identities
7. Combinatorial Identities
8. Recursion
9. Expected Value
10. Combo Team Game

Intermediate Number Theory Syllabus:

1. Divisibility, Divisor Counting
2. Modular Arithmetic I
3. Modular Arithmetic II (Chinese Remainder Theorem)
4. Combinatorial NT, Legendre
5. Modular Arithmetic III
6. Factoring is Factoring
7. Diophantine Equations
8. Base Number Model, Polynomials in NT
9. Modular Inverses, Sequences
10. Fermat's Little Theorem, Euler's Theorem, Wilson's Theorem.
Other Topics: Inverses, Divisibility rules base b

Intermediate Alg/Miscellaneous topics (2022)

1. Burnside's Lemma
2. Graph Theory part 1
3. Graph Theory part 2
4. Graph Theory part 3
5. Inequalities part 1
6. Inequalities part 2
7. Inequalities part 3
8. Conditional Probability
9. constructions and locus?
10. Generating functions
11. catalan numbers.

Advanced class:
Combinatorics – Enumerative Combinatorics is the study of enumerating certain combinatorial objects. For instance, we might ask, in how many ways can we tile a 1 x n rectangle with squares and dominos? In how many ways can we partition the set {1,2,3, …, n} into sets of size 2 or 3?

Advanced Combinatorics: (Tentative syllabus)

  1. Induction Proofs
  2. Fibonacci interpretation
  3. More combinatorial identities, Probability
  4. Stirling Numbers of the First, Second Kind
  5. Ramsey Theory, (and possibly More Graph Theory)
  6. Permutations, Groups
  7. Pattern Avoidance (and possibly graph avoidance)
  8. Combinatorial Configurations
  9. Pigeonhole Principle, Invariants
  10. Partitions, Compositions, Ferrers Diagrams, Young Tableaux



Number Theory: Number Theory is the study of the integers. Questions we might ask are: Are there infinitely many primes? Are there infinitely many primes of the form 4n+1?

Number Theory: (Tentative syllabus)

6 Proofs of the Infinitude of the Primes.

More Advanced Congruences

Lucas Theorem, Kummer Theorem

Primality Tests, Carmichael Numbers

Pell Equations

Legendre symbol, Jacobi Symbol

Quadratic Reciprocity

Intro Analytic Number Theory, RSA

Intro Algebraic Number Theory, Diffie Hellman

Bertrand’s Postulate, Wrap up, discussion of modern development in prime gaps.