CM601/CM602 – Contest Math Level 6 (AIME Advanced)

This is a TWO-YEAR (four semesters) class. It is is intended for students who have been active in math competitions for a few years and are willing to improve their scores by learning more in depths topics and techniques. CM 601/602 helps students improve their scores on AMC 10/12 and AIME.

Who should take this course? Students who wish to enroll this class are expected to have 7+ AIME score and/or have completed our CM501 course for both fall and spring semesters.

Homework: A 5-6 question problem set will be assigned by the end of each class. Students are expected to spend about 2-3 hours per week outside the class to finish the homework.

What will be covered:
Topics will be repeated every 2 years.
Year 1: 601A/601B
Year 2: 602A/602B

Who should skip this course: Students who scored 12+ on AIME

Topics are listed below:

Year 1 Fall (12 weeks): 601A

Week 1 Angles; Similarity of triangles;
Week 2 Pythagorean theorem, Length and area; Angle bisector theorem; Stewart theorem
Week 3 Cyclic quadrilaterals; Ptolemy theorem
Week 4 Law of Cosines; Law of Sines; applications of trigonometric identities
Week 5 Circles; Power of a point; Radical axes;
Week 6 Geometric transformations (reflection and rotation)
Week 7 Adv. topics I: Incenter-Excenter Lemma

Week 8 Euclidean algorithm; Prime factorization; powers of primes (perfect squares and cubes)
Week 9 Order modulo a prime; Fermat’s Little theorem
Week 10 Euler’s theorem; Arithmetic functions
Week 11 Congruences; Chinese remainder theorem
Week 12 Chicken McNugget theorem; Bezout theorem

Year 1 Spring (10 weeks): 601A

Week 1 Principle of Inclusion and Exclusion (PIE)
Week 2 Counting in Two Ways (Double Summation)
Week 3 Recursions (incl. Fibonacci/Catalan numbers)
Week 4 Expected Value and Linearity of Expectation
Week 5 Events with States (State diagrams)

Week 6-10 ALGEBRA
Week 6 Solving system of equations; algebraic manipulations
Week 7 Functions review and special functions
Week 8 Exponents and logarithms
Week 9 Polynomials I Division and Factor Theorem
Week 10 Polynomials II Vieta Formulas

Year 2 Fall (12 weeks): 602A

Week 1 Ratio lemma; Ceva theorem; Menelaus theorem
Week 2 Homothety (dilation); Nine-Point Circle; Special triangles
Week 3 Coordinates (distance formula; etc.)
Week 4 Solid (3D) geometry (dihedral angle, volume, …)
Week 5 Geometric inequalities and optimization (Leibniz and Euler inequalities)
Week 6 Adv. topics II: Simson line; Spiral similarity
Week 7 Adv. topics III: Projective Geometry

Week 8 Solving Diophantine equations
Week 9 Modular Arithmetic; Wilson’s theorem
Week 10 Quadratic Diphantine equations; Pell-type equations
Week 11 Legendre’s Function; lifting the exponent
Week 12 Quadratic residues; Legendre Symbol

Year 2 Spring (10 weeks): 602B

Week 1 Combinatorial identities
Week 2 1-1 Correspondences (Bijections)
Week 3 Distribution problems; Stirling numbers
Week 4 Conditional probability and Bayes’ formula
Week 5 Graphs; cycles; permutations

Week 6-10 ALGEBRA
Week 6 Quadratic Equations; conic sections; graphing parabola, hyperbola, and ellipse
Week 7 Complex Numbers and Roots of Unity
Week 8 Inequalities and Cauchy Schwarz Inequality
Week 9 Optimization: Maxima and minima
Week 10 Arithmetic and Geometric Sequences and Series