### 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 1-7 GEOMETRY**

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 8-12 NUMBER THEORY**

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 1-5 COMBINATORICS**

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 1-7 GEOMETRY**

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 8-12 NUMBER THEORY**

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 1-5 COMBINATORICS
**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