CM701/702 – Contest Math Level 7 (Advanced – Olympiad Math)

Description: This is a TWO-YEAR (four semesters) class. It 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. CM701/702 helps students improve their USA(J)MO scores.  This class is proof-based (not computational). If the student’s main goal is to improve AIME scores, he/she should take 601/602 instead.

Who should take this course? Students who wish to enroll this class are expected to have qualified for the USA(J)MO in the past.

Homework:
Mandatory homework: One Olympiad problem per week. Homework needs to be submitted in pdf format (in Canvas). It will be graded by the instructor and the feedback will be provided to the students.

Optional homework: Additional homework problems will be assigned. Solutions will be provided. These problems will not be graded.

Who should skip this course: Students who are USAJMO Winners or who scored 28+ on USAMO

What will be covered (Topics repeated every 2 years):
Year 1: 701A/701B
Year 2: 702A/702B

Topics:

Fall 2019:  701A (12 weeks)

Week 1-7 GEOMETRY

Week 1 Angle chasing; cyclic quads; the incenter-excenter lemma; phantom points
Week 2 Power of a point; radical axes and radical center
Week 3 Extended law of sines; Ceva and Menelaus theorems
Week 4 Homothety; the nine-point circle; Diameter of Incircle
Week 5 Simson lines; isogonal conjugates; symmedians
Week 6 Mixtilinear incircles; the HM point on the median
Week 7 Cartesian coordinates; trig bash; Ptolemy, Casey
Week 8 Spiral similarity and Miquel’s theorem

Week 9-12 NUMBER THEORY

Week 9 Orders Modulo a Prime; Primitive roots; Cyclotomic polynomials
Week 10 Euler, FLT and Wilson Theorems
Week 11 Diophantine Equations (Linear and Quadratic)
Week 12 Quadratic Residues and Legendre Symbol

Spring 2020: 701B (12 weeks)

Week 1-6 COMBINATORICS

Week 1 Algorithms and Invariants I (Greedy algorithm)
Week 2 Algorithms II (Various algorithms)
Week 3 Processes I (Bounding number of steps)
Week 4 Processes II (Vieta jumping; infinite descent)
Week 5 Games and optimal strategies
Week 6 Counting in Two Ways

Week 7-12 ALGEBRA

Week 7 Algebraic Inequalities I (Cauchy Schwarz; Jensen)
Week 8 Algebraic Inequalities II (Smoothing; Isolated fudging; and more)
Week 9 Algebraic Inequalities III (n-1 EV; Tangent line)
Week 10 Functional Equations I (Pointwise trap; Additive and multiplicative Cauchy-type FE’s)
Week 11 Functional Equations II (Injection; involutions)
Week 12 Functional Equations III (Recursive)

Fall 2020: 702A (12 weeks)

Week 1-7 GEOMETRY
Week 1 Geometry with Complex numbers I
Week 2 Geometry with Complex numbers II
Week 3 Inversion 1 (Inverting the incircle; Orthogonal circles)
Week 4 Inversion 2 (root BC inversion)
Week 5 Projective Geometry I (Cross ratios; Harmonic bundles; Apollonia circles)
Week 6 Projective Geometry II (Poles and Polars; Brocard’s Theorem; Pascal’s theorem)
Week 7 Gauss Bodenmiller theorem; Miquel points of cyclic quadrilaterals

Week 8-12 NUMBER THEORY
Week 8 Divisibility; Perfect squares and cubes
Week 9 Arithmetic Functions I
Week 10 Arithmetic Functions II
Week 11 Floor function and Fractional part
Week 12 Bases and decimal representations

Spring 2021: 702B (12 weeks)

Week 1-6 COMBINATORICS

Week 1 One-to-one Correspondences (Bijections)
Week 2 Extremal Combinatorics
Week 3 Graph Theory I (Eulerian and Hamiltonian paths)
Week 4 Graph Theory II (Bipartite graphs)
Week 5 Graph Theory III (Special topics)
Week 6 The probabilistic method

Week 7-12 ALGEBRA

Week 7 Geometric Inequalities I (Euler and Leibniz)
Week 8 Geometric Inequalities II (Erdos Mordell inequality)
Week 9 Geometric Inequalities III (Optimization)
Week 10 Functional Equations I (Substitutions; symmetry)
Week 11 Functional Equations II (Fixed points)
Week 12 Functional Equations III (Polynomial equations)

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