PUMaC普林斯顿大学数学竞赛备赛辅导课程

Princeton University Mathematics Competition普林斯顿大学数学竞赛(简称:PUMaC)自2006年起由Princeton University Math Club普林斯顿大学数学俱乐部全权举办,并由普林斯顿大学的学生进行组织。

该竞赛是一项面向国际高中生的世界范围内最顶尖的数学学科竞赛,每年都将汇聚世界各地热爱数学的优秀中学生,共同分享数学学科的乐趣,并进行专业竞技,角逐各大奖项。

PUMaC旨在激励并培养高中生对数学学科的兴趣和热爱,让学生接触到创造性思维和严谨的应用技能。同时,帮助学生们享受解决问题的乐趣,并发现传统意义上枯燥和深奥问题中所蕴含的趣味。

PUMaC普林斯顿大学数学竞赛备赛辅导课程大纲

1、Number Theory

Essential Topics(1):Prime factorization

Extended Details (Divison A and Beyond):

—Number of factors

—Sum/Product of factors

—LCM and GCD

—Euclidean Algorithm and Bézout's Theorem

Essential Topics(2):Congruence and Modular Algebra

Extended Details (Divison A and Beyond):

—Chinese Remainder Theorem(CRT)

—Euler’s Theorem/Fermat's Little Theorem

—Wilson's Theorem

Essential Topics(3):Diphantine Equations

Extended Details (Divison A and Beyond):

—*Pell's Equation and Approximation Method

 

2、Algebra

Essential Topics(1):Resursive Sequences

Extended Details (Divison A and Beyond):

—Characteristic Equation Method

—Conjecture and Mathematical Induction Proof

Essential Topics(2):Functions and Equations

Extended Details (Divison A and Beyond):

—Gaussian/Floor function

—Functional Equations

Essential Topics(3):Inequalities and Extreme Value Problems

Extended Details (Divison A and Beyond):

—Cauchy inequality

—Jensen's inequality

—Weighted AM-GM Inequality

—Rearrangement Inequality and Chebyshev Inequality

Essential Topics(4):Polynomials

Extended Details (Divison A and Beyond):

—Fundamental Theorem of Algebra (Polynomial Factorization)

—Generalized Remainder's Theorem

—Rational Root Theorem

—Vieta's Theorem and Newton's Sums

Essential Topics(5):Complex numbers

Extended Details (Divison A and Beyond):

—De Moivre's Theorem and Roots of unity

—Rotation, Translation and Complex Vector Method

 

3、Geometry

Essential Topics(1):Basic Geometry (Triangles and Polygons)

Extended Details (Divison A and Beyond):

—The Law of Sines, The Law of Cosines

—Area Method and Heron's formula

—Centers of triangle

—Menelaus's theorem, Ceva's theorem, Stewart Theorem

Essential Topics(2):Circles

Extended Details (Divison A and Beyond):

—Inscribed and circumscribed polygon/Circle, Cyclic Quadrilateral

—Ptolemy's theorem, Butterfly Theorem, Simson's Theorem

Essential Topics(3):Basic Analytic Geometry

Extended Details (Divison A and Beyond):

—Ellipse, *Parabola and Hyperbola

Essential Topics(4):Basic Solid Geometry

Extended Details (Divison A and Beyond):

—*Euler's Polyhedron Formula

 

 4、Combinatorics

Essential Topics(1):Basic Counting Principle

Extended Details (Divison A and Beyond):

—Basic Counting Principle

Essential Topics(2):Permutations and Combinations

Extended Details (Divison A and Beyond):

—Advanced problems in combinatorics

Essential Topics(3):Logic reasoning

Extended Details (Divison A and Beyond):

—Pigeonhole principle

Essential Topics(4):*Baisc Graph Theory

Extended Details (Divison A and Beyond):

—Ramsey's Theory

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