# 藤校学霸们都参加的美国四大数学顶尖夏校——ROSS数学营

67%的学生进入哈耶普斯麻

《藤校学霸们都参加的美国四大数学顶尖夏校——SUMaC斯坦福大学数学营》

《藤校学霸们都参加的美国四大数学顶尖夏校——PROMYS青少年科学家数学项目》

ROSS数学营

ROSS于1957年在圣母大学创立，并于1964年起与俄亥俄州立大学联合举办，是美国三大数学营（另外两个是PROYMS，斯坦福数学营）之首，在数学圈内名声极大。

ROSS数学营每年只接受40名第一年参与的学生以及10-15名青年辅导员（junior counselors)。其中青年辅导员是上一年在项目中表现优异，想重返参加的学生。

6人哈佛大学

5人麻省理工

3人耶鲁大学

2人普林斯顿

1人宾夕法尼亚大学

Ross数学测试有4道提供，每道题又包含很多小问题。注重引导申请者发现规律。小题目由浅入深，理论知识越来越难，需要总结的规律层次也越深。罗斯题目很开放，没有固定答案，在做题过程中可以得出很多结论，申请者需要证明自己的结论，不论正确与否。

Ross数学营的申请难度极大,招生比例不超过10%。Ross美国营每年只招60位新学员,中国学生的录取率则更低。所以充足的准备，是一定不能少的，毕竟进入Ross数学营相当于半只脚已经踏入了常春藤名校！

Euclid’s Algorithm.
Greatest common divisor. Diophantine equation ax + by = c.
Proof of unique factorization in Z.
Modular arithmetic.
Inverses. Solving congruences. Fermat’s Theorem. Chinese Remainder Theorem.
Hensel’s lemma for solving congruences (mod pm).
Binomial coefficients.
Pascal’s triangle. Binomial Theorem.
Arithmetic properties of binomial coefficients, like: (x+y)p = xp + yp (mod p).
Polynomials.
Division algorithm, Remainder Theorem, number of roots.
Polynomials in Zp[x]. Irreducibles and unique factorization.
Z[x] and Gauss’s Lemma.
Cyclotomic polynomials.
Orders of elements.
Units. The group Um. Computing orders.
Cyclicity of Up. For which m is Um cyclic?
Legendre symbols. Euler’s criterion. Gauss’s fourth proof of Reciprocity.
Jacobi symbols.
Continued fractions.
Computing convergents. |x – p/q| < 1/q2.
Best rational approximations. Pell’s equation.
Arithmetic functions.
phi(n), tau(n), sigma(n), and mu(n). Multiplicative functions.
Sum of f(d) as d divides n. Moebius Inversion.
Convolutions of functions.
Gaussian integers: Z[i].
Norms. Which rational primes have Gaussian factors? Division algorithm.
Unique factorization. Fermat’s two squares theorem.
Counting residues (mod a+bi).
Finite fields.
Characteristic. Frobenius map. Factoring xpn – x.
Counting irreducible polynomials.
Uniqueness Theorem for the field of pn elements.
Resultants.
Discriminant of a polynomial and formal derivatives.
Resultant of two polynomials and relation with Euclid’s algorithm.