竞赛营系列之——Ross 数学营

项目介绍

Ross数学营项目初始于1957年圣母大学，由Arnold Ross博士创办，并于1964年起与俄亥俄州立大学联合举办。Ross与“PROMYS”和“SUMAC”并称三大美国顶尖数学训练营,含金量非常高。获得Ross营的录取对于学生在大学申请中是极大的加分项。历来Ross营学员申请到哈耶普斯麻等美国顶尖名校的不在少数。高强度的Ross数学营旨在带领高中孩子探索数学之美。Think deeply about simple things！夏令营引导孩子们从极具创造力的角度思考他们闻所未闻的数学问题，带领孩子们学习他们从未见过的数学方法，培养并塑造孩子们的数学思维。数学教育的意义不仅仅在于获取计算能力，更在于通过数学，培养孩子们的批判思维。一个从来不会提出质疑的孩子将来不可能成为科学界的领头人，对于真正的科学人才来说，独立思考能力和批判质疑的态度是至关重要的，而这也是Ross数学营能够带给学生们的最重要，最核心的能力。Ross数学营的入学申请竞争非常激烈，通常，只有不到三分之一的申请人被录取。每个成功的申请者都有良好的高中成绩，并且在回答入学的数学测试问题上表现出色。2019年的入学测试题目请见文后。

举办地点

2019年Ross数学营在俄亥俄州立大学和中国江苏的大学校园进行。亚洲Ross数学营与美国数学营没有任何区别，学校将采用完全一致的形式和教学风格，所有的课程也都将用英语授课。

项目时间

Ross/美国	2019年6月23日周日—8月2日周五
Ross/亚洲	2019年7月7日周日—8月9日周五

课程设置

课程内容有：欧几里得算法、模运算、二项式系数、多项式、元素的阶、二次互反性、连分式、算术函数、高斯整数：Z[i]，有限域，结式、几何数论、二次数域等。每周8小时课时，包括5小时讲座和3小时研讨会；课余时间学生需要利用课上所学知识解决很多有挑战性的数学难题

招生对象

对数学与科学有浓厚兴趣的高中生，年龄15-18岁之间；

正常情况下，学校不会接收年龄太小或者太大的申请者；

学校会综合考虑申请者的在校成绩，教师评价及其学习目标，以决定是否录取，也会看申请者对于数学难题的解决能力。

学费

Ross/美国	5000美元
Ross/亚洲	35000人民币；如若需要，可以申请助学金

 申请条件

1. 高中成绩单
2. 两封推荐信
3. 个人陈述（学习兴趣与目标，需回答若干问题）
4. 数学测试(难度极大)5. 2019年1月开放申请，截止日期为4月1日

申请时间

3月1日开始接受申请，4月1日申请截止。此价格为夏校收取的学费，不包含：护照、签证费用、往返机票费用、保险费用、国际生费、申请服务费用、辅导员接送等费用，以及其它以上未提及的费用。

测试样题

Ross Program 2019 Application ProblemsPlease submit your own work on each of these problems.For each problem, explore the situation (with calculations, tables, pictures, etc.),observe patterns, make some guesses, test the truth of those guesses, and write logicalproofs when possible. Where were you led by your experimenting?Include your thoughts even though you may not have found a complete solution.If you’ve seen one of the problems before (e.g. in a class or online), please include areference along with your solution.We are not looking for quick answers written in minimal space. Instead, we hopeto see evidence of your explorations, conjectures, and proofs written in a readableformat.The quality of mathematical exposition, as well as the correctness andcompleteness of your solutions, are factors in admission decisions.Please convert your problem solutions into a PDF file. You may type the solutionsusing LATEX or a word processor, and convert the output to PDF format.Alternatively, you may scan or photograph your solutions from a handwritten papercopy, and convert the output to PDF. (Please use dark pencil or pen and write ononly one side of the paper.)*Note: Unlike the problems here, each Ross Program course concentrates deeply on one subject.These problems are intended to assess your general mathematical background and interests.

Problem 1 

What numbers can be expressed as an alternating-sum of an increasing sequence ofpowers of 2? To form such a sum, choose a subset of the sequence 1, 2, 4, 8, 16, 32, 64, . . .(these are the powers of 2). List the numbers in that subset in increasing order (norepetitions allowed), and combine them with alternating plus and minus signs. Forexample, 1 = −1 + 2; 2 = −2 + 4; 3 = 1 − 2 + 4; 4=−4+8; 5=1−4+8; 6=−2+8; etc. Note: The expression 5 = −1 − 2 + 8 is invalid because the signs are not alternating. (a)  Is every positive integer expressible in this fashion? If so, give a convincing proof.(b)  A number might have more than one expression of this type. For instance 3=1−2+4 and 3=−1+4.Given a number n, how many different ways are there to write n in this way?Prove that your answer is correct. (c)  Do other sequences (an) of integers have similar alternating-sum properties?Explore a sequence of your choice and make observations.One idea: Can every integer k be expressed as an alternating sum of an increasingsequence of Fibonacci numbers? Can some integers be expressed as such sums inmany different ways?Or you could explore some other sequence instead.

Problem 2 

A polynomial f(x) has the factor-square property (or FSP) if f(x) is a factor of f(x²).For instance, g(x) = x − 1 and h(x) = x have FSP, but k(x) = x + 2 does not.Reason:x−1 is a factor of x²−1,and x is a factor of x²,but x+2 is not a factor of x²+2.Multiplying by a nonzero constant “preserves” FSP, so we restrict attention to poly-nomials that are monic (i.e., have 1 as highest-degree coefficient).What patterns do monic FSP polynomials satisfy?
To make progress on this topic, investigate the following questions and justify youranswers.(a)  Are x and x − 1 the only monic FSP polynomials of degree 1?(b)  List all the monic FSP polynomials of degree 2.
To start, note that x², x² −1, x² −x, and x² +x+1 are on that list.
Some of them are products of FSP polynomials of smaller degree. For instance,x² and x² −x arise from degree 1 cases. However, x² −1 and x² +x+1 are new,not expressible as a product of two smaller FSP polynomials.
Which terms in your list of degree 2 examples are new?(c)  List all the monic FSP polynomials of degree 3. Which of those are new?Can you make a similar list in degree 4 ?(d)  Answers to the previous questions might depend on what coefficients are al-lowed. List the monic FSP polynomials of degree 3 that have integer coefficients.Separately list those (if any) with complex number coefficients that are not allintegers.Can you make similar lists for degree 4?
Are there examples of monic FSP polynomials with real number coefficients thatare not all integers?

Problem 3

 For a positive integer k, let Sκ be the set of numbers n > 1 that are expressible asn = kx + 1 for some positive integer x. The set Sκ is closed under multiplication.Thatis: Ifa,b∈Sκ thenab∈Sκ.Definition. Suppose n ∈ Sκ. If n is expressible as n = ab for some a, b ∈ Sκ, then nis called k-composite. Otherwise n is called a k-prime.For example, S₄ ={5,9,13,17,21,25,29,33,37,41,45,49,...}. The numbers25, 45, 65, 81, . . . are 4-composites, while 5, 9, 13, 17, 21, 29, . . . are 4-primes.

Which n ∈ S₄ are 4-primes? (Answer in terms of the standard prime factorization of n.)Show: Every n ∈ S₄ is either a 4-prime or a product of some 4-primes.But “unique factorization into 4-primes” fails. To prove that, find some
n = P₁P₂ ···Ps and n = q₁q₂ ···qt where each pj and qk is a 4-prime, but the list(q₁,...,qt) is not just a rearrangement of the list (P₁,...,Pr).

Which n ∈ S₃ are 3-primes? Is there unique factorization into 3-primes?

Suppose a positive integer k is given, along with its standard prime factorization. Which integers n ∈ Sκ are k-primes?
For which k does the system Sκ have unique factorization into k-primes?

Prove that your answers are correct.

Problem 4 

If S is a set of points in space, define its line-closure
L(S) = the union of all lines passing through two distinct points of S.That is: Point X lies in L(S) if there exist distinct points A, B ∈ S such that A, B, Xare collinear. Then S ⊆ L(S), provided S contains at least two points.For example, if points A, B, C do not lie in a line, then L({A, B, C}) is the union ofthree lines whose intersection points are A, B, C. In this case, L(L({A, B, C})) is thewhole plane containing those points.

1. When can a set S equal its own line-closure: L(S) = S ?Prove that your answer is correct.
2. Suppose A,B,C,D are points in 3-space that do not all lie in one plane. Describe the sets L({A, B, C, D}) and L(L({A, B, C, D})).
3. Suggest some ways that these ideas can be generalized.

We hope you enjoyed working on these problems! For more information aboutthis summer math program visit https://rossprogram.org/. You may email yourquestions and comments to ross@rossprogram.org.