競賽營系列之——Ross 數學營

項目介紹

Ross數學營項目初始于1957年圣母大學，由Arnold Ross博士創辦，并于1964年起與俄亥俄州立大學聯合舉辦。Ross與“PROMYS”和“SUMAC”并稱三大美國頂尖數學訓練營,含金量非常高。獲得Ross營的錄取對于學生在大學申請中是極大的加分項。歷來Ross營學員申請到哈耶普斯麻等美國頂尖名校的不在少數。高強度的Ross數學營旨在帶領高中孩子探索數學之美。Think deeply about simple things！夏令營引導孩子們從極具創造力的角度思考他們聞所未聞的數學問題，帶領孩子們學習他們從未見過的數學方法，培養并塑造孩子們的數學思維。數學教育的意義不僅僅在于獲取計算能力，更在于通過數學，培養孩子們的批判思維。一個從來不會提出質疑的孩子將來不可能成為科學界的領頭人，對于真正的科學人才來說，獨立思考能力和批判質疑的態度是至關重要的，而這也是Ross數學營能夠帶給學生們的最重要，最核心的能力。Ross數學營的入學申請競爭非常激烈，通常，只有不到三分之一的申請人被錄取。每個成功的申請者都有良好的高中成績，并且在回答入學的數學測試問題上表現出色。2019年的入學測試題目請見文后。

舉辦地點

2019年Ross數學營在俄亥俄州立大學和中國江蘇的大學校園進行。亞洲Ross數學營與美國數學營沒有任何區別，學校將采用完全一致的形式和教學風格，所有的課程也都將用英語授課。

項目時間

Ross/美國	?2019年6月23日周日—8月2日周五
Ross/亞洲	?2019年7月7日周日—8月9日周五

課程設置

課程內容有：歐幾里得算法、模運算、二項式系數、多項式、元素的階、二次互反性、連分式、算術函數、高斯整數：Z[i]，有限域，結式、幾何數論、二次數域等。每周8小時課時，包括5小時講座和3小時研討會；課余時間學生需要利用課上所學知識解決很多有挑戰性的數學難題

招生對象

對數學與科學有濃厚興趣的高中生，年齡15-18歲之間；

正常情況下，學校不會接收年齡太小或者太大的申請者；

學校會綜合考慮申請者的在校成績，教師評價及其學習目標，以決定是否錄取，也會看申請者對于數學難題的解決能力。

學費

Ross/美國	5000美元
Ross/亞洲	35000人民幣；如若需要，可以申請助學金

?申請條件

1. 高中成績單
2. 兩封推薦信
3. 個人陳述（學習興趣與目標，需回答若干問題）
4. 數學測試(難度極大)5. 2019年1月開放申請，截止日期為4月1日

申請時間

3月1日開始接受申請，4月1日申請截止。此價格為夏校收取的學費，不包含：護照、簽證費用、往返機票費用、保險費用、國際生費、申請服務費用、輔導員接送等費用，以及其它以上未提及的費用。

測試樣題

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, S? ={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 ∈ S? are 4-primes? (Answer in terms of the standard prime factorization of n.)Show: Every n ∈ S? 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 = P?P? ···Ps and n = q?q? ···qt?where each pj?and qk is a 4-prime, but the list(q?,...,qt) is not just a rearrangement of the list (P?,...,Pr).

Which n ∈ S? 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.