This document is one part of the application to the Ross Mathematics Program,and will remain posted at https://rossprogram.org/students/to-apply from January through March. The deadline for applications is March 15, 2024. The Admissions Committee will start reading applications on March 16.
You are not expected to answer every question perfectly; rather, take this application as an opportunity to explore some beautiful mathematics! We are interested in seeing how you approach unfamiliar, open-ended math problems, and we encourage you to write up your discoveries and conjectures, even if you can’t prove them.
We believe that the most valuable part of a problem is the time spent thinking on it, and your application should reflect this: we are not looking for quick answers written in minimal space. Instead, we hope to see evidence of your explorations, conjectures, proofs, and generalizations written in a readable format. If you make progress on these four problems (even if you don’t solve them completely), we encourage you to submit
your Ross application.
Submit your own work on these problems. If you’ve seen one of the problems before (e.g. in a class or online), please include a reference along with your solution.
Admission factors include the quality of mathematical exposition and the questions you pose, as well as the completeness and correctness of your solutions to those questions.
Work on the 2024 Application Problems which you can find at https://raw.githubusercontent.com/rossprogram/rossprogram.github.io/master/students/application-problems.pdf. There are four problems. Upload your carefully written solutions to those four problems.
- The solution to each problem must be uploaded as a separate PDF.
- Use the PDF file format. If your solution file is in some other fomat, please transform it to a PDF, check that the converted file is readable, and then upload that PDF.
- We welcome solutions that have been prepared with LaTeX or some similar typesetting program that produces PDFs. It is OK to type up your work using a word processor (like Microsoft Word), and then converting that file to a PDF. Please verify that the mathematical symbols you use are readable.
- You are allowed to write up your solutions by hand, provided you use black ink or very dark pencil. Scan each page, combine the pages for one problem into a single PDF, check that the file is readable, and then upload that file. Please be sure to reduce the file size to be less than five megabytes. The Ross system cannot accept any files of larger size.
The Admissions Committee is not looking for quick answers written in minimal space, but rather for readable mathematical expositions that includes evidence of your explorations, conjectures, and proofs.
Problem 1
A polynomial is integral when it has integer coefficients. The square root of 2 is a solution to the integral polynomial equation $x^2-2=0$.
A number is rational when it can be expressed as $\frac{a}{b}$ for integers $a$ and $b$ (with $b \neq 0$ ).
A number is irrational when it is not rational.
(a) Suppose $c$ is a non-square integer. (That is, $c \neq n^2$ for any $n$.) Explain why $\sqrt{c}$ is not rational. Similarly, if $c$ is a non-cube integer, does it follow that $\sqrt[3]{c}$ is irrational?
(b) Find an integral polynomial equation that has $\alpha=\sqrt{3}+\sqrt{5}$ as a solution. Show that $\alpha$ is irrational.
(c) Let $a$ and $b$ be integers. Find an integral polynomial equation which has $\sqrt{a}+\sqrt{b}$ as a solution. Must $\sqrt{a}+\sqrt{b}$ be irrational? If $a \neq b$, must $\sqrt{a}-\sqrt{b}$ be irrational?
(d) Formulate some generalizations. As a starting point, is $\beta=\sqrt{3}+\sqrt{5}+\sqrt{7}$ irrational? What about numbers like $\gamma=3 \sqrt{2}-2 \sqrt{3}-3 \sqrt{5}+\sqrt{6}$ and $\delta=\sqrt[3]{5}-\sqrt{2}$ ?
Problem 2
Here’s a (paraphrased) conversation that took place between two Ross students.
A: Hey, want to see a magic trick?
B: Sure, how does it go?
A: Think of any number. Any nonnegative integer, I should say.
B: Okay.
A: Multiply it by 3 .
B: Okay….
A: Now divide it by 2, and if you get a decimal, then round down. Tell me if you rounded down or not.
B: I did have to round down.
A: Multiply it by 3 then divide it by 2 again. Again, tell me if you rounded down.
B: Okay, hang on… I did not have to round down this time.
A: Great, now just tell me: how many times does 9 go into this last number?
B: You want me to divide by 9 ?
A: Yes, just the quotient.
B: Um, the quotient is 4 .
A: So your original number was 19 .
B: That’s right! How did you do that? Let me think….
(a) Figure out how A’s magic trick works, and write up a clear, mathematical explanation of how to perform it.
(b) What variants of this trick can you come up with using the same principles? Can you change the numbers 2,3 , and 9 in the trick, or maybe the operations involved? What about the information you ask for?
问题 1
当一个多项式有整数系数时，它就是积分多项式。2 的平方根是积分多项式方程 $x^2-2=0$ 的解。
当一个数可以用 $ rac{a}{b}$ 表示整数 $a$ 和 $b$（其中 $b
等于 0$ ）。
当一个数不是有理数时，它就是无理数。
(a) 假设 $c$ 是一个非平方整数。即
eq n^2$ for any $n$）。请解释为什么 $\sqrt{c}$ 不是有理数。同样，如果 $c$ 是一个非立方整数，那么 $\sqrt[3]{c}$ 是不是无理数呢？
(b) 找出一个以 $lpha=\sqrt{3}+\sqrt{5}$ 为解的积分多项式方程。证明 $lpha$ 是无理数。
(c) 设 $a$ 和 $b$ 为整数。求一个以 $\sqrt{a}+\sqrt{b}$ 为解的积分多项式方程。$sqrt{a}+\sqrt{b}$ 一定是无理数吗？如果 $a
eq b$，那么 $\sqrt{a}-\sqrt{b}$ 一定是无理数吗？
(d) 提出一些概括。作为一个起点，$ea=\sqrt{3}+\sqrt{5}+\sqrt{7}$是无理数吗？那么像 $\gamma=3 \sqrt{2}-2 \sqrt{3}-3 \sqrt{5}+\sqrt{6}$ 和 $\delta=sqrt[3]{5}-\sqrt{2}$ 这样的数呢？
问题 2
下面是罗斯大学两个学生之间的一段对话（转述）。
A: 嘿，想看魔术吗？
乙：当然，怎么变？
甲：随便想一个数。应该说是任何一个非负整数。
乙：好的。
甲：乘以 3 。
乙：好的….。
甲：现在除以 2，如果是小数，就四舍五入。告诉我你有没有四舍五入。
乙：我必须四舍五入。
甲：乘以 3，然后再除以 2。同样，告诉我你是否四舍五入了。
乙：好的，等一下……这次我不用四舍五入。
甲：很好，现在告诉我：最后这个数是 9 的几倍？
乙：你要我除以 9？
甲：是的，只要商。
乙：嗯，商是 4。
甲：所以你原来的数字是 19 。
乙：没错！你是怎么算出来的？让我想想….
(a) 找出 A 的魔术是如何变出来的，并写出清晰的数学解释。
(b) 利用同样的原理，你还能变出什么魔术？你能改变魔术中的数字 2、3 和 9 吗？你要求的信息是什么？
Ross数学夏令营2024选拔代写
斯坦福大学数学夏令营保录取Sumac代写2023 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。
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