# Competitions and Prizes

All Yale first-year and second-year students are eligible to take a competitive exam given at the end of the spring term. The **Benjamin F. Barge** prizes are distributed amongst the highest scorers: three prizes for first-year students, and three for second-year students.

#### More information about the 2022 Barge exam

The 2022 Barge exam will take place on Sunday, May 1, from 1pm to 4pm. Location is still to be determined, and will be posted on this site.

Signup for the test is through an online form. The deadline to sign up is April 21.

If you have been granted testing accommodations through the SAS, please forward the official SAS notification for this semester to math.dus@yale.edu by April 21.

The exam is similar in style to competitions for high school students such as the International math olympiad, or contests for undergraduate students such as the Putnam exam. (Both sites provide sample problems and solutions.)

Recommended mathematical background includes familiarity with mathematical proofs, calculus, some knowledge of probability. All solutions must be fully justified.

A few sample questions from past years:

1. Fix a real number x. Define a sequence by a_{0 }= 1, and a_{n} = (1+x^(2^n)) a_{n-1} for all n > 1. For what values of x does this sequence converge? When it converges, what is the limit?

2. Let p be a prime and k a positive integer with k < p. How many k-element subsets {a_{1}, …, a_{k}} of {1, …, p} are there such that a_{1} + … + a_{k} is divisible by p?

3. Determine all (if any) integers a, b, c satisfying the following equation:

a^{2c} + b^{4c} = 2011

4. Suppose you have an equilateral triangle with vertices ABC. Let P be a point in your triangle, with |PA|=5, |PB|=4, and |PC|=3. Determine the area of the triangle with vertices PBC.

5. Let a and b be natural numbers. For each natural number n, let p_{n} be the greatest prime factor of an + b. Prove that there is a number M such that p_{n} < M for infinitely many values of n.

**John Alan Lewis Summer Research Fellowship**will be awarded to an undergraduate student majoring in mathematics who wishes to pursue their studies over the summer.

#### More information about the Lewis prize

For 2022: Interested students should submit their proposals to the DUS by April 21. The proposal should describe an independent reading or research project which the student will undertake with the approval and guidance of a faculty member.

Please note that reading projects must be on topic(s) that are currently not taught in Yale classes. (There can be some small overlap - but we are not permitted to have an official reading project on the contents of Math 380, for example.) The same rule applies to Math 470 during the semester, and YSS independent research course during the summer.

Your proposal should include:

- Your name, and name of the faculty member who agreed to advise your project.
- Typically, a Lewis project would mean two months of full-time work. It could be a little shorter and more intensive, or a bit longer and less intensive. Please provide information about your plan in the proposal.
- A description of the project that you plan to work on. (There is some flexibility, to follow the project where it leads you in real time, but the proposal should describe the topic(s) and initial plan.)
- Information about relevant text(s), paper(s), or other references.

You must consult with your project adviser about the details of your proposal, especially the timeline (to make sure it fits their schedule) and the contents of your project. The adviser’s approval is an essential component of our selection process.

The **Anthony D. Stanley** , **George Beckwith** and **Deforest** prizes are divided annually among students chosen by the Department.

#### More information about the three prizes

General note: Each student is eligible to win one prize only, even across different years.

**ANTHONY D. STANLEY MEMORIAL** (1904). Walter S. Pitkin, B.A. 1858, in memory of his uncle, Professor Anthony D. Stanley, B.A. 1830. Awarded for excellence in pure and applied mathematics. (This prize is typically awarded to a junior mathematics major.)

**GEORGE BECKWITH** (1926). Dr. Glover C. Beckwith-Ewell of Hartford, Connecticut, in memory of his grandfather, George Beckwith, publisher of Beckwith’s Almanac. For the undergraduate most proficient in some branch of astronomy or mathematics.

**DEFOREST**(1855). Established by John DeForest, B.A. 1826, and increased in 1886 by his son, Erastus L. DeForest, B.A. 1854. Awarded to two seniors for proficiency in pure and applied mathematics.

The **William Lowell Putnam Mathematical Competition** is a U.S. and Canada-wide undergraduate competition. The competition consists of two three-hour sessions generally held on the first Saturday in December.