# First-Year Student Prize Exam

## Congratulations to David Connelly, Dylan DuBeau, Daniel Longenecker, Mikhail Molodyk, Shuhao Qing, Linus Setiabrata, and Jiazhen Tan for co-winning the 2017 First-Year Student Prize!

Here are the 2006-2017 exams with hints for solution:

Cash prizes for the top finishers often total around \$500!

The exam is limited to first year students, so questions generally do not assume Mathematics beyond first year calculus. A wide range of topics appear. Beyond calculus and analysis based questions, recent exams have had questions from areas such as probability, graph theory, number theory, combinatorics, algebra, and geometry. Exam length recently has been 5-7 questions to be answered in one and a half or two hours. Questions usually require ingenuity; some are theoretical, others computational.

Below are some older sample questions. Note that the actual exams with solutions for the last ten years are above, so they may be more indicative of recent exams.

Here's a pdf file with these sample questions.

#### Problem 1.

A box contains white balls and black balls. If balls are drawn from the box at random, what is the probability of drawing white balls and black ball?

#### Problem 2.

The lines normal to the parabola through the points , , intersect the -axis at a set of points . What is the largest value of that is NOT in the set .

#### Problem 3.

added 3/30/10: There is a typo in this problem. A right hand side like (csubn+2/csubn)/2 was intended.

Put and define . Find .

#### Problem 4.

How many 0's does end in?

#### Problem 5.

Find the sum of all positive integral divisors of .

#### Problem 6.

A rectangle with sides parallel to the axes is drawn in the first quadrant region bounded by the -axis and the curve . The rectangle is then rotated about the -axis to form a solid. What is the maximum possible volume of this solid?

#### Problem 7.

The UN invited diplomats for a dinner. Each of them has at most enemies. Show that you can seat them at a round table so that nobody sits next to an enemy. (We assume that being an enemy is symmetric: if is an enemy of , then is an enemy of .)

#### Problem 8.

Every point in the plane is colored either blue, red, or green. Show that there is a rectangle all of whose corners have the same color.

#### Problem 9.

The number 313726685568359708377 is the power of some number . Find .

#### Problem 10.

Starting with any real number , a sequence of numbers is defined by where is measured in radians. Show that

exists and is independent of the initial value .

#### Problem 11.

The sum of the positive integers is . What is the largest possible product that can be formed under this condition?

#### Problem 12.

A path from to in the figure is valid if it does not cross itself, and never moves downwards. How many valid paths are there?

#### Problem 13.

Prove the inequality

#### Problem 14.

Show that

exists and is finite.

#### Problem 15.

Let p be a prime number.
(a)
Show that divides .
(b)
Show that divides .

#### Problem 16.

Given are points in space: . Pick a point and then pick points such that for each the line segments and have common mid points.

Show that provided that the number of points is even.

Questions can be directed to the organizers: Inna Zakharevich, Florian Frick, , and Allen Back.
E-mail to all:
putnam@math.cornell.edu

Last Update: May 19, 2017