# First-Year Student Prize Exam

### The First-Year Student Prize Exam (formerly called the Freshman Prize Exam)
is generally given in March or April of every year.

## 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.

And Brief Sample Problem Solution Notes.

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?

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 .

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
.

How many 0's does end in?

Find the sum of all positive integral divisors of .

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?

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 .)

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.

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

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 .

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

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?

Prove the inequality

Show that
exists and is finite.

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

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*