Freshman Prize Exam

The Freshman Prize Exam is generally given in March or April of every year.

Congratulations to the Freshman Prize Winners for 2012-2013:

Louis Brown (1st)

and

Matthew Persons (2nd)

Here are the 2006-2013 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 seven 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.

Problem 1.

A box contains $ 7$ white balls and $ 8$ black balls. If $ 3$ balls are drawn from the box at random, what is the probability of drawing $ 2$ white balls and $ 1$ black ball?


Problem 2.

The lines normal to the parabola $ y=x^2$ through the points $ (x,x^2)$, $ x \neq 0$, intersect the $ y$-axis at a set of points $ Y$. What is the largest value of $ y$ that is NOT in the set $ Y$.


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 $ c_1=1$ and define $ c_{n+1}=c_n+\frac{1}{c_n}$. Find $ lim_{n \to \infty} c_n$.


Problem 4.

How many 0's does $ 2000!$ end in?


Problem 5.

Find the sum of all positive integral divisors of $ 9600$.


Problem 6.

A rectangle with sides parallel to the axes is drawn in the first quadrant region bounded by the $ x$-axis and the curve $ y=\frac{x}{x^2+1}$. The rectangle is then rotated about the $ x$-axis to form a solid. What is the maximum possible volume of this solid?


Problem 7.

The UN invited $ 2n$ diplomats for a dinner. Each of them has at most $ n-1$ 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 $ A$ is an enemy of $ B$, then $ B$ is an enemy of $ A$.)


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 $ 11^{\rm th}$ power of some number $ n$. Find $ n$.


Problem 10.

Starting with any real number $ x_0$, a sequence of numbers $ \{x_n\}_{n=0}^{\infty}$ is defined by $ x_{n+1}=\cos(x_n)$ where $ x_n$ is measured in radians. Show that

$\displaystyle \lim_{n \to \infty} x_n $

exists and is independent of the initial value $ x_0$.

Problem 11.

The sum of the positive integers $ \{ a_1, \ldots, a_r \}$ is $ a_1 + \ldots + a_r = 2005$. What is the largest possible product $ a_1 \cdot \ldots \cdot a_r$ that can be formed under this condition?


Problem 12.

A path from $ A$ to $ B$ in the figure is valid if it does not cross itself, and never moves downwards. How many valid paths are there?
\epsfbox{prob1_graph.eps}


Problem 13.

Prove the inequality

$\displaystyle \left( \frac{ a }{ b } \right)^{b} \leq \left( \frac{ a+1 }{ b+1 } \right)^{b+1} \qquad \text{\ for\ } a>0,\,\, b>0 . $


Problem 14.

Show that

$\displaystyle lim_{R \to \infty} \int_0^R \sin{(t^2)} \ dt $

exists and is finite.


Problem 15.

Let p be a prime number.
(a)
Show that $ p$ divides $ 2^p-2$.
(b)
Show that $ p^2$ divides $ 2^{p^2}-2^{p}$.


Problem 16.

Given are $ n$ points in space: $ A_0, A_1, \dots A_n=A_0$. Pick a point $ B_0$ and then pick points $ B_1, B_2, \dots B_n$ such that for each $ k=1,2,\dots, n$ the line segments $ A_{k-1}A_k $ and $ B_{k-1}B_k $ have common mid points.

Show that $ B_0 = B_n$ provided that the number of points $ n$ is even.


Questions can be directed to the organizers, Camil Muscalu and Allen Back.
E-mail to all:
putnam@math.cornell.edu

Last Update: April 16, 2013