Freshman Prize Exam

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

Freshman Prize Winners for 2007-2008

First Prize:
Kyu Seob Kim
Second Prize Tie:
Pakawat Phalitnonkiat and Michael Shu

The 2008 Freshman Prize exam will be given on Wednesday April 9 at 5:45 in Malott 253. (1 1/2 hours)

Here are the 2006, 2007, and 2008 exams with solutions:

Cash prizes for the top three finishers in 2008 (as well as for the top five finishers in 2007) totaled $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 samples. (Here's a pdf file with the sample questions.)

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.

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, John Hubbard, Etienne Rassart, Erin Nevo, or Allen Back.
E-mail to all four:
putnam@math.cornell.edu

Last Update: May 5, 2008