## Sufficient Condition to be a Polynomial?

Appropriate Level:  Real analysis.

The Question:  Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is infinitely differentiable and that for each $x \in \mathbb{R}$ there is some $n \in \mathbb{N}$ for which the $n$-th derivative of $f$ at $x$ equals 0.  Must $f$ be a polynomial?

## Cyclotomic Polynomials

Appropriate Level:  Abstract Algebra

The Question:  Recall that the cyclotomic polynomials $\Phi _n (x)$ are defined for $n \geq 1$ so that $\prod_{d|n} \Phi_d (x) = x^n -1$.  Is it true that every coefficient of every cyclotomic polynomial is either 0, 1, or -1?

Appropriate Level:  Trigonometry

Background:  Here is a wonderful photograph:

This picture was taken by an astronaut onboard the Apollo 17 spacecraft in 1972.  This was the last manned spaceflight to the moon and the last time someone was far enough away from earth to take a picture like this.

The Question:  When the astronaut took this picture, how far from earth was he?

Source:  The very nice blog:  http://sucolex.blogspot.com/

## Coloring the Coordinate Plane

Appropriate Level:  Geometry and Beyond

The Warm-up Question:  If every point of the plane is colored either red or blue, do there necessarily exist two points of the same color that are exactly 1 unit apart?

The Question:  If every point of the plane is colored either red, blue, or green, do there necessarily exist two points of the same color that are exactly 1 unit apart?

Source:  The main question is part (a) of problem A4 of the 1988 Putnam Exam

## Watching the Sunrise (and figuring the radius of the earth)

Appropriate Level:  Trigonometry

The Story:  I spent a summer in Chicago and early one morning I watched the sun rise over Lake Michigan (the lake is large enough that you cannot see across it, at least from Chicago).  I stood at the waters edge and waited for the first rays of sunlight to peak over the water.  When that happened I immediately laid down on the sand near the waters edge, and could no longer see the sun.  It took about 10 seconds before the “first” rays of sun were again visible. By the way, when I am standing, my eyes are roughly 5 ft above the ground.

The Question:  What is the radius of the earth?

## Cutting a Triangle from a Square

Appropriate Level:  Calculus (1 or 2, depending on how your courses are divided).

The Question: You start with a unit square (say a piece of paper).  Choose two adjacent sides of the square and randomly pick a point on each.  Connect the two points with a line and cut along that line, thus removing a triangle (it could be a degenerate triangle) from the piece of paper.  Let A be the area of the triangle that has been removed.

What is the smallest A can be?

What is the largest A can be?

What is the probability that A is less than 1/4?  In particular, is this probability equal to 50%? (after all, 1/4 is half way between 0 and 1/2).

If you wish, give them the following estimates:  Google says the mass of a proton is about $2 \times 10 ^{-27}$ kilograms.  The wikipedia page on the “Observable Universe” says it has a mass of about $2 \times 10 ^{53}$ kilograms.