Math Club Problems

Welcome to the SUNY Brockport Math Club!

This is our problem page- if you have any good problem to contribute or want to check your solutions please contact Dr. Skogman.

1) We color the points of the plane in 2 different colors. Show that no matter how the coloring is done there are two points of the same color exactly one mile apart.
2) We color the points of the plane in two colors. Show that no matter how the coloring is done there is an equilateral triangle having all vertices of the same color.
3) We color a line using two colors. Show that we can always find 2 points of the same color such that the middle point also has the same color.
4) Show that it is possible to color the plane in four colors such that there are no two points of the same color one mile apart.
5) We color the points of the plane in three colors - Show that no matter how the coloring is done, there are two points of the same color one mile apart.
6) Solve the following equation: (1/6)(6^x+6^(-x))=2.
7) Show that there exists an infinite number of angles between 0 and Pi/2 whose sine and cosine are rational.
8) 100 ants on a meter stick begin traveling to the right or left at one meter per minute. Colliding ants instantaneously reverse direction; when an ant reaches either end of the stick it falls off.What is the longest amount of time one must wait to be sure that no ants remain?
9) 100 ants are placed uniformly randomly on a meter stick. An extra ant Alice is placed exactly in the middle of the stick. Each ant begins moving randomly right or left at one meter per minute and reverses direction on collision or if they reach the end of the stick. What is the probablility that Alice is exactly at the center of the one meter stick?
10) (Proposed by Dr. Michaels) We take a normal deck of 52 cards and flip them over one by one scoring one point every time we flip two cards of the same rank over consecutively, two points for every 3 consecutive cards of the same rank, and 3 points if 4 of the same rank are flipped consecutively. What is the expected value of the total number of points from flipping through the deck once?

For more information on joining the Math Club contact Dr. Howard Skogman.