Lower Division Problems

1. It is well known that . Find a similar expression for
.

2. Two datasets are “statistically equivalent” if they have the same number of data points, the same mean, and the same standard deviation. Find all unique datasets of positive integers that are statistically equivalent to . (Note: would not be unique – at least one number must be different)

Upper Division Problems

1. One hundred armed rats enter a bar containing one hundred fat cats. After the cats are lined up against the wall, the first rat marches down the line, taking one dollar from every cat. On his return, he passes the $100 that he collected to the second rat who proceeds to give one dollar to every second cat. On his return, he gives the remaining $50 to the third rat, who collects one dollar from every third cat. This continues with the rats alternatively giving and taking until the 100th rat gives one dollar to the 100th cat. At this point, the rats and the money they have taken disappear in a cloud of smoke.

a) How many cats profited? How many lost money? How many broke even?
b) Which cat(s) profited the most and how much did they get?
c) How much money did the rats leave with?

2. Prove that if one of the coordinates of the center of a circle is irrational, then the circle contains at most two points both of whose coordinates are rational.