Chotw to Dec. 3, 2010

A fair die is rolled up to three times. At the end of each roll, you have the option to roll again or take the number showing in dollars. What is the optimal strategy that maximizes your expected earnings? What are the expected earnings?

Chotw to Nov. 12, 2010

Two people decide to meet for lunch between 12:00 and 1:00, but they don't set a fixed time. They randomly select a time independent of each other to arrive at the restaurant. Each of them will wait exactly 15 minutes for the other to arrive before leaving. What is the probability that they meet?

Chotw to Nov. 5, 2010

Can the sum of the reciprocals of a finite number of distinct positive integers be equal to 2009/2010? If yes, show such integers and if not, prove that such integers do not exist.

Chotw to Oct. 29, 2010

Four race car drivers participate in a race on a loop track. All four are going at a constant speed. Assume that they make a flying start. That is, all four crossed the starting line at the same instant while each was going their constant speed. Then they continue driving forever and it is the case that for any three of the cars there is a moment in time, after the start, when these three cars are located at the same point along the track (all three are passing each other). Prove that there is a moment in time, after the start, when all four cars are located at the same point along the track.

Chotw to Sept. 22, 2010

Two carts, A and B, are connected by a rope 28 ft long that passes over a pulley P (see the figure). The point Q lies 12 ft directly below P at the same height at which the ropes are attached to the carts. Cart A is being pulled away from Q at the rate of 2 feet per second. How fast is cart B moving towards Q at the instant when cart A is 9 ft away from Q?

Chotw to Oct. 15, 2010

Is it possible to put 12 positive integers along a circle so that the ratio of any two neighboring integers is a prime number? (We take the ratio of the bigger number to the smaller number.)

Is this possible with 13 positive integers?

Chotw to Oct. 1, 2010

The integer lattice points are the points in the plane that have integer coordinates.

Consider all squares in the plane with corners that are integer lattice points. For example, the points (0, 0), (1, 1), (2, 0), and (1, -1) form a square.

a) Is it possible for such a square to have area 51?

b) Find all possible areas of such squares that have less than area 51.

Chotw to Sept. 24, 2010

Some number of students, all of different heights, are lined up on a rectangular grid in rows and columns. The shortest students from each row forms a set that we will call the shortest students.

The tallest students from each column forms a set that we will call the tallest students. Then let T be the tallest student that is in the set of shortest students and let S be the shortest student that is in the set of tallest students. Which student is taller, T or S?

Chotw to Sept. 17, 2010

Three steel balls are inside a cylindrical tube filling it tightly from the bottom to the top. Each ball and the tube have the same diameter. The steel balls are removed from the tube and you are given the tube filled with water.

You also have two additional containers, each with a larger volume than the tube, but these containers have an unknown volume and there are no markings on them so you cannot use them to make any measurements.

Explain how to use the steel balls, cylinder, and extra containers to divide the given water from the tube into three exactly equal portions?

Challenge the Week to Sept 10, 2010

a) Every point in the plane has been colored either red or blue. No matter how the plane has been colored, show that there always exist two points in the plane with the same color exactly 1 unit distance apart.

b) Every point in the plane has been colored either red, white, or blue. No matter how the plane has been colored, show that there always exist two points in the plane with the same color exactly 1 unit distance apart.