Every week on the department monitors will be a Math problem of the week. Do you have what it takes to solve these problems? Soultions and past problems will be posted here every week.

**April 25, 2022**

Suppose you have a circle, a diameter, and a point inside the circle not on the diameter. Using only a straightedge, construct the perpendicular from the point to the diameter.

**April 11, 2022**

Evaluate the 123rd derivative of 1/(x^2-x) at x=2

Answer: Write 1/(x^2 - x) = 1/(x-1) - 1/x. Differentiate each term separately, then evaluate at x=2 to get (123!)(-1 + 1/2^124).

**April 4, 2022**

You have one flask that holds 11 cups of water and another flask that holds 19 cups. You need to measure 1 cup of water for the recipe you are making. How can you do this?

Answer: Notice that 7x11 - 4x19 = 1, so if you fill the 11-cup flask 7 times and empty the 19-cup flask 4 times, you'll have 1 cup remaining. So do the following: Let's call the 11-cup flask E and the 19-cup flask N. Fill E and pour it into N.

Then fill E again and pour as much as possible, namely 8 cups, into N, leaving 3 cups in E. Empty N and pour these 3 cups into N. Fill E and pour into N. Then fill E again and pour as much as possible, namely 5 cups, into N. This leaves 6 in E. Empty N and pour in these 6 cups. Now fill E and pour it into N, giving 17 cups there. Then fill E and pour 2 more cups into N, leaving 9 in E. Empty N, pour in these 9 cups, refill E, and pour as much as possible, namely 10 cups, into N. This leaves one cup in E.

**March 14, 2022**

Let f(x) = x + sin(x). Compute f(1), then f(f(1)), then f(f(f(1))), etc. What is the limit of this sequence?

Answer: The fourth term of the series is already 3.1415926. The main step in general is to use Taylor series with remainder to show that f(x) = x + sin(x) differs from pi by at most (1/6)(x - pi)^3. Therefore, the sequence gets closer and closer to pi.

**March 7, 2022**

Cut a round pizza by five straightline cuts. Moving pieces is not allowed between cuts. What is the largest number of pieces you can get?

Answer: The *n*^{th} cut can cross up to *n*-1 of the previous cuts, which means it can divide at most n regions into 2 pieces, thus adding *n* regions to the total. There is one region to start with, so the largest possible total after 5 cuts is 1+1+2+3+4+5=16.

**February 28, 2022**

University of Maryland uniforms come in four different colors. In how many ways can a team suit up for five consecutive games so that no adjacent games use the same color? (everyone on the team wears one color for a game).

Answer: There are 4 possibilities for the first game, and then 3 for each of the next four games. This gives:

4*3*3*3*3 = 324 ways

**February 22, 2022**

Look at the last two digits of the powers of 2: 01, 02, 04, 08, 16, etc.

How many distinct numbers do you get?

Answer : 22

Explaination: One way is to list the powers until they repeat: 01, 02, 04, 08, 16, . . . , 76, 52, 04. Another way is to use Euler's theorem from number theory to conclude that the powers of 2 repeat mod 25 every 20 steps. After 01 and 02, all the numbers are multiples of 4, so the powers of 2 repeat mod 100 every 20 steps after the initial two powers.