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 nth 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.
