A fun math challenge for you all?

1. In a certain year, October 13 happens to be a Friday. On what day of the week does


Halloween fall?


2. Professor Euler wants a date for Valentine鈥檚 Day. He sends invitations to n women and


knows that the probability that any one woman will accept his offer is 2/9. How many


invitations should he send out to maximize his chances of getting exactly one woman?


3. On Presidents鈥?Day, 100 Abraham Lincoln impersonators walk into the Bakery of


the Executive Office. To commemorate Lincoln鈥檚 200th birthday, the Baker General,


a George Washington impersonator, asks them to play a game, and they can get a


lifetime supply of cherry pie if they win. In the game, they stand in a circle, and he


places a stovepipe hat on each of them. The hat can have one of four different colors:


red, white, blue, or black. The colors do not have to be evenly distributed, and not all


four are necessarily used. Each Lincoln impersonator can see everyone else鈥檚 hat but


not his own. They can formulate a strategy beforehand, but once the game begins,


they are not allowed to communicate (verbally or nonverbally, including gesturing or


moving around). In order for them to win, everyone must correctly write down the


color of his hat (others cannot see his writing); otherwise, they get nothing. How do


they maximize their probability of winning? What is that probability?


4. For the annual Pi Day celebration, Math Club orders a certain number of pies from the


Einstein Bakery, where Alice and Bob work. Working alone, Alice can fulfill the order


in 6 hours, while Bob can fulfill the order in 8 hours. When they work together, they


are more efficient, so their combined output is increased by 2 pies per hour. Working


together, they fulfill the order in 3 hours. How many pies were ordered?


5. The evil Dr. Tony is planning to take over the world. While making a rocket engine,


he needed to take the tangent of a certain angle. However, he did not know whether


the angle was measured in degrees or radians, so he set his calculator in both modes


and took the tangent using both settings. To his surprise, the calculator gave the same


answer both times! Find the smallest positive angle for which this is possible.


Page 1 of 2





6. (Calculus) A pie is modeled as the region bounded by y = 4 鈭?x, y = x 鈭?4, x = 1,


and x = 2 (measured in inches), rotated about the x-axis. An aluminum pan is used


to cover the side and bottom (smaller circular base) of the pie. What surface area of


aluminum is needed?


7. (Statistics) The Baker General (oh no, not him again!) claims that on average, he can


make pies with diameter 20 in. One day, you walk into his store and order a simple


random sample of 5 freshly made pies. The pies you receive have diameters 19.0, 19.3,


19.5, 19.7, and 20.0 in. Is this baker as honest as George Washington or Honest Abe?


Be sure to follow the appropriate significance test procedures.


8. (Physics) Mr. Holloway is standing at the top of a very large hemispheric hill. He is


holding a pie (modeled as a uniform cylindrical disk with mass m and radius r) in his


hand. He sets it on the ground and lets go of it with a negligible initial velocity, and


it rolls without slipping down the hill. If the hill has a height of R, at what (vertical)


distance below the top of the hill does the pie leave the surface of the hill?


9. (Chemistry) On Mole Day, Mr. Rollins puts a piece of solid dry ice at 鈭?8.5 C (the


sublimation point of carbon dioxide) in 1 kg of water at 20 C. The specific heat of


CO2 gas is 0.79 J/g C, and the heat of vaporization of solid CO2 is 571 J/g. The


specific heat of liquid H2O is 4.19 J/g C. What is the largest mass of dry ice that Mr.


Rollins can insert such that the water does not begin to freeze? Assume that there is


no transfer of heat between the system and the environment.


10. (Biology) On the remote island Einsteinia, an explorer finds a group of people called


the Mathematica. Interestingly, the Mathematica have the same blood types as most


other humans: A, B, AB, and O. If 39% of them have type A blood, 24% have B, 12%


have AB, and 25% have O, what are the allele frequencies of IA, IB, and i? Assume


that the Mathematica mate randomly.|||1. If the 13th is a Friday, then so is the 20th and 27th.


28 - Sat


29 - Sun


30 - Mon


31 - Halloween on Tuesday





2. I recently answered this as a separate question.


The answer is 4.


Open Yahoo Questions: qid= 20090305220402AA5EzLr





See also the Dr Tony question in there.





3. no idea, since they can't communicate at all.


I recently saw a variation of this where they at least get to hear


the other guesses.





4. Alice 6 hours, Bob 8 hours.


Without the stated "efficiency boost", it would take them


48/14 hours = 3.535


But with extra 2 pies per hour that goes to 3 hours.


Let their base rate be r pies per hour:


So pies / r = 48/14


but pies / (r + 2) = 3





r = p * 14/48


r + 2 = p / 3


p * 14/48 + 2 = p / 3


14p + 96 = 16 p


2p = 96


p = 48





Normal combined rate: 14 pies per hour, 48/14 hours


Boosted rate: 16 pies per hour, 48/16 = 3 hours





5. Tan (x degrees) = tan (x radians)


x = 3.1974 (approximately)


See other question for more details





6. Too small pies:


sounds like the Anti-Lake Wobegone Pie Shop:


all the pies are below average.





I don't know the statistical details.





This concludes the Math Portion of the Contest.