Math Week Online Problems (for AHS MAO members only)

* These problems are to be emailed back to me at ciaconna74@yahoo.com no later than 12 midnight on Thursday, March 11. If you are not able to solve all the problems though, just go ahead and email what answers you have (hey, you never know). First 20 people who email me back with the most number of questions correct will be awarded points. Points will be awarded accordingly and announced at the next meeting, March 22. Have a great spring break!
***IMPORTANT NOTE: the problems that were sent to everyone's emails has an error in #5...Emerson ate 10 more pies than Ralph, not 20... sorrie about this error, it's corrected below:

1. To test whether an integer, n, is prime, it is enough to be sure that none of the primes less than the square root of n divide n. If you want to check that a number between 900 and 950 is prime with this rule, what is the largest prime divisor you need to test?
2. The length, width, and height of a rectangular box are each decreased by 50%. By what percent, to the nearest tenth, is the volume of the box decreased?
3. Four standard 6-sided dice are rolled. The product of the four numbers rolled is 144. How many different sums of four such numbers are possible?
4. Bob and Drake are both freshman in college, and have just received back their scores from their advanced differential equations test. Bob is very unorganized and forgot that there was a test that day, so he ended up cramming 15 minutes before the test. Drake on the other hand, is much more responsible and keeps a planner with all the test dates highlighted well in advance. He not only studied 3 hours for this test, but he also spent 2 hours in tutorials to make sure that he completely understood the test material. Ironically though, Bob scored a 78, which was 130% of Drake’s score. What did poor Drake score on his advanced differential equations test?
5. At the previous Mu Alpha Theta meeting, the officers all brought in pies for everyone to eat afterwards, in celebration of “ð Day.” However, some people were extremely hungry and ate all the pies before the other people who waited patiently could even get a slice of pie. Ralph ate 450 slices. Emerson ate 10 more than Ralph and twice as many as Waldo. How many slices did they eat altogether?
6. If each pie could have been cut up into 12 slices, how many pies were there?
7. There actually was one more pie originally than the answer obtained in #5, but during the meeting, one sneaky person stole a pie and ran off quickly. If each pie cost $6.99, and assuming that each officer bought the same number of pies, how much did each officer spend on buying pies? (there are 6 officers)
8. After having to deal with so many pies in one single day, all the officers became quite sick of pie and decided to boycott eating pie. However, since pie is so irresistible and tempting, they decided that they would not take such drastic measures and decided to reduce the number of pies that they consume each day. After day 1, they will wait a full day before having another. Then they’ll wait two more days, then three, and so on, extending their waiting period by one day each. In how many years (to the nearest year) will they be eating a pie only once every 60 days?
9. The sum of an integer and its square is 6 less than the square of the next greater integer. What is the value of the integer?
10. A fraction is equivalent to 3/5. Its denominator is 60 more than its numerator. What is the numerator of this fraction?