Saturday, January 12, 2008

First 10

1. Find the length of the hypotenuse of a the triangle. (1 Point)

2. How many 10-digit numbers can be written by using all 10 digits 0-9. (Numbers starting with 0 do not count) (1 Point)

3. Of 9 coins of the same denomination, 8 weight the same, and one, a counterfeit, is lighter than the rest. Find the counterfeit using a balance only two times without weights. (1 Point)

4. Same as three only the counterfeit may be light or heavy, there are 12 coins, and you may use the balance three times. (3 Points)

5. How far does a ball travel if it is dropped from the Leaning Tower of Pisa (179 Feet) if it rebounds 10% of its height after each bounce. (1 Point for nearest foot, 2 points exact)

6. Express 100 in three different ways with five 5s. You may use these symbols + - / * ( ). (1 Point)

7. 2+2 = 2*2, 1+2+3 = 1*2*3. Find four positive integers whose sum equals their product. (1 Point)

8. Same as seven with 5 numbers. (1 Point)

9. There are 100 lockers in a row that are all closed at the beginning. There are 100 students. Each student makes a pass. The first student opens the every locker door. The second student toggles (if the door is closed, you open it, if its open, you close it) every 2nd locker door (#2, #4, #6 etc) starting with locker #2. The third student toggles every 3rd locker door starting with locker #3. The nth student toggles every nth locker door starting with nth locker.
Which lockers will be open at the end? (1 Point)

10. A mathematician attends a dinner party with his wife and four additional couples. When the guest arrive various hand shakes take place with no one shaking their own hand nor that of their spouse. When they sit down for dinner the mathematician asks each of the others (including his wife) how many hands they shook. To his surprise, he found that each had shook a different number of hands! How many hands did his wife shake? (1 Point)


Shyama Mandal said...

Problem 7: Do you mean positive integers.

Justin said...

Thanks, thats what I meant.

Shubhankar said...

For problem no 3, if we don't know whether counterfeit coin is lighter or heavier then is it possible to find the counterfeit using a balance only two times without weights?