6

Part of 1999 Harvard-MIT Mathematics Tournament

Problems(5)

1999 Advanced Topics #6: Always One Left Over

Source:

Matt has somewhere between

1000

2000

pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries

2

3

4

5

6

7

8

piles but ends up with one sheet left over each time. How many piles does he need?

1999 Calculus #6: Evaluation of a Derivative

Source:

\dfrac{d}{dx}\left(\sin x - \dfrac{4}{3}\sin^3 x\right)

x=15

calculusderivativetrigonometry

1999 HMMT Geometry # 6

Source:

A sphere of radius

1

is covered in ink and rolling around between concentric spheres of radii

3

5

. If this process traces a region of area 1 on the larger sphere, what is the area of the region traced on the smaller sphere?

1999 HMMT Team #6

Source:

Reduce the number

\sqrt[3]{2 +\sqrt5} + \sqrt[3]{2 -\sqrt5}

1999 Oral #6: Sorting Using Blocks

Source:

You want to sort the numbers 5 4 3 2 1 using block moves. In other words, you can take any set of numbers that appear consecutively and put them back in at any spot as a block. For example, 6 5 3 4 2 1 -> 4 2 6 5 3 1 is a valid block move for 6 numbers. What is the minimum number of block moves necessary to get 1 2 3 4 5?