MathDB

p1. The final Big

10

standings for the

1996

Women's Softball season were
1. Michigan 2. Minnesota З. Iowa 4. Indiana 5. Michigan State 6. Purdue 7. Northwestern 8. Ohio State 9. Penn State 10. Wisconsin (Illinois does not participate in Women's Softball.)
When you compare the

1996

final standings (above) to the final standings for the

1999

season, you find that the following pairs of teams changed order relative to each other from

1996

1999

(there are no ties, and no other pairs changed places):
(Iowa, Michigan State) (Indiana, Penn State) (Purdue, Wisconsin) (Iowa, Penn State) (Indiana, Wisconsin) (Northwestern, Penn State) (Indiana, Michigan State) (Michigan State, Penn State) (Northwestern, Wisconsin) (Indiana, Purdue) (Purdue, Northwestern) (Ohio State, Penn State) (Indiana, Northwestern) (Purdue, Penn State) (Ohio State, Penn State) (Indiana, Ohio State)
Determine as much as you can about the final Big

10

standings for the

1999

Women's Softball season. If you cannot determine the standings, explain why you do not have enough information. You must justify your answer.
p2. a) Take as a given that any expression of the form

A \sin t + B \cos t

(

A>0

) can be put in the form

C \sin (t + D)

, where

C>0

and

-\pi /2 <D <\pi /2

. Determine

C

and

D

in terms of

A

and

B

. b) For the values of

C

and

D

found in part a), prove that

A \sin t + B \cos t = C \sin (t + D)

. c) Find the maximum value of

3 \sin t +2 \cos t

.
pЗ. А

6

-bу-

6

checkerboard is completelу filled with

18

dominoes (blocks of size

1

-bу-

2

). Prove that some horizontal or vertical line cuts the board in two parts but does not cut anу of the dominoes.
p4. a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon? Justify your answer and simplify as much as possible. b) The midpoints of the sides of a regular

n

-gon (

n >2

) are the vertices of a new

n

-gon. What is the ratio of the area of the new

n

-gon to that of the old? Justify your answer and simplify as much as possible.
p5. You run a boarding house that has

90

rooms. You have

100

guests registered, but on any given night only

90

of these guests actually stay in the boarding house. Each evening a different random set of

90

guests will show up. You don't know which

90

it will be, but they all arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of

90

guests, you can assign each to a private room without any switching of keys. a) You could give every guest a key to every room. But this requires

9000

keys. Find a way to hand out fewer than

9000

keys so that each guest will have a key to a private room. b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement