MathDB

p1. What is the slope of the line perpendicular to the the graph

\frac{x}{4}+\frac{y}{9}= 1

(0, 9)

?
p2. A boy is standing in the middle of a very very long staircase and he has two pogo sticks. One pogo stick allows him to jump

220

steps up the staircase. The second pogo stick allows him to jump

125

steps down the staircase. What is the smallest positive number of steps that he can reach from his original position by a series of jumps?
p3. If you roll three fair six-sided dice, what is the probability that the product of the results will be a multiple of

3

?
p4. Right triangle

ABC

has squares

ABXY

and

ACWZ

drawn externally to its legs and a semicircle drawn externally to its hypotenuse

BC

. If the area of the semicircle is

18\pi

and the area of triangle

ABC

30

, what is the sum of the areas of squares

ABXY

and

ACWZ

? https://cdn.artofproblemsolving.com/attachments/5/1/c9717e7731af84e5286335420b73b974da2643.png
p5. You have a bag containing

3

types of pens: red, green, and blue.

30\%

of the pens are red pens, and

20\%

are green pens. If, after you add

10

blue pens,

60\%

of the pens are blue pens, how many green pens did you start with?
p6. Canada gained partial independence from the United Kingdom in

1867

, beginning its long role as the headgear of the United States. It gained its full independence in

1982

. What is the last digit of

1867^{1982}

?
p7. Bacon, Meat, and Tomato are dealing with paperwork. Bacon can fill out

5

forms in

3

minutes, Meat can fill out

7

forms in

5

minutes, and Tomato can staple

3

forms in

1

minute. A form must be filled out and stapled together (in either order) to complete it. How long will it take them to complete

105

forms?
p8. Nice numbers are defined to be

7

-digit palindromes that have no

3

identical digits (e.g.,

1234321

5610165

but not

7427247

). A pretty number is a nice number with a

7

in its decimal representation (e.g.,

3781873

). What is the

7^{th}

pretty number?
p9. Let

O

be the center of a semicircle with diameter

AD

and area

2\pi

. Given square

ABCD

drawn externally to the semicircle, construct a new circle with center

B

and radius

BO

. If we extend

BC

, this new circle intersects

BC

P

. What is the length of

CP

? https://cdn.artofproblemsolving.com/attachments/b/1/be15e45cd6465c7d9b45529b6547c0010b8029.png
p10. Derek has

10

American coins in his pocket, summing to a total of

53

cents. If he randomly grabs

3

coins from his pocket, what is the probability that they're all different?
p11. What is the sum of the whole numbers between

6\sqrt{10}

and

7\pi

?
p12. What is the volume of a cylinder whose radius is equal to its height and whose surface area is numerically equal to its volume?
p13.

15

people, including Luke and Matt, attend a Berkeley Math meeting. If Luke and Matt sit next to each other, a fight will break out. If they sit around a circular table, all positions equally likely, what is the probability that a fight doesn't break out?
p14. A non-degenerate square has sides of length

s

, and a circle has radius

r

. Let the area of the square be equal to that of the circle. If we have a rectangle with sides of lengths

r

s

, and its area has an integer value, what is the smallest possible value for

s

?
p15. How many ways can you arrange the letters of the word "

BERKELEY

" such that no two

E

's are next to each other?
p16. Kim, who has a tragic allergy to cake, is having a birthday party. She invites

12

people but isn't sure if

11

12

will show up. However, she needs to cut the cake before the party starts. What is the least number of slices (not necessarily of equal size) that she will need to cut to ensure that the cake can be equally split among either

11

12

guests with no excess?
p17. Tom has

2012

blue cards,

2012

red cards, and

2012

boxes. He distributes the cards in such a way such that each box has at least

1

card. Sam chooses a box randomly, then chooses a card randomly. Suppose that Tom arranges the cards such that the probability of Sam choosing a blue card is maximized. What is this maximum probability?
p18. Allison wants to bake a pie, so she goes to the discount market with a handful of dollar bills. The discount market sells different fruit each for some integer number of cents and does not add tax to purchases. She buys

22

apples and

7

boxes of blueberries, using all of her dollar bills. When she arrives back home, she realizes she needs more fruit, though, so she grabs her checkbook and heads back to the market. This time, she buys

31

apples and

4

boxes of blueberries, for a total of

60

cents more than her last visit. Given she spent less than

100

dollars over the two trips, how much (in dollars) did she spend on her first trip to the market?
p19. Consider a parallelogram

ABCD

. Let

k

be the line passing through A and parallel to the bisector of

\angle ABC

, and let

\ell

be the bisector of

\angle BAD

. Let

k

intersect line

CD

E

and

\ell

intersect line

CD

F

. If

AB = 13

and

BC = 37

, find the length

EF

.
p20. Given for some real

a, b, c, d,

P(x) = ax^4 + bx^3 + cx^2 + dx

P(-5) = P(-2) = P(2) = P(5) = 1

Find

P(10).

PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement