MathDB

p1. Evaluate

1! + 2! + 3! + 4! + 5!

(where

n!

is the product of all integers from

1

n

, inclusive).
p2. Harold opens a pack of Bertie Bott's Every Flavor Beans that contains

10

blueberry,

10

watermelon,

3

spinach and

2

earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored?
p3. Find the sum of the positive factors of

32

(including

32

itself).
p4. Carol stands at a flag pole that is

21

feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked

10

feet, her shadow passes the flag's shadow. Given that Carol is exactly

5

feet tall, how long in feet is her shadow?
p5. A solid metal sphere of radius

7

cm is melted and reshaped into four solid metal spheres with radii

1

5

6

, and

x

cm. What is the value of

x

?
p6. Let

A = (2,-2)

and

B = (-3, 3)

. If

(a,0)

and

(0, b)

are both equidistant from

A

and

B

, then what is the value of

a + b

?
p7. For every flip, there is an

x^2

percent chance of flipping heads, where

x

is the number of flips that have already been made. What is the probability that my first three flips will all come up tails?
p8. Consider the sequence of letters

Z\,\,W\,\,Y\,\,X\,\,V

. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order?
p9. A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is

169

, then find the side length of the square.
p10. If

A = 50\sqrt3

B = 60\sqrt2

, and

C = 85

, then order

A

B

, and

C

from least to greatest.
p11. How many ways are there to arrange the letters of the word

RACECAR

? (Identical letters are assumed to be indistinguishable.)
p12. A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let

r

be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find

r^2

.
p13. Given that

x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10

, find all possible values of

x +\frac{1}{x}

.
p14. Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions

3\times 2\times 2

. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible.
p15. Triangle

ABC

has

AB = 4

BC = 3

, and

AC = 5

. Point

B

is reflected across

\overline{AC}

to point

B'

. The lines that contain

AB'

and

BC

are then drawn to intersect at point

D

. Find

AD

.
p16. Consider a rectangle

ABCD

with side lengths

5

and

12

. If a circle tangent to all sides of

\vartriangle ABD

and a circle tangent to all sides of

\vartriangle BCD

are drawn, then how far apart are the centers of the circles?
p17. An increasing geometric sequence

a_0, a_1, a_2,...

has a positive common ratio. Also, the value of

a_3 + a_2 - a_1 - a_0

is equal to half the value of

a_4 - a_0

. What is the value of the common ratio?
p18. In triangle

ABC

AB = 9

BC = 11

, and

AC = 16

. Points

E

and

F

are on

\overline{AB}

and

\overline{BC}

, respectively, such that

BE = BF = 4

. What is the area of triangle

CEF

?
p19. Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After

8

minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in

12

minutes. After how many seconds since the three began running did Zach first pass Yuna?
p20. How many unit fractions are there such that their decimal equivalent has a cycle of

6

repeating integers? Exclude fractions that repeat in cycles of

1

2

, or

3

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

Problem Statement