MathDB

p1. Sujay sees a shooting star go across the night sky, and took a picture of it. The shooting star consists of a star body, which is bounded by four quarter-circle arcs, and a triangular tail. Suppose

AB = 2

AC = 4

. Let the area of the shooting star be

X

. If

6X = a-b\pi

for positive integers

a, b

, find

a + b

. https://cdn.artofproblemsolving.com/attachments/0/f/f9c9ff23416565760df225c133330e795b9076.png
p2. Assuming that each distinct arrangement of the letters in

DISCUSSIONS

is equally likely to occur, what is the probability that a random arrangement of the letters in

DISCUSSIONS

has all the

S

’s together?
p3. Evaluate

\frac{(1 + 2022)(1 + 2022^2)(1 + 2022^4) ... (1 + 2022^{2^{2022}})}{1 + 2022 + 2022^2 + ... + 2022^{2^{2023}-1}} .

p4. Dr. Kraines has

27

unit cubes, each of which has one side painted red while the other five are white. If he assembles his cubes into one

3 \times 3 \times 3

cube by placing each unit cube in a random orientation, what is the probability that the entire surface of the cube will be white, with no red faces visible? If the answer is

2^a3^b5^c

for integers

a

b

c

, find

|a + b + c|

.
p5. Let S be a subset of

\{1, 2, 3, ... , 1000, 1001\}

such that no two elements of

S

have a difference of

4

7

. What is the largest number of elements

S

can have?
p6. George writes the number

1

. At each iteration, he removes the number

x

written and instead writes either

4x+1

8x+1

. He does this until

x > 1000

, after which the game ends. What is the minimum possible value of the last number George writes?
p7. List all positive integer ordered pairs

(a, b)

satisfying

a^4 + 4b^4 = 281 \cdot 61

.
p8. Karthik the farmer is trying to protect his crops from a wildfire. Karthik’s land is a

5 \times 6

rectangle divided into

30

smaller square plots. The

5

plots on the left edge contain fire, the

5

plots on the right edge contain blueberry trees, and the other

5 \times 4

plots of land contain banana bushes. Fire will repeatedly spread to all squares with bushes or trees that share a side with a square with fire. How many ways can Karthik replace

5

of his

20

plots of banana bushes with firebreaks so that fire will not consume any of his prized blueberry trees?
p9. Find

a_0 \in R

such that the sequence

\{a_n\}^{\infty}_{n=0}

defined by

a_{n+1} = -3a_n + 2^n

is strictly increasing.
p10. Jonathan is playing with his life savings. He lines up a penny, nickel, dime, quarter, and half-dollar from left to right. At each step, Jonathan takes the leftmost coin at position

1

and uniformly chooses a position

2 \le k \le 5

. He then moves the coin to position

k

, shifting all coins at positions

2

through

k

leftward. What is the expected number of steps it takes for the half-dollar to leave and subsequently return to position

5

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

Problem Statement