
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
**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  $a + b$ .
![Image](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  $4$  or  $7$ . 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  $4x+1$  or  $8x+1$ . 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  $5$ ?


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