
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.** Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted  $1/4$  of the time, out of  $120$  random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted?**p2.** Andy thinks of four numbers  $27$ ,  $81$ ,  $36$ , and  $41$  and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once)**p3.** Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by  $2$  and increases the width of his garden by  $3$ . If the new area of his garden is  $182$ , then what was the original area of his garden?**p4.** Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of  $6$ , there are  $2$   flute players left over. When he arranges his players into rows of  $13$ , there are  $10$  flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors?**p5.** On the AMC  $9$  (Acton Math Competition  $9$ ),  $5$  points are given for a correct answer,  $2$  points are given for a blank answer and  $0$  points are given for an incorrect answer. How many possible scores are there on the AMC  $9$ , a  $15$  problem contest?**p6.** Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of  $10$ . The total area occupied by  $1$  of each of the different types of CDs can be expressed in the form  $a + b\pi + c\sqrt{d}$  where  $d$  is not divisible by the square of any prime. Find  $a + b + c + d$ .**p7.** You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you  $10$  dollars and each bushel of strawberries earns you  $8$  dollars. However your cart can only fit  $24$  bushels total and has a weight limit of  $100$  lbs. If a bushel of blueberries weighs  $8$  lbs and each bushel of strawberries weighs  $6$  lbs, what is your maximum profit. (You can only pick an integer number of bushels)**p8.** The number   $$ \sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}} $$   can be expressed in the form  $a\sqrt5 + b\sqrt7 + c\sqrt{11}$  for positive integers  $a, b, c$ . Find  $abc$ .**p9.** Let  $(x, y)$  be a point such that no circle passes through the three points  $(9,15)$ ,  $(12, 20)$ ,  $(x, y)$ , and no circle passes through the points  $(0, 17)$ ,  $(16, 19)$ ,  $(x, y)$ . Given that  $x - y = -\frac{p}{q}$  for relatively prime positive integers  $p$ ,  $q$ , Find  $p + q$ .**p10.** How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a  $6$  person table if no more than three consecutive people can be in alphabetical order (clockwise)?


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

Present the answer in LaTex format: \boxed{Your answer}
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