
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.** Will is distributing his money to three friends. Since he likes some friends more than others, the amount of money he gives each is in the ratio of  $5 : 3 : 2$ . If the person who received neither the least nor greatest amount of money was given  $42$  dollars, how many dollars did Will distribute in all?**p2.** Fan, Zhu, and Ming are driving around a circular track. Fan drives  $24$  times as fast as Ming and Zhu drives  $9$  times as fast as Ming. All three drivers start at the same point on the track and keep driving until Fan and Zhu pass Ming at the same time. During this interval, how many laps have Fan and Zhu driven together?**p3.** Mr. DoBa is playing a game with Gunga the Gorilla. They both agree to think of a positive integer from  $1$  to  $120$ , inclusive. Let the sum of their numbers be  $n$ . Let the remainder of the operation  $\frac{n^2}{4}$  be  $r$ . If  $r$  is  $0$  or  $1$ , Mr. DoBa wins. Otherwise, Gunga wins. Let the probability that Mr. DoBa wins a given round of this game be  $p$ . What is  $120p$ ?**p4.** Let S be the set  $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ . How many subsets of  $S$  are there such that if  $a$  is the number of even numbers in the subset and  $b$  is the number of odd numbers in the subset, then  $a$  and  $b$  are either both odd or both even? By definition, subsets of  $S$  are unordered and only contain distinct elements that belong to  $S$ .**p5.** Phillips Academy has five clusters,  $WQN$ ,   $WQS$ ,   $PKN$ ,   $FLG$  and  $ABB$ . The Blue Key heads are going to visit all five clusters in some order, except  $WQS$  must be visited before  $WQN$ . How many total ways can they visit the five clusters?**p6.** An astronaut is in a spaceship which is a cube of side length  $6$ . He can go outside but has to be within a distance of  $3$  from the spaceship, as that is the length of the rope that tethers him to the ship. Out of all the possible points he can reach, the surface area of the outer surface can be expressed as  $m+n\pi$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m + n$ ?**p7.** Let  $ABCD$  be a square and  $E$  be a point in its interior such that  $CDE$  is an equilateral triangle. The circumcircle of  $CDE$  intersects sides  $AD$  and  $BC$  at  $D$ ,  $F$  and  $C$ ,  $G$ , respectively. If  $AB = 30$ , the area of  $AFGB$  can be expressed as  $a-b\sqrt{c}$ , where  $a$ ,  $b$ , and  $c$  are positive integers and c is not divisible by the square of any prime. Find  $a + b + c$ .**p8.** Suppose that  $x, y, z$  satisfy the equations   $$ x + y + z = 3 $$    $$ x^2 + y^2 + z^2 = 3 $$    $$ x^3 + y^3 + z^3 = 3 $$   Let the sum of all possible values of  $x$  be  $N$ . What is  $12000N$ ?**p9.** In circle  $O$  inscribe triangle  $\vartriangle ABC$  so that  $AB = 13$ ,  $BC = 14$ , and  $CA = 15$ . Let  $D$  be the midpoint of arc  $BC$ , and let  $AD$  intersect  $BC$  at  $E$ . Determine the value of  $DE \cdot DA$ .**p10.** How many ways are there to color the vertices of a regular octagon in  $3$  colors such that no two adjacent  vertices have the same color?


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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