
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.** Point  $P$  has coordinate value  $(3,2)$ , and point  $Q$  has coordinate value  $(6, x)$ . For what value of  $x$  does Line  $PQ$  have a slope of  $6$ ?**p2.** I have fewer than one hundred pens. When my pens are grouped into nines, I have one pen left over. When grouped into fives, I have three left over. When grouped into pairs, I have one left over. How many do I have?**p3.** How many unique rearrangements of the word “COMBO” exist?**p4.** A right triangle has sides of length  $4x+1$ ,  $2x-1$ , and  $x+9$ , where  $x$  is a positive integer, and the side of length  $4x+1$  is the hypotenuse. What is the area of the triangle?**p5.** A parabola has vertex  $(4,-2)$  and roots  $(8,0)$  and  $(0,0)$ . What is the coefficient of the linear term when the equation of the parabola is expanded?**p6.** The terms  $2x-1$ ,  $x + 4$ ,  $4x + 1$  form an arithmetic sequence while the terms  $x$ ,  $x+4$ ,  $xy^2$  form a geometric sequence. Evaluate  $x+y$ .**p7.** A sphere is inscribed inside a cube that is inscribed inside another sphere. The larger sphere has volume  $972\pi$ . Calculate the volume of the space inside the larger sphere but outside the smaller sphere. Express your answer in simplest radical form and in terms of  $\pi$ .**p8.** A computer battery loses  $50\%$  of its remaining power every hour. Assuming the computer begins at full battery, how long will it take for the computer to reach  $10\%$  battery? Express your answer in minutes and round your answer to the nearest hundredth.**p9.** How many two-digit numbers have six factors?**p10.**Thomas is graduating. He decides to attend a college that offers him a sizable scholarship. The number of dollar he receives is an eight digit number using the digits  $0$  through  $7$  without repetition and follows the following criteria: $\bullet$  The number is divisible by  $5$ . $\bullet$  The last four digits are prime numbers. $\bullet$  A three digit number comprised of the last three digits in order is divisible by  $11$ . $\bullet$  A four digit number comprised of the first four digits in order is divisible by  $205$ .
How much money is Thomas receiving in scholarship?


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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