
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.** Triangle  $ABC$  has side lengths  $AB = 3^2$  and  $BC = 4^2$ . Given that  $\angle ABC$  is a right angle, determine the length of  $AC$ .**p2.** Suppose  $m$  and  $n$  are integers such that  $m^2+n^2 = 65$ . Find the largest possible value of  $m-n$ .**p3.** Six middle school students are sitting in a circle, facing inwards, and doing math problems. There is a stack of nine math problems. A random student picks up the stack and, beginning with himself and proceeding clockwise around the circle, gives one problem to each student in order until the pile is exhausted. Aditya falls asleep and is therefore not the student who picks up the pile, although he still receives problem(s) in turn. If every other student is equally likely to have picked up the stack of problems and Vishwesh is sitting directly to Aditya’s left, what is the probability that Vishwesh receives exactly two problems?**p4.** Paul bakes a pizza in  $15$  minutes if he places it  $2$  feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza  $2$  feet from the fire at  $10:30$ . Later, he makes another pizza, puts it  $2$  feet away from the fire, and moves the first pizza to a distance of  $3$  feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done?**p5.** You have  $n$  coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between  $10$  and  $20$  cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of  $n$  such that you can be certain of appeasing Charon?**p6.** Let  $a, b$ , and  $c$  be positive integers such that  $gcd(a, b)$ ,  $gcd(b, c)$  and  $gcd(c, a)$  are all greater than  $1$ , but  $gcd(a, b, c) = 1$ . Find the minimum possible value of  $a + b + c$ .**p7.** Let  $ABC$  be a triangle inscribed in a circle with  $AB = 7$ ,  $AC = 9$ , and  $BC = 8$ . Suppose  $D$  is the midpoint of minor arc  $BC$  and that  $X$  is the intersection of  $\overline{AD}$  and  $\overline{BC}$ . Find the length of  $\overline{BX}$ .**p8.** What are the last two digits of the simplified value of  $1! + 3! + 5! + · · · + 2009! + 2011!$  ?**p9.** How many terms are in the simplified expansion of  $(L + M + T)^{10}$  ?**p10.** Ben draws a circle of radius five at the origin, and draws a circle with radius  $5$  centered at  $(15, 0)$ . What are all possible slopes for a line tangent to both of the circles?


PS. You had better use hide for answers.

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