
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
<u>Round 1</u>**p1.** If  $2m = 200 cm$  and  $m \ne 0$ , find  $c$ .**p2.** A right triangle has two sides of lengths  $3$  and  $4$ . Find the smallest possible length of the third side.**p3.** Given that  $20(x + 17) = 17(x + 20)$ , determine the value of  $x$ .


<u>Round 2</u>**p4.** According to the Egyptian Metropolitan Culinary Community, food service is delayed on  $\frac23$  of flights departing from Cairo airport. On average, if  flights with delayed food service have twice as many passengers per flight as those without, what is the probability that a passenger departing from Cairo airport experiences delayed food service?**p5.** In a positive geometric sequence  $\{a_n\}$ ,  $a_1 = 9$ ,  $a_9 = 25$ . Find the integer  $k$  such that  $a_k = 15$ **p6.** In the Delicate, Elegant, and Exotic Music Organization, pianist Hans is selling two types of  owers with different prices (per  ower): magnolias and myosotis. His friend Alice originally plans to buy a bunch containing  $50\%$  more magnolias than myosotis for  $\$ 50 $, but then she realizes that if she buys $ 50\% $ less magnolias and $ 50\% $ more myosotis than her original plan, she would still need to pay the same amount of money. If instead she buys $ 50\% $ more magnolias and $ 50\% $ less myosotis than her original plan, then how much, in dollars, would she need to pay?


<u>Round 3</u>**p7.** In square $ ABCD $, point $ P $ lies on side $ AB $ such that $ AP = 3 $,$ BP = 7 $. Points $ Q,R, S $ lie on sides $ BC,CD,DA $ respectively such that $ PQ = PR = PS = AB $. Find the area of quadrilateral $ PQRS $.**p8.** Kristy is thinking of a number $ n < 10^4 $ and she says that $ 143 $ is one of its divisors. What is the smallest number greater than $ 143 $ that could divide $ n $?**p9.** A positive integer $ n $ is called *special* if the product of the $ n $ smallest prime numbers is divisible by the sum of the $ n $ smallest prime numbers. Find the sum of the three smallest special numbers.


<u>Round 4</u>**p10.**  In the diagram below, all adjacent points connected with a segment are unit distance apart. Find the number of squares whose vertices are among the points in the diagram and whose sides coincide with the drawn segments.
![Image](https://cdn.artofproblemsolving.com/attachments/b/a/923e4d2d44e436ccec90661648967908306fea.png)**p11.** Geyang tells Junze that he is thinking of a positive integer. Geyang gives Junze the following clues:
$ \bullet $ My number has three distinct odd digits.
$ \bullet $ It is divisible by each of its three digits, as well as their sum.
What is the sum of all possible values of Geyang's number?**p12.** Regular octagon $ ABCDEFGH $ has center $ O $ and side length $ 2 $. A circle passes through $ A,B $, and $ O$. What is the area of the part of the circle that lies outside of the octagon?


PS. You should use hide for answers. Rounds 5-8 have been posted [here](https://artofproblemsolving.com/community/c3h2936505p26278645). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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