
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.** Let  $ABCDEF$  be a regular hexagon. How many acute triangles have all their vertices among the vertices of  $ABCDEF$ ?**p2.** A rectangle has a diagonal of length  $20$ . If the width of the rectangle is doubled, the length of the diagonal becomes  $22$ . Given that the width of the original rectangle is  $w$ , compute  $w^2$ .**p3.** The number  $\overline{2022A20B22}$  is divisible by 99. What is  $A + B$ ?


<u>Round 2</u>**p4.** How many two-digit positive integers have digits that sum to at least  $16$ ?**p5.** For how many integers  $k$  less than  $10$  do there exist positive integers x and y such that  $k =x^2 - xy + y^2$ ?**p6.** Isosceles trapezoid  $ABCD$  is inscribed in a circle of radius  $2$  with  $AB \parallel CD$ ,  $AB = 2$ , and one of the interior angles of the trapezoid equal to  $110^o$ . What is the degree measure of minor arc  $CD$ ?


<u>Round 3</u>**p7.** In rectangle  $ALEX$ , point  $U$  lies on side  $EX$  so that  $\angle AUL = 90^o$ . Suppose that  $UE = 2$  and  $UX = 12$ . Compute the square of the area of  $ALEX$ .**p8.** How many digits does  $20^{22}$  have?**p9.** Compute the units digit of  $3 + 3^3 + 3^{3^3} + ... + 3^{3^{...{^3}}}$  , where the last term of the series has  $2022$   $3$ s.


<u>Round 4</u>**p10.** Given that  $\sqrt{x - 1} + \sqrt{x} = \sqrt{x + 1}$  for some real number  $x$ , the number  $x^2$  can be written as  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .**p11.**  Eric the Chicken Farmer arranges his  $9$  chickens in a  $3$ -by- $3$  grid, with each chicken being exactly one meter away from its closest neighbors. At the sound of a whistle, each chicken simultaneously chooses one of its closest neighbors at random and moves  $\frac12$  of a unit towards it. Given that the expected number of pairs of chickens that meet can be written as  $\frac{p}{q}$  , where  $p$  and  $q$  are relatively prime positive integers, compute  $p + q$ .**p12.** For a positive integer  $n$ , let  $s(n)$  denote the sum of the digits of  $n$  in base  $10$ . Find the greatest positive integer  $n$  less than  $2022$  such that  $s(n) = s(n^2)$ .


PS. You should use hide for answers. Rounds 5-8 have been posted [here](https://artofproblemsolving.com/community/c3h2949432p26408285). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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