
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>Set 7</u>**G19.** How many ordered triples  $(x, y, z)$  with  $1 \le x, y, z \le 50$  are there such that both  $x + y + z$  and  $xy + yz + zx$  are divisible by $ 6$ ?**G20.** Triangle  $ABC$  has orthocenter  $H$  and circumcenter  $O$ . If  $D$  is the foot of the perpendicular from  $A$  to  $BC$ , then  $AH = 8$  and  $HD = 3$ . If  $\angle AOH = 90^o$ , find  $BC^2$ .**G21.** Nate flips a fair coin until he gets two heads in a row, immediately followed by a tails. The probability that he flips the coin exactly  $12$  times is  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


<u>Set 8</u>**G22.** Let  $f$  be a function defined by  $f(1) = 1$  and   $$ f(n) = \frac{1}{p}f\left(\frac{n}{p}\right)f(p) + 2p - 2, $$   where  $p$  is the least prime dividing  $n$ , for all integers  $n \ge 2$ . Find  $f(2022)$ .**G23.** Jessica has  $15$  balls numbered  $1$  through  $15$ . With her left hand, she scoops up  $2$  of the balls. With her right hand, she scoops up  $2$  of the remaining balls. The probability that the sum of the balls in her left hand is equal to the sum of the balls in her right hand can be expressed as  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .**G24.** Let  $ABCD$  be a cyclic quadrilateral such that its diagonal  $BD = 17$  is the diameter of its circumcircle. Given  $AB = 8$ ,  $BC = CD$ , and that a line  $\ell$  through A intersects the incircle of  $ABD$  at two points  $P$  and  $Q$ , the maximum area of  $CP Q$  can be expressed as a fraction  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Find  $m + n$ .


<u>Set 9</u>

*This set consists of three estimation problems, with scoring schemes described.***G25.** Estimate  $N$ , the total number of participants (in person and online) at MOAA this year. An estimate of  $e$  gets a total of max  $ \left( 0, \lfloor 150 \left( 1- \frac{|N-e|}{N}\right) \rfloor -120 \right)$  points.**G26.** If  $A$  is the the total number of in person participants at MOAA this year, and  $B$  is the total number of online participants at MOAA this year, estimate  $N$ , the product  $AB$ . An estimate of  $e$  gets a total of max  $(0, 30 - \lceil \log10(8|N - e| + 1)\rceil )$  points.**G27.** Estimate  $N$ , the total number of letters in all the teams that signed up for MOAA this year, both in person and online. An estimate of e gets a total of max  $(0, 30 - \lceil 7 log5(|N - E|)\rceil )$  points.
 

PS. You should use hide for answers. Sets 1-3 have been posted [here ](https://artofproblemsolving.com/community/c3h3131303p28367061) and 4-6 [here](https://artofproblemsolving.com/community/c3h3131305p28367080). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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