
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 6</u>**p16.** Triangle  $ABC$  with  $AB < AC$  is inscribed in a circle. Point  $D$  lies on the circle and point  $E$  lies on side  $AC$  such that  $ABDE$  is a rhombus. Given that  $CD = 4$  and  $CE = 3$ , compute  $AD^2$ .**p17.**  Wam and Sang are walking on the coordinate plane. Both start at the origin. Sang walks to the right at a constant rate of  $1$  m/s. At any positive time  $t$  (in seconds),Wam walks with a speed of  $1$  m/s with a direction of  $t$  radians clockwise of the positive  $x$ -axis. Evaluate the square of the distance betweenWamand Sang in meters after exactly  $5\pi$  seconds.**p18.**  Mawile is playing a game against Salamance. Every turn,Mawile chooses one of two moves: Sucker Punch or IronHead, and Salamance chooses one of two moves: Dragon Dance or Earthquake. Mawile wins if the moves used are Sucker Punch and Earthquake, or Iron Head and Dragon Dance. Salamance wins if the moves used are Iron Head and Earthquake. If the moves used are Sucker Punch and Dragon Dance, nothing happens and a new turn begins. Mawile can only use Sucker Punch up to  $8$  times. All other moves can be used indefinitely. Assuming bothMawile and Salamance play optimally, find the probability thatMawile wins.


<u>Round 7</u>**p19.** Ephram is attempting to organize what rounds everyone is doing for the NEAML competition. There are  $4$  rounds, of which everyone must attend exactly  $2$ . Additionally, of the 6 people on the team(Ephram,Wam, Billiam, Hacooba,Matata, and Derke), exactly  $3$  must attend every round. In how many different ways can Ephram organize the teams like this?**p20.** For some  $4$ th degree polynomial  $f (x)$ , the following is true: $\bullet$   $f (-1) = 1$ . $\bullet$   $f (0) = 2$ . $\bullet$   $f (1) = 4$ . $\bullet$   $f (-2) = f (2) = f (3)$ .
Find  $f (4)$ .**p21.** Find the minimum value of the expression  $\sqrt{5x^2-16x +16}+\sqrt{5x^2-18x +29}$  over all real  $x$ .


<u>Round 8</u>**p22.**  Let  $O$  and  $I$  be the circumcenter and incenter, respectively, of  $\vartriangle ABC$  with  $AB = 15$ ,  $BC = 17$ , and  $C A = 16$ . Let  $X \ne A$  be the intersection of line  $AI$  and the circumcircle of  $\vartriangle ABC$ . Find the area of  $\vartriangle IOX$ .**p23.**  Find the sum of all integers  $x$  such that there exist integers  $y$  and  $z$  such that   $$ x^2 + y^2 = 3(2016^z )+77. $$  **p24.**  Evaluate   $$  \left \lfloor \sum^{2022}_{i=1} \frac{1}{\sqrt{i}} \right \rfloor = \left \lfloor \frac{1}{\sqrt{1}} +\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+ \frac{1}{\sqrt{2022}}\right \rfloor $$  <u>Round 9</u> **p25.**Either:
1. Submit  $-2$  as your answer and you’ll be rewarded with two points OR
2. Estimate the number of teams that choose the first option. If your answer is within  $1$  of the correct answer, you’ll be rewarded with three points, and if you are correct, you’ll receive ten points.**p26.** Jeff is playing a turn-based game that starts with a positive integer  $n$ .
Each turn, if the current number is  $n$ , Jeff must choose one of the following:
1. The number becomes the nearest perfect square to  $n$ 2. The number becomes  $n-a$ , where  $a$  is the largest digit in  $n$ Let  $S(k)$  be the least number of turns Jeff needs to get from the starting number  $k$  to  $0$ . Estimate   $$ \sum^{2023}_{k=1}S(k). $$   If your estimation is  $E$  and the actual answer is  $A$ , you will receive  $\max \left( \left \lfloor 10 - \left| \frac{E-A}{6000} \right| \right \rfloor , 0 \right)$  points.**p27.**  Estimate the smallest positive integer n such that if  $N$  is the area of the  $n$ -sided regular polygon with circumradius  $100$ ,  $10000\pi -N < 1$  is true.
If your estimation is  $E$  and the actual answer is  $A$ , you will receive  $ \max \left \lfloor \left( 10 - \left| 10 \cdot \log_3 \left( \frac{A}{E}\right) \right|\right| ,0\right \rfloor.$  points.


PS. You should use hide for answers. Rounds 1-5 have been posted [here](https://artofproblemsolving.com/community/c3h3167360p28825713). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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