
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 9</u>**p25.** Circle  $\omega_1$  has radius  $1$  and diameter  $AB$ . Let circle  $\omega_2$  be a circle withm aximum radius such that it is tangent to  $AB$  and internally tangent to  $\omega_1$ . A point  $C$  is then chosen such that  $\omega_2$  is the incircle of triangle  $ABC$ . Compute the area of  $ABC$ .**p26.** Two particles lie at the origin of a Cartesian plane. Every second, the first particle moves from its initial position  $(x, y)$  to one of either  $(x +1, y +2)$  or  $(x -1, y -2)$ , each with probability  $0.5$ . Likewise, every second the second particle moves from it’s initial position  $(x, y)$  to one of either  $(x +2, y +3)$  or  $(x -2, y -3)$ , each with probability  $0.5$ . Let  $d$  be the distance distance between the two particles after exactly one minute has elapsed. Find the expected value of  $d^2$ .**p27.** Find the largest possible positive integer  $n$  such that for all positive integers  $k$  with  $gcd (k,n) = 1$ ,  $k^2 -1$  is a multiple of  $n$ .


<u>Round 10</u>**p28.**  Let  $\vartriangle ABC$  be a triangle with side lengths  $AB = 13$ ,  $BC = 14$ ,  $C A = 15$ . Let  $H$  be the orthcenter of   $\vartriangle ABC$ ,  $M$  be the midpoint of segment  $BC$ , and  $F$  be the foot of altitude from  $C$  to  $AB$ . Let  $K$  be the point on line  $BC$  such that  $\angle MHK = 90^o$ . Let  $P$  be the intersection of  $HK$  and  $AB$ . Let  $Q$  be the intersection of circumcircle of  $\vartriangle FPK$  and  $BC$ . Find the length of  $QK$ .**p29.** Real numbers  $(x, y, z)$  are chosen uniformly at random from the interval  $[0,2\pi]$ . Find the probability that   $$ \cos (x) \cdot \cos (y)+ \cos(y) \cdot \cos (z)+ \cos (z) \cdot \cos(x) + \sin (x) \cdot  \sin (y)+ \sin (y) \cdot  \sin (z)+ \sin (z) \cdot \sin (x)+1 $$   is positive.**p30.** Find the number of positive integers where each digit is either  $1$ ,  $3$ , or  $4$ , and the sum of the digits is  $22$ .


<u>Round 11</u>**p31.**  In  $\vartriangle ABC$ , let  $D$  be the point on ray   $\overrightarrow{CB}$  such that  $AB = BD$  and let  $E$  be the point on ray  $\overrightarrow{AC}$  such that  $BC =CE$ . Let  $L$  be the intersection of  $AD$  and circumcircle of  $\vartriangle ABC$ . The exterior angle bisector of  $\angle C$  intersects  $AD$  at  $K$  and it follows that  $AK = AB +BC +C A$ . Given that points  $B$ ,  $E$ , and  $L$  are collinear, find  $\angle C AB$ .**p32.** Let  $a$  be the largest root of the equation  $x^3 -3x^2 +1  0$ . Find the remainder when  $\lfloor a^{2019} \rfloor$  is divided by  $17$ .**p33.**  For all  $x, y \in Q$ , functions  $f , g ,h : Q \to Q$  satisfy  $f (x + g (y)) = g (h( f (x)))+ y$ . If  $f (6)=2$ ,  $g\left( \frac12 \right) = 2$ , and  $h \left( \frac72 \right)= 2$ , find all possible values of  $f (2019)$ .


<u>Round 12</u>  **p34.**  An  $n$ -polyomino is formed by joining  $n$  unit squares along their edges. A free polyomino is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translations, rotations, and reflections that turns one into the other. Let  $P(n)$  be the number of free  $n$ -polyominos. For example,  $P(3) = 2$  and  $P(4) = 5$ . Estimate  $P(20)+P(19)$ . If your estimate is  $E$  and the actual value is  $A$ , your score for this problem will be   $$ \max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right). $$  **p35.**  Estimate   $$ \sum^{2019}_{k=1} sin(k), $$  where  $k$  is measured in radians. If your estimate is  $E$  and the actual value is  $A$ , your score for this problem will be  $\max \, (0,15-10 \cdot |E - A|)$  .**p36.** For a positive integer  $n$ , let  $r_{10}(n)$  be the number of  $10$ -tuples of (not necessarily positive) integers  $(a_1,a_2,... ,a_9,a_{10})$  such that   $$ a^2_1 +a^2_2+ ...+a^2_9+a^2_{10}= n. $$   Estimate  $r_{10}(20)+r_{10}(19)$ . If your estimate is  $E$  and the actual value is  $A$ , your score for this problem will be  $$ \max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right). $$  PS. You should use hide for answers. Rounds 1-4 have been posted [here ](https://artofproblemsolving.com/community/c3h3165997p28809441) and 5-8 [here](https://artofproblemsolving.com/community/c3h3166012p28809547).Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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