
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.** Given that  $x$  and  $y$  are positive real numbers such that  $x^3+y = 20$ , the maximum possible value of  $x + y$  can be written as  $\frac{a\sqrt{b}}{c}$  +d where  $a$ ,  $b$ ,  $c$ , and  $d$  are positive integers such that  $gcd(a,c) = 1$  and  $b$  is square-free. Find  $a +b +c +d$ .**p17.** In  $\vartriangle DRK$  ,  $DR = 13$ ,  $DK = 14$ , and  $RK = 15$ . Let  $E$  be the intersection of the altitudes of  $\vartriangle DRK$ . Find the value of  $\lfloor DE +RE +KE \rfloor$ .**p18.** Subaru the frog lives on lily pad  $1$ . There is a line of lily pads, numbered  $2$ ,  $3$ ,  $4$ ,  $5$ ,  $6$ , and  $7$ . Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either  $1$  or  $2$  greater, chosen at random from valid possibilities. There are alligators on lily pads  $2$  and  $5$ . If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number  $1$ . Find the expected number of jumps it takes Subaru to reach pad  $7$ .


<u>Round 7</u>

This set has problems whose answers depend on one another.**p19.** Let  $B$  be the answer to Problem  $20$  and let  $C$  be the answer to Problem  $21$ . Given that   $$ f (x) = x^3-Bx-C = (x-r )(x-s)(x-t ) $$   where  $r$ ,  $s$ , and  $t$  are complex numbers, find the value of  $r^2+s^2+t^2$ .**p20.** Let  $A$  be the answer to Problem  $19$  and let  $C$  be the answer to Problem  $21$ . Circles  $\omega_1$  and  $\omega_2$  meet at points  $X$  and  $Y$  . Let point  $P \ne Y$  be the point on  $\omega_1$  such that  $PY$  is tangent to  $\omega_2$ , and let point  $Q  \ne Y$  be the point on  $\omega_2$  such that  $QY$  is tangent to  $\omega_1$ . Given that  $PX = A$  and  $QX =C$ , find  $XY$  .**p21.** Let  $A$  be the answer to Problem  $19$  and let  $B$  be the answer to Problem  $20$ . Given that the positive difference between the number of positive integer factors of  $A^B$  and the number of positive integer factors of  $B^A$  is  $D$ , and given that the answer to this problem is an odd prime, find  $\frac{D}{B}-40$ .


<u>Round 8</u>**p22.** Let  $v_p (n)$  for a prime  $p$  and positive integer  $n$  output the greatest nonnegative integer  $x$  such that  $p^x$  divides  $n$ . Find   $$ \sum^{50}_{i=1}\sum^{i}_{p=1} { v_p (i )+1 \choose 2}, $$   where the inner summation only sums over primes  $p$  between  $1$  and  $i$  .**p23.** Let  $a$ ,  $b$ , and  $c$  be positive real solutions to the following equations.   $$ \frac{2b^2 +2c^2 -a^2}{4}= 25 $$    $$ \frac{2c^2 +2a^2 -b^2}{4}= 49 $$    $$ \frac{2a^2 +2b^2 -c^2}{4}= 64 $$   The area of a triangle with side lengths  $a$ ,  $b$ , and  $c$  can be written as  $\frac{x\sqrt{y}}{z}$  where  $x$  and  $z$  are relatively prime positive integers and  $y$  is square-free. Find  $x + y +z$ .**p24.** Alan, Jiji, Ina, Ryan, and Gavin want to meet up. However, none of them know when to go, so they each pick a random  $1$  hour period from  $5$  AM to  $11$  AM to meet up at Alan’s house. Find the probability that there exists a time when all of them are at the house at one time.**Round 9** **p25.** Let  $n$  be the number of registered participantsin this  $LMT$ . Estimate the number of digits of  $\left[ {n \choose 2} \right]$  in base  $10$ . If your answer is  $A$  and the correct answer is  $C$ , then your score will be  $$ \left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor. $$  **p26.** Let  $\gamma$  be theminimum value of  $x^x$  over all real numbers  $x$ . Estimate  $\lfloor 10000\gamma \rfloor$ . If your answer is  $A$  and the correct answer is  $C$ , then your score will be  $$ \left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor. $$  **p27.**  Let   $$ E = \log_{13} 1+log_{13}2+log_{13}3+...+log_{13}513513. $$   Estimate  $\lfloor E \rfloor$ . If your answer is  $A$  and the correct answer is  $C$ , your score will be   $$ \left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor. $$  PS. You should use hide for answers. Rounds 1-5 have been posted [here](https://artofproblemsolving.com/community/c3h3167127p28823220). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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