
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 5</u>**p13.**  Two concentric circles have radii  $1$  and  $3$ . Compute the length of the longest straight line segment that can be drawn froma point on the circle of radius  $1$  to a point on the circle of radius  $3$  if the segment cannot intersect the circle of radius  $1$ .**p14.**  Find the value of  $\frac{1}{3} + \frac29+\frac{3}{27}+\frac{4}{81}+\frac{5}{243}+...$ **p15.** Bob is trying to type the word "welp". However, he has a  $18$  chance ofmistyping each letter and instead typing one of four adjacent keys, each with equal probability. Find the probability that he types "qelp" instead of "welp".


<u>Round 6</u>**p16.**  How many ways are there to tile a  $2\times 12$  board using an unlimited supply of  $1\times 1$  and  $1\times 3$  pieces?**p17.**  Jeffrey and Yiming independently choose a number between  $0$  and  $1$  uniformly at random. What is the probability that their two numbers can formthe sidelengths of a triangle with longest side of length  $1$ ?**p18.** On  $\vartriangle ABC$  with  $AB = 12$  and  $AC = 16$ , let  $M$  be the midpoint of  $BC$  and  $E$ , $F$  be the points such that  $E$  is on  $AB$ ,  $F$  is on  $AC$ , and  $AE = 2AF$ . Let  $G$  be the intersection of  $EF$  and  $AM$ . Compute  $\frac{EG}{GF}$  .


<u>Round 7</u>**p19.** Find the remainder when  $2019x^{2019} -2018x^{2018}+ 2017x^{2017}-...+x$  is divided by  $x +1$ .**p20.** Parallelogram  $ABCD$  has  $AB = 5$ ,  $BC = 3$ , and  $\angle ABC = 45^o$ . A line through C intersects  $AB$  at  $M$  and  $AD$  at  $N$  such that  $\vartriangle BCM$  is isosceles. Determine the maximum possible area of  $\vartriangle MAN$ .**p21**. Determine the number of convex hexagons whose sides only lie along the grid shown below.
![Image](https://cdn.artofproblemsolving.com/attachments/2/9/93cf897a321dfda282a14e8f1c78d32fafb58d.png)

<u>Round 8</u>**p22.**  Let  $\vartriangle ABC$  be a triangle with side lengths  $AB = 4$ ,  $BC = 5$ , and  $C A = 6$ . Extend ray  $\overrightarrow{AB}$  to a point  $D$  such that  $AD = 12$ , and similarly extend ray   $\overrightarrow{CB}$  to point  $E$  such that  $CE = 15$ . Let  $M$  and  $N$  be points on the circumcircles of  $ABC$  and  $DBE$ , respectively, such that line  $MN$  is tangent to both circles. Determine the length of  $MN$ .**p23.** A volcano will erupt with probability  $\frac{1}{20-n}$  if it has not erupted in the last  $n$  years. Determine the expected number of years between consecutive eruptions.**p24.**  If  $x$  and  $y$  are integers such that  $x+ y = 9$  and  $3x^2+4x y = 128$ , find  $x$ .


PS. You should use hide for answers. Rounds 1-4 have been posted [here ](https://artofproblemsolving.com/community/c3h3165997p28809441) and 9-12 [here](https://artofproblemsolving.com/community/c3h3166099p28810427).Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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