
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 1</u>**p1.** How many powers of  $2$  are greater than  $3$  but less than  $2013$ ?**p2.** What number is equal to six greater than three times the answer to this question? **p3.** Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a styrofoam cup. The cup is a cylinder, open on one end, with base radius  $3$  centimeters and height  $10$  centimeters. What is the exterior surface area of the styrofoam cup?


<u>Round 2</u>**p4.** Andrew has two  $6$ -foot-length sticks that he wishes to make into two of the sides of the entrance to his fort, with the ground being the third side. If he wants to make his entrance in the shape of a triangle, what is the largest area that he can make the entrance?**p5.** Ethan and Devin met a fairy who told them “if you have less than  $15$  dollars, I will give you cake”. If both had integral amounts of dollars, and Devin had 5 more dollars than Ethan, but only Ethan got cake, how many different amounts of money could Ethan have had?**p6.** If  $2012^x = 2013$ , for what value of  $a$ , in terms of  $x$ , is it true that  $2012^a = 2013^2$ ?


<u>Round 3</u>**p7.** Find the ordered triple  $(L, M, T)$  of positive integers that makes the following equation true:   $$ 1 + \dfrac{1}{L + \dfrac{1}{M+\dfrac{1}{T}}}=\frac{79}{43}. $$  **p8.** Jonathan would like to start a banana plantation so he is saving up to buy an acre of land, which costs  $\$ 600,000 $. He deposits $ \ $300,000$  in the bank, which gives  $20\%$  interest compounded at the end of each year. At this rate, how many years will Jonathan have to wait until he can buy the acre of land?**p9.** Arul and Ethan went swimming at their town pool and started to swim laps to see who was in better shape. After one hour of swimming at their own paces, Ethan completed  $32$  more laps than Arul. However, after that, Ethan got tired and swam at half his original speed while Arul’s speed didn’t change. After one more hour, Arul swam a total of  $320$  laps. How many laps did Ethan swim after two hours?


<u>Round 4</u>**p10.** A right triangle with a side length of  $6$  and a hypotenuse of 10 has circles of radius  $1$  centered at each vertex. What is the area of the space inside the triangle but outside all three circles?**p11.** In isosceles trapezoid  $ABCD$ ,  $\overline{AB} \parallel\overline{CD}$  and the lengths of  $\overline{AB}$  and  $\overline{CD}$  are  $2$  and  $6$ , respectively. Let the diagonals of the trapezoid intersect at point  $E$ . If the distance from  $E$  to  $\overline{CD}$  is  $9$ , what is the area of triangle  $ABE$ ?**p12.** If  $144$  unit cubes are glued together to form a rectangular prism and the perimeter of the base is  $54$  units, what is the height?


 PS. You should use hide for answers. Rounds 6-8 are [here](https://artofproblemsolving.com/community/c3h3136014p28427163) and 9-12 [here](https://artofproblemsolving.com/community/c3h3137069p28442224).  Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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