
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
**p1.** How many lines of symmetry does a square have?**p2.** Compute $ 1/2 + 1/6 + 1/12 + 1/4$ .**p3.** What is the maximum possible area of a rectangle with integer side lengths and perimeter  $8$ ?**p4.** Given that  $1$  printer weighs  $400000$  pennies, and  $80$  pennies weighs  $2$  books, what is the weight of a printer expressed in books?**p5.** Given that two sides of a triangle are  $28$  and  $3$  and all three sides are integers, what is the sum of the possible lengths of the remaining side?**p6.** What is half the sum of all positive integers between  $1$  and  $15$ ,  inclusive, that have an even number of positive divisors?**p7.** Austin the Snowman has a very big brain. His head has radius  $3$ , and the volume of his torso is one third of his head, and the volume of his legs combined is one third of his torso. If Austin's total volume is  $a\pi$  where  $a$  is an integer, what is  $a$ ?**p8.** Neethine the Kiwi says that she is the eye of the tiger, a fighter, and that everyone is gonna hear her roar. She is standing at point  $(3, 3)$ . Neeton the Cat is standing at  $(11,18)$ , the farthest he can stand from Neethine such that he can still hear her roar. Let the total area of the region that Neeton can stand in where he can hear Neethine's roar be  $a\pi$  where  $a$  is an integer. What is  $a$ ?**p9.** Consider  $2018$  identical kiwis. These are to be divided between  $5$  people, such that the first person gets  $a_1$  kiwis, the second gets  $a_2$  kiwis, and so forth, with  $a_1 \le a_2 \le a_3 \le a_4 \le a_5$ . How many tuples  $(a_1, a_2, a_3, a_4, a_5)$  can be chosen such that they form an arithmetic sequence?**p10.** On the standard  $12$  hour clock, each number from  $1$  to  $12$  is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between  $7$  and  $6$  while the minute hand is pointing at  $15$ ?**p11.** In equiangular hexagon  $ABCDEF$ ,  $AB = 7$ ,  $BC = 3$ ,  $CD = 8$ , and  $DE = 5$ . The area of the hexagon is in the form  $\frac{a\sqrt{b}}{c}$  with  $b$  square free and  $a$  and  $c$  relatively prime. Find  $a+b+c$  where  $a, b,$  and  $c$  are integers.**p12.** Let  $\frac{p}{q} = \frac15 + \frac{2}{5^2} + \frac{3}{5^3} + ...$  . Find  $p + q$ ,  where  $p$  and  $q$  are relatively prime positive integers.**p13.** Two circles  $F$  and  $G$  with radius  $10$  and  $4$  respectively are externally tangent. A square  $ABMC$  is inscribed in circle  $F$  and equilateral triangle  $MOP$  is inscribed in circle  $G$  (they share vertex  $M$ ). If the area of pentagon  $ABOPC$  is equal to  $a + b\sqrt{c}$ , where  $a$ ,  $b$ ,  $c$  are integers  $c$  is square free, then find  $a + b + c$ .**p14.** Consider the polynomial  $P(x) = x^3 + 3x^2 + ax + 8$ . Find the sum of all integer  $a$  such that the sum of the squares of the roots of  $P(x)$  divides the sum of the coecients of  $P(x)$ .**p15.** Nithin and Antonio play a number game. At the beginning of the game, Nithin picks a prime  $p$  that is less than  $100$ . Antonio then tries to find an integer  $n$  such that  $n^6 + 2n^5 + 2n^4 + n^3 + (n^2 + n + 1)^2$  is a multiple of  $p$ . If Antonio can find such a number n, then he wins, otherwise, he loses. Nithin doesn't know what he is doing, and he always picks his prime randomly while Antonio always plays optimally. The probability of Antonio winning is  $a/b$  where  $a$  and  $b$  are relatively prime positive integers. Find $a + b$ .


PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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