
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
<details><summary>R stands for Ramanujan  , P stands for Pascal</summary>they had two problem sets under those two names</details> 

<u>Set 3</u>**P3.11** Find all possible values of  $c$  in the following system of equations:  $$ a^2 + ab + c^2 = 31 $$    $$ b^2 + ab - c^2 = 18 $$    $$ a^2 - b^2 = 7 $$  **P3.12 / R5.25**  In square  $ABCD$  with side length  $13$ , point  $E$  lies on segment  $CD$ . Segment  $AE$  divides  $ABCD$  into triangle  $ADE$  and quadrilateral  $ABCE$ . If the ratio of the area of  $ADE$  to the area of  $ABCE$  is  $4 : 11$ , what is the ratio of the perimeter of  $ADE$  to the perimeter of $ ABCE$ ?**P3.13** Thomas has two distinct chocolate bars. One of them is  $1$  by  $5$  and the other one is  $1$  by  $3$ . If he can only eat a single  $1$  by  $1$  piece off of either the leftmost side or the rightmost side of either bar at a time, how many different ways can he eat the two bars?**P3.14** In triangle  $ABC$ ,  $AB = 13$ ,  $BC = 14$ , and  $CA = 15$ . The entire triangle is revolved about side  $BC$ . What is the volume of the swept out region?**P3.15** Find the number of ordered pairs of positive integers  $(a, b)$  that satisfy the equation  $a(a -1) + 2ab + b(b - 1) = 600$ .


<u>Set 4</u>**P4.16** Compute the sum of the digits of  $(10^{2017} - 1)^2$  .**P4.17** A right triangle with area  $210$  is inscribed within a semicircle, with its hypotenuse coinciding with the diameter of the semicircle.  $2$  semicircles are constructed (facing outwards) with the legs of the triangle as their diameters. What is the area inside the  $2$  semicircles but outside the first semicircle?**P4.18**  Find the smallest positive integer  $n$  such that exactly  $\frac{1}{10}$  of its positive divisors are perfect squares.**P4.19**  One day, Sambuddha and Jamie decide to have a tower building competition using oranges of radius  $1$  inch. Each player begins with  $14$  oranges. Jamie builds his tower by making a  $3$  by  $3$  base, placing a  $2$  by  $2$  square on top, and placing the last orange at the very top. However, Sambuddha is very hungry and eats  $4$  of his oranges. With his remaining  $10$  oranges, he builds a similar tower, forming an equilateral triangle with  $3$  oranges on each side, placing another equilateral triangle with  $2$  oranges on each side on top, and placing the last orange at the very top. What is the positive difference between the heights of these two towers?**P4.20**  Let  $r, s$ , and  $t$  be the roots of the polynomial  $x^3 - 9x + 42$ . Compute the value of  $(rs)^3 + (st)^3 + (tr)^3$ .


<u>Set 5</u>**P5.21**  For all integers  $k > 1$ ,  $\sum_{n=0}^{\infty}k^{-n} =\frac{k}{k -1}$ .
There exists a sequence of integers  $j_0, j_1, ...$  such that  $\sum_{n=0}^{\infty}j_n k^{-n} =\left(\frac{k}{k -1}\right)^3$  for all integers  $k > 1$ . Find  $j_{10}$ .**P5.22** Nimi is a triangle with vertices located at  $(-1, 6)$ ,  $(6, 3)$ , and  $(7, 9)$ . His center of mass is tied to his owner, who is asleep at  $(0, 0)$ , using a rod. Nimi is capable of spinning around his center of mass and revolving about his owner. What is the maximum area that Nimi can sweep through?**P5.23**  The polynomial  $x^{19} - x - 2$  has  $19$  distinct roots. Let these roots be  $a_1, a_2, ..., a_{19}$ . Find  $a^{37}_1 + a^{37}_2+...+a^{37}_{19}$ .**P5.24** I start with a positive integer  $n$ . Every turn, if  $n$  is even, I replace  $n$  with  $\frac{n}{2}$ , otherwise I replace  $n$  with  $n-1$ . Let  $k$  be the most turns required for a number  $n < 500$  to be reduced to  $1$ . How many values of  $n < 500$  require k turns to be reduced to  $1$ ?**P5.25**  In triangle  $ABC$ ,  $AB = 13$ ,  $BC = 14$ , and  $AC = 15$ . Let  $I$  and  $O$  be the incircle and circumcircle of  $ABC$ , respectively. The altitude from  $A$  intersects  $I$  at points  $P$  and  $Q$ , and  $O$  at point  $R$ , such that  $Q$  lies between  $P$  and  $R$ . Find  $PR$ .


PS. You should use hide for answers.  R1-15 /P1-5 have been posted [here](https://artofproblemsolving.com/community/c3h2786721p24495629), and R16-30 /P6-10/ P26-30 [here](https://artofproblemsolving.com/community/c3h2786837p24497019) Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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