[hide=R stands for Ramanujan  , P stands for Pascal]they had two problem sets under those two names[/hide] 

[u]Set 3[/u]


[b]P3.11[/b] Find all possible values of $c$ in the following system of equations:
$$a^2 + ab + c^2 = 31$$
$$b^2 + ab - c^2 = 18$$
$$a^2 - b^2 = 7$$


[b]P3.12 / R5.25[/b]  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$?


[b]P3.13[/b] 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?


[b]P3.14[/b] 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?


[b]P3.15[/b] 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]


[b]P4.16[/b] Compute the sum of the digits of $(10^{2017} - 1)^2$ .


[b]P4.17[/b] 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?


[b]P4.18[/b]  Find the smallest positive integer $n$ such that exactly $\frac{1}{10}$ of its positive divisors are perfect squares.


[b]P4.19[/b]  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?


[b]P4.20[/b]  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]


[b]P5.21[/b]  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 $j_{10}$.


[b]P5.22[/b] 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?


[b]P5.23[/b]  The polynomial $x^{19} - x - 2$ has $19$ distinct roots. Let these roots be $a_1, a_2, ..., a_{19}$. Find $a^{37}_1 + a^{37}_2+...+a^{37}_{19}$.


[b]P5.24[/b] 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 $1$?


[b]P5.25[/b]  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 [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and R16-30 /P6-10/ P26-30 [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find all possible values of \( c \) in the given system of equations:
\[
a^2 + ab + c^2 = 31
\]
\[
b^2 + ab - c^2 = 18
\]
\[
a^2 - b^2 = 7
\]

1. **Add the first two equations:**
   \[
   (a^2 + ab + c^2) + (b^2 + ab - c^2) = 31 + 18
   \]
   Simplifying, we get:
   \[
   a^2 + b^2 + 2ab = 49
   \]
   Notice that this can be rewritten as:
   \[
   (a + b)^2 = 49
   \]
   Therefore:
   \[
   a + b = 7 \quad \text{or} \quad a + b = -7
   \]

2. **Use the third equation:**
   \[
   a^2 - b^2 = 7
   \]
   This can be factored using the difference of squares:
   \[
   (a - b)(a + b) = 7
   \]

3. **Case 1: \( a + b = 7 \)**
   \[
   (a - b) \cdot 7 = 7 \implies a - b = 1
   \]
   Solving the system:
   \[
   a + b = 7
   \]
   \[
   a - b = 1
   \]
   Adding these two equations:
   \[
   2a = 8 \implies a = 4
   \]
   Subtracting the second from the first:
   \[
   2b = 6 \implies b = 3
   \]

4. **Substitute \( a = 4 \) and \( b = 3 \) into the first equation:**
   \[
   4^2 + 4 \cdot 3 + c^2 = 31
   \]
   Simplifying:
   \[
   16 + 12 + c^2 = 31 \implies 28 + c^2 = 31 \implies c^2 = 3
   \]
   Therefore:
   \[
   c = \pm \sqrt{3}
   \]

5. **Case 2: \( a + b = -7 \)**
   \[
   (a - b) \cdot (-7) = 7 \implies a - b = -1
   \]
   Solving the system:
   \[
   a + b = -7
   \]
   \[
   a - b = -1
   \]
   Adding these two equations:
   \[
   2a = -8 \implies a = -4
   \]
   Subtracting the second from the first:
   \[
   2b = -6 \implies b = -3
   \]

6. **Substitute \( a = -4 \) and \( b = -3 \) into the first equation:**
   \[
   (-4)^2 + (-4)(-3) + c^2 = 31
   \]
   Simplifying:
   \[
   16 + 12 + c^2 = 31 \implies 28 + c^2 = 31 \implies c^2 = 3
   \]
   Therefore:
   \[
   c = \pm \sqrt{3}
   \]

Since both cases yield the same values for \( c \), the only solutions are:
\[
\boxed{\pm \sqrt{3}}
\]