[b]p1.[/b] Martin’s car insurance costed $\$6000$ before he switched to Geico, when he saved $15\%$ on car insurance. When Mayhem switched to Allstate, he, a safe driver, saved $40\%$ on car insurance. If Mayhem and Martin are now paying the same amount for car insurance, how much was Mayhem paying before he switched to Allstate?


[b]p2.[/b] The $7$-digit number $N$ can be written as $\underline{A} \,\, \underline{2}  \,\,\underline{0}  \,\,\underline{B}  \,\,\underline{2} \,\, \underline{1}  \,\,\underline{5}$. How many values of $N$ are divisible by $9$?


[b]p3.[/b] The solutions to the equation $x^2-18x-115 = 0$ can be represented as $a$ and $b$. What is $a^2+2ab+b^2$?


[b]p4.[/b] The exterior angles of a regular polygon measure to $4$ degrees. What is a third of the number of sides of this polygon?


[b]p5.[/b] Charlie Brown is having a thanksgiving party.
$\bullet$ He wants one turkey, with three different sizes to choose from.
$\bullet$  He wants to have two or three vegetable dishes, when he can pick from Mashed Potatoes, Saut´eed Brussels Sprouts, Roasted Butternut Squash, Buttery Green Beans, and Sweet Yams;
$\bullet$  He wants two desserts out of Pumpkin Pie, Apple Pie, Carrot Cake, and Cheesecake.
How many different combinations of menus are there?


[b]p6.[/b] In the diagram below, $\overline{AD} \cong \overline{CD}$ and $\vartriangle DAB$ is a right triangle with $\angle DAB = 90^o$. Given that the radius of the circle is $6$ and $m \angle ADC = 30^o$, if the length of minor arc $AB$ is written as $a\pi$, what is $a$?
[img]https://cdn.artofproblemsolving.com/attachments/d/9/ea57032a30c16f4402886af086064261d6828b.png[/img]

[b]p7.[/b] This Halloween, Owen and his two friends dressed up as guards from Squid Game. They needed to make three masks, which were black circles with a white equilateral triangle, circle, or square inscribed in their upper halves. Resourcefully, they used black paper circles with a radius of $5$ inches and white tape to create these masks. Ignoring the width of the tape, how much tape did they use? If the length can be expressed $a\sqrt{b}+c\sqrt{d}+ \frac{e}{f} \pi$ such that $b$ and $d$ are not divisible by the square of any prime, and $e$ and $f$ are relatively prime, find $a + b + c + d + e + f$.
[img]https://cdn.artofproblemsolving.com/attachments/0/c/bafe3f9939bd5767ba5cf77a51031dd32bbbec.png[/img]

[b]p8.[/b] Given $LCM (10^8, 8^{10}, n) = 20^{15}$, where $n$ is a positive integer, find the total number of possible values of $n$.


[b]p9.[/b] If one can represent the infinite progression $\frac{1}{11} + \frac{2}{13} + \frac{3}{121} + \frac{4}{169} + \frac{5}{1331} + \frac{6}{2197}+ ...$ as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, what is $a$?


[b]p10.[/b] Consider a tiled $3\times 3$ square without a center tile. How many ways are there to color the squares such that no two colored squares are adjacent (vertically or horizontally)? Consider rotations of an configuration to be the same, and consider the no-color configuration to be a coloring.


[b]p11.[/b] Let $ABC$ be a triangle with $AB = 4$ and $AC = 7$. Let $AD$ be an angle bisector of triangle $ABC$. Point $M$ is on $AC$ such that $AD$ intersects $BM$ at point $P$, and $AP : PD = 3 : 1$. If the ratio $AM : MC$ can be expressed as $\frac{a}{b}$ such that $a$, $b$ are relatively prime positive integers, find $a + b$.


[b]p12.[/b] For a positive integer $n$, define $f(n)$ as the number of positive integers less than or equal to $n$ that are coprime with $n$. For example, $f(9) = 6$ because $9$ does not have any common divisors with $1$,  $2$, $4$, $5$, $7$, or $8$. Calculate: $$\sum^{100}_{i=2} \left( 29^{f(i)}\,\,\, mod \,\,i \right).$$


[b]p13.[/b] Let $ABC$ be an equilateral triangle. Let $P$ be a randomly selected point in the incircle of $ABC$. Find $a+b+c+d$ if the probability that $\angle BPC$ is acute can be expressed as $\frac{a\sqrt{b} -c\pi}{d\pi }$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c, d) = 1$ and $b$ is not divisible by the square of any prime.


[b]p14.[/b] When the following expression is simplified by expanding then combining like terms, how many terms are in the resulting expression? $$(a + b + c + d)^{100} + (a + b - c - d)^{100}$$


[b]p15.[/b] Jerry has a rectangular box with integral side lengths. If $3$ units are added to each side of the box, the volume of the box is tripled. What is the largest possible volume of this box?


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. By the Angle Bisector Theorem, we know that the ratio of the segments created by the angle bisector is equal to the ratio of the other two sides of the triangle. Therefore, we have:
   \[
   \frac{BD}{DC} = \frac{AB}{AC} = \frac{4}{7}
   \]

2. Given that \(\frac{AP}{PD} = \frac{3}{1}\), we can infer that the area ratio of \(\triangle APB\) to \(\triangle PBD\) is also \(\frac{3}{1}\). This is because the areas of triangles with a common height are proportional to their bases.

3. Let us draw the third cevian \(CP\). We need to find the ratio \(\frac{[APB]}{[BPC]}\). Since \(\frac{BD}{DC} = \frac{4}{7}\), we can write:
   \[
   \frac{BD}{DC} = \frac{1}{\frac{7}{4}}
   \]

4. Using the area ratios, we have:
   \[
   \frac{[APB]}{[BPC]} = \frac{3}{\frac{11}{4}} = \frac{12}{11}
   \]

5. Let \(D\) and \(E\) be the feet of the perpendiculars from \(A\) and \(C\) to \(BM\) and \(BM\) produced, respectively. Triangles \(AMP\) and \(CME\) are similar, so:
   \[
   \frac{AD}{CE} = \frac{12}{11}
   \]

6. Therefore, the ratio \(\frac{AM}{MC}\) is:
   \[
   \frac{AM}{MC} = \frac{12}{11} = \frac{a}{b}
   \]

7. Given that \(a\) and \(b\) are relatively prime, we have \(a = 12\) and \(b = 11\). Thus:
   \[
   a + b = 12 + 11 = 23
   \]

The final answer is \( \boxed{ 23 } \).