[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$?


[b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails?


[b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$.


[b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$?


[b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit  9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average.


[b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number.


[b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle.


[b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all  fit inside the rectangle without overlapping.


[b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$.


[b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there?


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. We start with the given equation involving the smallest nonzero real number \( x \) such that the reciprocal of the number is equal to the number minus the square root of the square of the number. Mathematically, this can be written as:
   \[
   \frac{1}{x} = x - \sqrt{x^2}
   \]
2. Since \(\sqrt{x^2} = |x|\), the equation becomes:
   \[
   \frac{1}{x} = x - |x|
   \]
3. We need to consider the cases for \( x \):
   - If \( x > 0 \), then \(|x| = x\), and the equation simplifies to:
     \[
     \frac{1}{x} = x - x = 0
     \]
     This is not possible since \(\frac{1}{x}\) cannot be zero.
   - If \( x < 0 \), then \(|x| = -x\), and the equation simplifies to:
     \[
     \frac{1}{x} = x - (-x) = x + x = 2x
     \]
     Thus, we have:
     \[
     \frac{1}{x} = 2x
     \]
4. Solving for \( x \), we multiply both sides by \( x \):
   \[
   1 = 2x^2 \implies x^2 = \frac{1}{2} \implies x = \pm \frac{1}{\sqrt{2}}
   \]
5. Since \( x < 0 \), we take the negative value:
   \[
   x = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}
   \]
6. We are given that \( x = -\frac{\sqrt{p}}{q} \). Comparing this with \( x = -\frac{\sqrt{2}}{2} \), we identify \( p = 2 \) and \( q = 2 \).
7. Finally, we need to find \( p + q \):
   \[
   p + q = 2 + 2 = 4
   \]

The final answer is \( \boxed{4} \).