[b]p1.[/b] Point $P$ has coordinate value $(3,2)$, and point $Q$ has coordinate value $(6, x)$. For what value of $x$ does Line $PQ$ have a slope of $6$?


[b]p2.[/b] I have fewer than one hundred pens. When my pens are grouped into nines, I have one pen left over. When grouped into fives, I have three left over. When grouped into pairs, I have one left over. How many do I have?


[b]p3.[/b] How many unique rearrangements of the word “COMBO” exist?


[b]p4.[/b] A right triangle has sides of length $4x+1$, $2x-1$, and $x+9$, where $x$ is a positive integer, and the side of length $4x+1$ is the hypotenuse. What is the area of the triangle?


[b]p5.[/b] A parabola has vertex $(4,-2)$ and roots $(8,0)$ and $(0,0)$. What is the coefficient of the linear term when the equation of the parabola is expanded?


[b]p6.[/b] The terms $2x-1$, $x + 4$, $4x + 1$ form an arithmetic sequence while the terms $x$, $x+4$, $xy^2$ form a geometric sequence. Evaluate $x+y$.


[b]p7.[/b] A sphere is inscribed inside a cube that is inscribed inside another sphere. The larger sphere has volume $972\pi$. Calculate the volume of the space inside the larger sphere but outside the smaller sphere. Express your answer in simplest radical form and in terms of $\pi$.


[b]p8.[/b] A computer battery loses $50\%$ of its remaining power every hour. Assuming the computer begins at full battery, how long will it take for the computer to reach $10\%$ battery? Express your answer in minutes and round your answer to the nearest hundredth.


[b]p9.[/b] How many two-digit numbers have six factors?


[b]p10. [/b]Thomas is graduating. He decides to attend a college that offers him a sizable scholarship. The number of dollar he receives is an eight digit number using the digits $0$ through $7$ without repetition and follows the following criteria:
$\bullet$ The number is divisible by $5$.
$\bullet$ The last four digits are prime numbers.
$\bullet$ A three digit number comprised of the last three digits in order is divisible by $11$.
$\bullet$ A four digit number comprised of the first four digits in order is divisible by $205$.
How much money is Thomas receiving in scholarship?


PS. You should use hide for answers.  Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Let \( X \) be the total number of pens.
2. We are given the following congruences:
   \[
   \begin{align*}
   X &\equiv 1 \pmod{9}, \\
   X &\equiv 3 \pmod{5}, \\
   X &\equiv 1 \pmod{2}.
   \end{align*}
   \]
3. First, we solve the congruences \( X \equiv 1 \pmod{9} \) and \( X \equiv 1 \pmod{2} \). Since \( 9 \) and \( 2 \) are coprime, we can combine these congruences:
   \[
   X \equiv 1 \pmod{\text{lcm}(9, 2)} = X \equiv 1 \pmod{18}.
   \]
4. Now, we need to solve the system:
   \[
   \begin{align*}
   X &\equiv 1 \pmod{18}, \\
   X &\equiv 3 \pmod{5}.
   \end{align*}
   \]
5. We express \( X \) in terms of the modulus 18:
   \[
   X = 18k + 1 \quad \text{for some integer } k.
   \]
6. Substitute \( X \) into the second congruence:
   \[
   18k + 1 \equiv 3 \pmod{5}.
   \]
7. Simplify the congruence:
   \[
   18k + 1 \equiv 3 \pmod{5} \implies 3k + 1 \equiv 3 \pmod{5} \implies 3k \equiv 2 \pmod{5}.
   \]
8. To solve \( 3k \equiv 2 \pmod{5} \), we find the multiplicative inverse of 3 modulo 5, which is 2 (since \( 3 \cdot 2 \equiv 1 \pmod{5} \)):
   \[
   k \equiv 2 \cdot 2 \pmod{5} \implies k \equiv 4 \pmod{5}.
   \]
9. Express \( k \) in terms of the modulus 5:
   \[
   k = 5m + 4 \quad \text{for some integer } m.
   \]
10. Substitute \( k \) back into the expression for \( X \):
    \[
    X = 18(5m + 4) + 1 = 90m + 73.
    \]
11. Therefore, \( X \equiv 73 \pmod{90} \).
12. Since \( X \) must be fewer than 100, the only solution in the range \( [0, 99] \) is:
    \[
    \boxed{73}.
    \]