[b]p1.[/b] Kevin plants corn and cotton. Once he harvests the crops, he has $30$ pounds of corn and $x$ pounds of cotton. Corn sells for $\$5$ per pound and cotton sells for $\$10$ per pound. If Kevin sells all his corn and cotton for a total of $\$640$, then compute $x$.


[b]p2.[/b] $ABCD$ is a square where $AB =\sqrt{2016}$. Let $X$ be a point on $AB$ and $Y$ be a point on $CD$ such that $AX = CY$ . Compute the area of trapezoid $AXYD$.


[b]p3.[/b] If the integer $n$ leaves a remainder of 4 when divided by $5$, then what is the largest possible remainder when $2n$ is divided by $15$?


[b]p4.[/b] Let $d(n)$ represent the sum of the digits of the integer $n$. For example, $d(2016) = 2 + 0 + 1 + 6 = 9$. For how many positive 3-digit integers k is it true that $d(k) > d(k + 1)$?


[b]p5.[/b] Let $A, B$, and $C$ be three points on circle $O$ such that $AC$ is a diameter of $O$. Extend line $AC$ to a point $D$ such that $DB$ is tangent to $O$ at $B$, and suppose that $\angle ADB = 20^o$. Compute  $\angle ACB$.


[b]p6.[/b] A group of n people, including Caroline and her best friend Catherine, stand in random order around a circle, with each order being equally likely. If the probability that Catherine is adjacent to Caroline is $1/9$ , then what is the value of $n$?


[b]p7.[/b] The polynomial $P(x) = x^4 + 4x^3 + 8x^2 + 8x + 4$ is the square of another polynomial $Q(x) = ax^2 + bx + c$ with positive coefficients $a, b$, and $c$. Compute $4a + 2b + c$.


[b]p8.[/b] A physics class has $25$ students, two of whom are Alex and Justin. Three students are chosen uniformly at random for a lab demonstration. What is the probability that at least one of Alex and Justin is chosen?


[b]p9.[/b] Natasha carries a paintbrush of length $1$ and walks around the perimeter of a regular $2016$-gon with side length $1$, painting all of the area outside the polygon that she can reach. What is the area that Natasha paints?


[b]p10.[/b] Let $S$ be the set of values which can be written as the sum of five consecutive perfect squares. What is the smallest element of $S$ which is divisible by $17$?


[b]p11.[/b] Consider $6$ points, two of which are $A$ and $B$, such that each pair of points is connected by a line segment and no three points are collinear. Compute the number of distinct paths from $A$ to $B$ such that no point is visited more than once.


[b]p12.[/b] Let $f(x) = 3x^2 + 2x+ 6$. Two distinct lines $l_1$ and $l_2$ exist that are tangent to $f(x)$ and intersect at the origin. Given that l1 is tangent to $f(x)$ at $(x_1, f(x_1))$ and that $l_2$ is tangent to $f(x)$ at $(x_2, f(x_2))$, compute $x_1x_2$.


[b]p13.[/b] One day, Connie plays a game with a fair $6$-sided die. Connie rolls the die until she rolls a $6$, at which point the game ends. If she rolls a $6$ on her first turn, Connie wins $6$ dollars. For each subsequent turn, Connie wins $1/6$ of the amount she would have won the previous turn. What is Connie’s expected earnings from the game?


[b]p14.[/b] Calculate the positive root of the polynomial $(x - 1)(x - 2)(x - 3)(x - 4) - 63$.


[b]p15.[/b] Consider an equilateral triangle 4ABC with unit side length. Let $M$ be the midpoint of side $AB$. A ball is released in a straight line from $M$ and bounces off the side $BC$ at a point $D$. Then, it bounces off the side $CA$ at a point $E$ and lands exactly at $B$. By the law of reflection, we have $\angle BDM = \angle CDE$ and $\angle CED = \angle AEB$. Calculate $MD + DE + EB$, the distance that the ball travels before reaching $B$.


PS. You should use hide for answers. Problems 16-27 have been posted [url=https://artofproblemsolving.com/community/c4h2780524p24413447]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. **Problem 1:**
   Kevin plants corn and cotton. Once he harvests the crops, he has 30 pounds of corn and \( x \) pounds of cotton. Corn sells for \$5 per pound and cotton sells for \$10 per pound. If Kevin sells all his corn and cotton for a total of \$640, then compute \( x \).

   Let's denote the total revenue from selling corn and cotton as \( R \). We know:
   \[
   R = 5 \times 30 + 10 \times x = 640
   \]
   Simplifying the equation:
   \[
   150 + 10x = 640
   \]
   Subtract 150 from both sides:
   \[
   10x = 490
   \]
   Divide by 10:
   \[
   x = 49
   \]

The final answer is \( \boxed{ 8 } \)