
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
**p1.** 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 $.**p2.** $ 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 $.**p3.** 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 $?**p4.** 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) $?**p5.** 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 $.**p6.** 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 $?**p7.** 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 $.**p8.** 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?**p9.** 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?**p10.** 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 $?**p11.** 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.**p12.** 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 $.**p13.** 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?**p14.** Calculate the positive root of the polynomial $ (x - 1)(x - 2)(x - 3)(x - 4) - 63 $.**p15.** 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 [here](https://artofproblemsolving.com/community/c4h2780524p24413447). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Present the answer in LaTex format: \boxed{Your answer}
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