
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
*20 problems for 25 minutes.***p1.** Matt has a twenty dollar bill and buys two items worth  $\$ 7:99 $ each. How much change does he receive, in dollars?**p2.** The sum of two distinct numbers is equal to the positive difference of the two numbers. What is the product of the two numbers?**p3.** Evaluate  $  $\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}.$  $ **p4.** A sphere with radius $ r $ has volume $ 2\pi $. Find the volume of a sphere with diameter $ r $.**p5.** Yannick ran $ 100 $ meters in $ 14.22 $ seconds. Compute his average speed in meters per second, rounded to the nearest integer.**p6.** The mean of the numbers $ 2, 0, 1, 5, $ and $ x $ is an integer. Find the smallest possible positive integer value for $ x $.**p7.** Let $ f(x) =\sqrt{2^2 - x^2} $. Find the value of $ f(f(f(f(f(-1))))) $.**p8.** Find the smallest positive integer $ n $ such that $ 20 $ divides $ 15n $ and $ 15 $ divides $ 20n $.**p9.** A circle is inscribed in equilateral triangle $ ABC $. Let $ M $ be the point where the circle touches side $ AB $ and let $ N $ be the second intersection of segment $ CM $ and the circle. Compute the ratio $ \frac{MN}{CN} $ .**p10.** Four boys and four girls line up in a random order. What is the probability that both the first and last person in line is a girl?**p11.** Let $ k $ be a positive integer. After making $ k $ consecutive shots successfully, Andy's overall shooting accuracy increased from $ 65\% $ to $ 70\% $. Determine the minimum possible value of $ k $.**p12.** In square $ ABCD $, $ M $ is the midpoint of side $ CD $. Points $ N $ and $ P $ are on segments $ BC $ and $ AB $ respectively such that $  \angle AMN = \angle MNP = 90^o $. Compute the ratio $ \frac{AP}{PB} $ .**p13.** Meena writes the numbers $ 1, 2, 3 $, and $ 4 $ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $ 1 $, $ 2 $, $ 3 $, $ 4 $. How many sequences could she have written?**p14.** Find the smallest positive integer $ N $ such that $ 2N $ is a perfect square and $ 3N $ is a perfect cube.**p15.** A polyhedron has $ 60 $ vertices, $ 150 $ edges, and $ 92 $ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $ 92 $ faces are pentagons?**p16.** All positive integers relatively prime to $ 2015 $ are written in increasing order. Let the twentieth number be $ p $. The value of $ \frac{2015}{p}-1 $ can be expressed as $ \frac{a}{b} $ , where $ a $ and $ b $ are relatively prime positive integers. Compute $ a + b $.**p17.** Five red lines and three blue lines are drawn on a plane. Given that $ x $ pairs of lines of the same color intersect and $ y $ pairs of lines of different colors intersect, find the maximum possible value of $ y - x $.**p18.** In triangle $ ABC $, where $ AC > AB $, $ M $ is the midpoint of $ BC $ and $ D $ is on segment $ AC $ such that $ DM $ is perpendicular to $ BC $. Given that the areas of $ MAD $ and $ MBD $ are $ 5 $ and $ 6 $, respectively, compute the area of triangle $ ABC $.**p19.** For how many ordered pairs $ (x, y) $ of integers satisfying $ 0 \le x, y \le 10 $ is $ (x + y)^2 + (xy - 1)^2 $ a prime number?**p20.** A solitaire game is played with $ 8 $ red, $ 9 $ green, and $ 10 $ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $ c $, he gains a number of points equal to the number of cards that are not of color $ c$ in his hand. Find the maximum number of points that he can obtain by the end of the game.


PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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