
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.** What is  $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$ ?**p2.** How many integers  $n$  have exactly  $\sqrt{n}$  factors?**p3.** A triangle has distinct angles  $3x+10$ ,  $2x+20$ , and  $x+30$ . What is the value of  $x$ ?**p4.** If  $4$  people of the Math Club are randomly chosen to be captains, and Henry is one of the  $30$  people eligible to be chosen, what is the probability that he is not chosen to be captain?**p5.**  $a, b, c, d$  is an arithmetic sequence with difference  $x$  such that  $a, c, d$  is a geometric sequence. If  $b$  is  $12$ , what is  $x$ ? (Note: the difference of an aritmetic sequence can be positive or negative, but not  $0$ )**p6.** What is the smallest positive integer that contains only  $0$ s and  $5$ s that is a multiple of  $24$ .**p7.** If  $ABC$  is a triangle with side lengths  $13$ ,  $14$ , and  $15$ , what is the area of the triangle made by connecting the points at the midpoints of its sides?**p8.** How many ways are there to order the numbers  $1,2,3,4,5,6,7,8$  such that  $1$  and  $8$  are not adjacent?**p9.** Find all ordered triples of nonnegative integers  $(x, y, z)$  such that  $x + y + z = xyz$ .**p10.** Noah inscribes equilateral triangle  $ABC$  with area  $\sqrt3$  in a cricle. If  $BR$  is a diameter of the circle, then what is the arc length of Noah's  $ARC$ ?**p11.** Today,  $4/12/14$ , is a palindromic date, because the number without slashes  $41214$  is a palindrome. What is the last palindromic date before the year  $3000$ ?**p12.** Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon?**p13.** How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important?**p14.** Find all functions  $f$  from  $R \to R$  such that  $f(x + y) + f(x - y) = x^2 + y^2$ .**p15.** What are the last four digits of  $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$ /**p16.** In how many distinct ways can a regular octagon be divided up into  $6$  non-overlapping triangles?**p17.** Find the sum of the solutions to the equation  $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$  .**p18.** How many integers  $n$  have the property that  $(n+1)(n+2)(n+3)(n+4)$  is a perfect square of an integer?**p19.** A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals?**p20.** In blindfolded solitary tic-tac-toe, a player starts with a blank  $3$ -by- $3$  tic-tac-toe board. On each turn, he randomly places an " $X$ " in one of the open spaces on the board. The game ends when the player gets  $3$   $X$ s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only  $X$ s are used, not  $O$ s). What fraction of games will run the maximum  $7$  amount of moves?


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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