
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  $20\times 19 + 20 \div (2 - 7)$ ?**p2.** Will has three spinners. The first has three equally sized sections numbered  $1$ ,  $2$ ,  $3$ ; the second has four equally sized sections numbered  $1$ ,  $2$ ,  $3$ ,  $4$ ; and the third has five equally sized sections numbered  $1$ ,  $2$ ,  $3$ ,  $4$ ,  $5$ . When Will spins all three spinners, the probability that the same number appears on all three spinners is  $p$ . Compute  $\frac{1}{p}$ .**p3.** Three girls and five boys are seated randomly in a row of eight desks. Let  $p$  be the probability that the students at the ends of the row are both boys. If  $p$  can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , compute  $m + n$ .**p4.** Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was  $.300$ . (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting  $10$  home runs and striking out zero times in the last week, Jaron has now raised his batting average to  $.310$ . How many home runs has Jaron now hit?**p5.** Suppose that the sum   $$ \frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100} $$   is expressible as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m + n$ .**p6.** Let  $ABCD$  be a unit square with center  $O$ , and  $\vartriangle OEF$  be an equilateral triangle with center  $A$ . Suppose that  $M$  is the area of the region inside the square but outside the triangle and  $N$  is the area of the region inside the triangle but outside the square, and let  $x = |M -N|$  be the positive difference between  $M$  and  $N$ . If   $$ x =\frac1 8(p -\sqrt{q}) $$   for positive integers  $p$  and  $q$ , find  $p + q$ .**p7.** Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by  $3$ . For example, the number  $1212121$  satisfies this property.**p8.** There is a unique positive integer  $x$  such that  $x^x$  has  $703$  positive factors. What is  $x$ ?**p9.** Let  $x$  be the number of digits in  $2^{2019}$  and let  $y$  be the number of digits in  $5^{2019}$ . Compute  $x + y$ .**p10.** Let  $ABC$  be an isosceles triangle with  $AB = AC = 13$  and  $BC = 10$ . Consider the set of all points  $D$  in three-dimensional space such that  $BCD$  is an equilateral triangle. This set of points forms a circle  $\omega$ . Let  $E$  and  $F$  be points on  $\omega$  such that  $AE$  and  $AF$  are tangent to  $\omega$ . If  $EF^2$  can be expressed in the form  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers, determine  $m + n$ .


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

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