
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 the smallest positive integer  $x$  such that  $\frac{1}{x} <\sqrt{12011} - \sqrt{12006}$ ?**p2.** Two soccer players run a drill on a  $100$  foot by  $300$  foot rectangular soccer eld. The two players start on two different corners of the rectangle separated by  $100$  feet, then run parallel along the long edges of the eld, passing a soccer ball back and forth between them. Assume that the ball travels at a constant speed of  $50$  feet per second, both players run at a constant speed of  $30$  feet per second, and the players lead each other perfectly and pass the ball as soon as they receive it, how far has the ball travelled by the time it reaches the other end of the eld?**p3.** A trapezoid  $ABCD$  has  $AB$  and  $CD$  both perpendicular to  $AD$  and  $BC =AB + AD$ . If  $AB = 26$ , what is  $\frac{CD^2}{AD+CD}$  ?**p4.** A hydrophobic, hungry, and lazy mouse is at  $(0, 0)$ , a piece of cheese at  $(26, 26)$ , and a circular lake of radius  $5\sqrt2$  is centered at  $(13, 13)$ . What is the length of the shortest path that the mouse can take to reach the cheese that also does not also pass through the lake?**p5.** Let  $a, b$ , and  $c$  be real numbers such that  $a + b + c = 0$  and  $a^2 + b^2 + c^2 = 3$ . If  $a^5 + b^5 + c^5\ne 0$ , compute  $\frac{(a^3+b^3+c^3)(a^4+b^4+c^4)}{a^5+b^5+c^5}$ .**p6.** Let  $S$  be the number of points with integer coordinates that lie on the line segment with endpoints  $\left( 2^{2^2}, 4^{4^4}\right)$  and   $\left(4^{4^4}, 0\right)$ . Compute  $\log_2 (S - 1)$ .**p7.** For a positive integer  $n$  let  $f(n)$  be the sum of the digits of  $n$ . Calculate   $$ f(f(f(2^{2006}))) $$  **p8.** If  $a_1, a_2, a_3, a_4$  are roots of  $x^4 - 2006x^3 + 11x + 11 = 0$ , find  $|a^3_1 + a^3_2 + a^3_3 + a^3_4|$ .**p9.** A triangle  $ABC$  has  $M$  and  $N$  on sides  $BC$  and  $AC$ , respectively, such that  $AM$  and  $BN$  intersect at  $P$  and the areas of triangles  $ANP$ ,  $APB$ , and  $PMB$  are  $5$ ,  $10$ , and  $8$  respectively. If  $R$  and  $S$  are the midpoints of  $MC$  and  $NC$ , respectively, compute the area of triangle  $CRS$ .**p10.** Jack's calculator has a strange button labelled ''PS.'' If Jack's calculator is displaying the positive integer  $n$ , pressing PS will cause the calculator to divide  $n$  by the largest power of  $2$  that evenly divides  $n$ , and then adding 1 to the result and displaying that number. If Jack randomly chooses an integer  $k$  between  $ 1$  and  $1023$ , inclusive, and enters it on his calculator, then presses the PS button twice, what is the probability that the number that is displayed is a power of  $2$ ?


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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