
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**. Find the maximum value of  $e^{\sin x \cos x \tan x}$ .**p2.** A fighter pilot finds that the average number of enemy ZIG planes she shoots down is  $56z -4z^2$ , where  $z$  is the number of missiles she fires. Intending to maximize the number of planes she shoots down, she orders her gunner to “Have a nap ... then fire  $z$  missiles!” where  $z$  is an integer. What should  $z$  be?**p3.** A sequence is generated as follows: if the  $n^{th}$  term is even, then the  $(n+ 1)^{th}$  term is half the  $n^{th}$  term; otherwise it is two more than twice the  $n^{th}$  term. If the first term is  $10$ , what is the  $2008^{th}$  term?**p4.** Find the volume of the solid formed by rotating the area under the graph of  $y =\sqrt{x}$  around the  $x$ -axis, with  $0 \le x \le 2$ .**p5.** Find the volume of a regular octahedron whose vertices are at the centers of the faces of a unit cube.**p6.** What is the area of the triangle with vertices  $(x, 0, 0)$ ,  $(0, y, 0)$ , and  $(0, 0, z)$ ?**p7.** Daphne is in a maze of tunnels shown below. She enters at  $A$ , and at each intersection, chooses a direction randomly (including possibly turning around). Once Daphne reaches an exit, she will not return into the tunnels. What is the probability that she will exit at  $A$ ?
![Image](https://cdn.artofproblemsolving.com/attachments/c/0/0f8777e9ac9cbe302f042d040e8864d68cadb6.png)**p8.** In triangle  $AXE$ ,  $T$  is the midpoint of  $\overline{EX}$ , and  $P$  is the midpoint of  $\overline{ET}$ . If triangle  $APE$  is equilateral, find  $\cos(m \angle XAE)$ .**p9.** In rectangle  $XKCD$ ,  $J$  lies on  $\overline{KC}$  and  $Z$  lies on  $\overline{XK}$ . If  $\overline{XJ}$  and  $\overline{KD}$  intersect at  $Q$ ,  $\overline{QZ} \perp \overline{XK}$ , and  $\frac{KC}{KJ} = n$ , find  $\frac{XD}{QZ}$  .**p10.** Bill the magician has cards  $ A$ ,  $ B$ , and  $C$  as shown. For his act, he asks a volunteer to pick any number from  $ 1$  through  $ 8$  and tell him which cards among  $ A$ ,  $ B$ , and  $C$  contain it. He then uses this information to guess the volunteer’s number (for example, if the volunteer told Bill “ $A$  and  $C$ ”, he would guess “3”). 
One day, Bill loses card  $C$  and cannot remember which numbers were on it. He is in a hurry and randomly chooses four different numbers from  $ 1$  to  $ 8$  to write on a card. What is the probability Bill will still be able to do his trick? $A$ : 2 3 5 7 $B$ : 2 4 6 7 $C$ : 2 3 6 1**p11.** Given that  $f(x, y) = x^7y^8 + x^4y^{14} + A$  has root  $(16, 7)$ , find another root.**p12.** How many nonrectangular trapezoids can be formed from the vertices of a regular octagon?**p13.** If  $re^{i\theta}$  is a root of  $x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = 0$ ,  $r > 0$ , and  $0 \le \theta < 360$  with  $\theta$  in degrees, find all possible values of  $\theta$ .**p14.** For what real values of  $n$  is   $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\tan(x))^n dx$  defined?**p15.** A parametric graph is given by   $$ \begin{cases} y = \sin t  x = \cos t +\frac12 t \end{cases} $$  How many times does the graph intersect itself between  $x = 1$  and  $x = 40$ ?


PS. You had better use hide for answers.

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