MathDB

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

? 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

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

\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.

Problem Statement