MathDB

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 [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement