MathDB

p1. Find the perimeter of a regular hexagon with apothem

3

.
p2. Concentric circles of radius

1

and r are drawn on a circular dartboard of radius

5

. The probability that a randomly thrown dart lands between the two circles is

0.12

. Find

r

.
p3. Find all ordered pairs of integers

(x, y)

with

0 \le x \le 100

0 \le y \le 100

satisfying

xy = (x - 22) (y + 15) .

p4. Points

A_1

A_2

...

A_{12}

are evenly spaced around a circle of radius

1

, but not necessarily in order. Given that chords

A_1A_2

A_3A_4

, and

A_5A_6

have length

2

and chords

A_7A_8

and

A_9A_{10}

have length

2 sin (\pi / 12)

, find all possible lengths for chord

A_{11}A_{12}

.
p5. Let

a

be the number of digits of

2^{1998}

, and let

b

be the number of digits in

5^{1998}

. Find

a + b

.
p6. Find the volume of the solid in

R^3

defined by the equations

x^2 + y^2 \le 2

x + y + |z| \le 3.

p7. Positive integer

n

is such that

3n

has

28

positive divisors and

4n

has

36

positive divisors. Find the number of positive divisors of

n

.
p8. Define functions

f

and

g

f (x) = x +\sqrt{x}

and

g (x) = x + 1/4

. Compute

g(f(g(f(g(f(g(f(3)))))))).

(Your answer must be in the form

a + b \sqrt{ c}

where

a

b

, and

c

are rational.)
p9. Sequence

(a_1, a_2,...)

is defined recursively by

a_1 = 0

a_2 = 100

, and

a_n = 2a_{n-1}-a_{n-2}-3

. Find the greatest term in the sequence

(a_1, a_2,...)

.
p10. Points

X = (3/5, 0)

and

Y = (0, 4/5)

are located on a Cartesian coordinate system. Consider all line segments which (like

\overline{XY}

) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point

P

\overline{XY}

such that none of these line segments (except

\overline{XY}

itself) pass through

P

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement