MathDB

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
C1. A circle has circumference

6\pi

. Find the area of this circle.
C2 / G2. Points

A

B

, and

C

are on a line such that

AB = 6

and

BC = 11

. Find all possible values of

AC

.
C3. A trapezoid has area

84

and one base of length

5

. If the height is

12

, what is the length of the other base?
C4 / G1.

27

cubes of side length 1 are arranged to form a

3 \times 3 \times 3

cube. If the corner

1 \times 1 \times 1

cubes are removed, what fraction of the volume of the big cube is left?
C5. There is a

50

-foot tall wall and a

300

-foot tall guard tower

50

feet from the wall. What is the minimum

a

such that a flat “

X

” drawn on the ground

a

feet from the side of the wall opposite the guard tower is visible from the top of the guard tower?
C6. Steven’s pizzeria makes pizzas in the shape of equilateral triangles. If a pizza with side length 8 inches will feed 2 people, how many people will a pizza of side length of 16 inches feed?
C7 / G3. Consider rectangle

ABCD

, with

1 = AB < BC

. The angle bisector of

\angle DAB

intersects

\overline{BC}

E

and

\overline{DC}

F

. If

FE = FD

, find

BC

.
C8 / G6.

\vartriangle ABC

. is a right triangle with

\angle A = 90^o

. Square

ADEF

is drawn, with

D

\overline{AB}

F

\overline{AC}

, and

E

inside

\vartriangle ABC

. Point

G

is chosen on

\overline{BC}

such that

EG

is perpendicular to

BC

. Additionally,

DE = EG

. Given that

\angle C = 20^o

, find the measure of

\angle BEG

.
G4. Consider a lamp in the shape of a hollow cylinder with the circular faces removed with height

48

cm and radius

7

cm. A point source of light is situated at the center of the lamp. The lamp is held so that the bottom of the lamp is at a height

48

cm above an infinite flat sheet of paper. What is the area of the illuminated region on the flat sheet of paper, in

cm^2

? https://cdn.artofproblemsolving.com/attachments/c/6/6e5497a67ae5ff5a7bff7007834a4271ce3ca7.png
G5. There exist two triangles

ABC

such that

AB = 13

BC = 12\sqrt2

, and

\angle C = 45^o

. Find the positive difference between their areas.
G7. Let

ABC

be an equilateral triangle with side length

2

. Let the circle with diameter

AB

\Gamma

. Consider the two tangents from

C

\Gamma

, and let the tangency point closer to

A

D

. Find the area of

\angle CAD

.
G8. Let

ABC

be a triangle with

\angle A = 60^o

AB = 37

AC = 41

. Let

H

and

O

be the orthocenter and circumcenter of

ABC

, respectively. Find

OH

. The orthocenter of a triangle is the intersection point of the three altitudes. The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement