MathDB

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
D1. A Giant Hopper is

200

meters away from you. It can hop

50

meters. How many hops would it take for it to reach you?
D2. A rope of length

6

is used to form the edges of an equilateral triangle (a triangle with equal side lengths). What is the length of one of these edges?
D3 / Z1. Point

E

is on side

AB

of rectangle

ABCD

. Find the area of triangle

ECD

divided by the area of rectangle

ABCD

.
D4 / Z2. Garb and Grunt have two rectangular pastures of area

30

. Garb notices that his has a side length of

3

, while Grunt’s has a side length of

5

. What’s the positive difference between the perimeters of their pastures?
D5. Let points

A

and

B

be on a circle with radius

6

and center

O

. If

\angle AOB = 90^o

, find the area of triangle

AOB

.
D6 / Z3. A scalene triangle (the

3

side lengths are all different) has integer angle measures (in degrees). What is the largest possible difference between two angles in the triangle?
D7. Square

ABCD

has side length

6

. If triangle

ABE

has area

9

, find the sum of all possible values of the distance from

E

to line

CD

.
D8 / Z4. Let point

E

be on side

\overline{AB}

of square

ABCD

with side length

2

. Given

DE = BC+BE

, find

BE

.
Z5. The two diagonals of rectangle

ABCD

meet at point

E

. If

\angle AEB = 2\angle BEC

, and

BC = 1

, find the area of rectangle

ABCD

.
Z6. In

\vartriangle ABC

, let

D

be the foot of the altitude from

A

BC

. Additionally, let

X

be the intersection of the angle bisector of

\angle ACB

and

AD

. If

BD = AC = 2AX = 6

, find the area of

ABC

.
Z7. Let

\vartriangle ABC

have

\angle ABC = 40^o

. Let

D

and

E

be on

\overline{AB}

and

\overline{AC}

respectively such that DE is parallel to

\overline{BC}

, and the circle passing through points

D

E

, and

C

is tangent to

\overline{AB}

. If the center of the circle is

O

, find

\angle DOE

.
Z8. Consider

\vartriangle ABC

with

AB = 3

BC = 4

, and

AC = 5

. Let

D

be a point of

AC

other than

A

for which

BD = 3

, and

E

be a point on

BC

such that

\angle BDE = 90^o

. Find

EC

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

Problem Statement