MathDB

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
D1. Triangle

ABC

has

AB = 3

BC = 4

, and

\angle B = 90^o

. Find the area of triangle

ABC

.
D2 / L1. Let

ABCDEF

be a regular hexagon. Given that

AD = 5

, find

AB

.
D3. Caroline glues two pentagonal pyramids to the top and bottom of a pentagonal prism so that the pentagonal faces coincide. How many edges does Caroline’s figure have?
D4 / L3. The hour hand of a clock is

6

inches long, and the minute hand is

10

inches long. Find the area of the region swept out by the hands from

8:45

AM to

9:15

AM of a single day, in square inches.
D5 / L2. Circles

A

B

, and

C

are all externally tangent, with radii

1

10

, and

100

, respectively. What is the radius of the smallest circle entirely containing all three circles?
D6. Four parallel lines are drawn such that they are equally spaced and pass through the four vertices of a unit square. Find the distance between any two consecutive lines.
D7 / L4. In rectangle

ABCD

AB = 2

and

AD > AB

. Two quarter circles are drawn inside of

ABCD

with centers at

A

and

C

that pass through

B

and

D

, respectively. If these two quarter circles are tangent, find the area inside of

ABCD

that is outside both of the quarter circles.
D8 / L6. Triangle

ABC

is equilateral. A circle passes through

A

and is tangent to side

BC

. It intersects sides AB and

AC

again at

E

and

F

, respectively. If

AE = 10

and

AF = 11

, find

AB

.
L5. Find the area of a triangle with side lengths

\sqrt{2}

\sqrt{58}

, and

2\sqrt{17}

.
L7. Triangle

ABC

has area

80

. Point

D

is in the interior of

\vartriangle ABC

such that

AD =6

BD = 4

CD = 16

, and the area of

\vartriangle ADC = 48

. Determine the area of

\vartriangle ADB

.
L8. Given two points

A

and

B

in the plane with

AB = 1

, define

f(C)

to be the circumcenter of triangle

ABC

, if it exists. Find the number of points

X

so that

f^{2019}(X) = X

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

Problem Statement