MathDB

[hide=E stands for Euclid, L stands for Lobachevsky]they had two problem sets under those two names
E1. What is the perimeter of a rectangle if its area is

24

and one side length is

6

?
E2. John moves 3 miles south, then

2

miles west, then

7

miles north, and then

5

miles east. What is the length of the shortest path, in miles, from John's current position to his original position?
E3. An equilateral triangle

ABC

is drawn with side length

2

. The midpoints of sides

AB

BC

, and

CA

are constructed, and are connected to form a triangle. What is the perimeter of the newly formed triangle?
E4. Let triangle

ABC

have sides

AB = 74

and

AC = 5

. What is the sum of all possible integral side lengths of BC?
E5. What is the area of quadrilateral

ABCD

on the coordinate plane with

A(1, 0)

B(0, 1)

C(1, 3)

, and

D(5, 2)

?
E6 / L1. Let

ABCD

be a square with side length

30

. A circle centered at the center of

ABCD

with diameter

34

is drawn. Let

E

and

F

be the points at which the circle intersects side

AB

. What is

EF

?
E7 / L2. What is the area of the quadrilateral bounded by

|2x| + |3y| = 6

?
E8. A circle

O

with radius

2

has a regular hexagon inscribed in it. Upon the sides of the hexagon, equilateral triangles of side length

2

are erected outwards. Find the area of the union of these triangles and circle

O

.
L3. Right triangle

ABC

has hypotenuse

AB

. Altitude

CD

divides

AB

into segments

AD

and

DB

, with

AD = 20

and

DB = 16

. What is the area of triangle

ABC

?
L4. Circle

O

has chord

AB

. Extend

AB

past

B

to a point

C

. A ray from

C

is drawn, and this ray intersects circle

O

. Let point

D

be the point of intersection of the ray and the circle that is closest to point

C

. Given

AB = 20

BC = 16

, and

OA = \frac{201}{6}

, find the longest possible length of

CD

.
L5. Consider a circular cone with vertex

A

. The cone's height is

4

and the radius of its base is

3

. Inscribe a sphere inside the cone. Find the ratio of the volume of the cone to the volume of the sphere.
L6. A disk of radius

\frac12

is randomly placed on the coordinate plane. What is the probability that it contains a lattice point (point with integer coordinates)?
L7. Let

ABC

be an equilateral triangle of side length

2

. Let

D

be the midpoint of

BC

, and let

P

be a variable point on

AC

. By moving

P

along

AC

, what is the minimum perimeter of triangle

BDP

?
L8. Let

ABCD

be a rectangle with

AB = 8

and

BC = 9

. Let

DEFG

be a rhombus, where

G

is on line

BC

and

A

is on line

EF

. If

m\angle EFG = 30^o, what is

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

Problem Statement