MathDB

[hide=R stands for Ramanujan, P stands for Pascal]they had two problem sets under those two names
R1. What is the distance between the points

(6, 0)

and

(-2, 0)

?
R2 / P1. Angle

X

has a degree measure of

35

degrees. What is the supplement of the complement of angle

X

? The complement of an angle is

90

degrees minus the angle measure. The supplement of an angle is

180

degrees minus the angle measure.
R3. A cube has a volume of

729

. What is the side length of the cube?
R4 / P2. A car that always travels in a straight line starts at the origin and goes towards the point

(8, 12)

. The car stops halfway on its path, turns around, and returns back towards the origin. The car again stops halfway on its return. What are the car’s final coordinates?
R5. A full, cylindrical soup can has a height of

16

and a circular base of radius

3

. All the soup in the can is used to fill a hemispherical bowl to its brim. What is the radius of the bowl?
R6. In square

ABCD

, the numerical value of the length of the diagonal is three times the numerical value of the area of the square. What is the side length of the square?
R7. Consider triangle

ABC

with

AB = 3

BC = 4

, and

AC = 5

. The altitude from

B

AC

intersects

AC

H

. Compute

BH

.
R8. Mary shoots

5

darts at a square with side length

2

. Let

x

be equal to the shortest distance between any pair of her darts. What is the maximum possible value of

x

?
P3. Let

ABC

be an isosceles triangle such that

AB = BC

and all of its angles have integer degree measures. Two lines,

\ell_1

and

\ell_2

, trisect

\angle ABC

\ell_1

and

\ell_2

intersect

AC

at points

D

and

E

respectively, such that

D

is between

A

and

E

. What is the smallest possible integer degree measure of

\angle BDC

?
P4. In rectangle

ABCD

AB = 9

and

BC = 8

W

X

Y

, and

Z

are on sides

AB

BC

CD

, and

DA

, respectively, such that

AW = 2WB

CX = 3BX

CY = 2DY

, and

AZ = DZ

. If

WY

and

XZ

intersect at

O

, find the area of

OWBX

.
P5. Consider a regular

n

-gon with vertices

A_1A_2...A_n

. Find the smallest value of

n

so that there exist positive integers

i, j, k \le n

with

\angle A_iA_jA_k = \frac{34^o}{5}

.
P6. In right triangle

ABC

with

\angle A = 90^o

and

AB < AC

D

is the foot of the altitude from

A

BC

, and

M

is the midpoint of

BC

. Given that

AM = 13

and

AD = 5

, what is

\frac{AB}{AC}

?
P7. An ant is on the circumference of the base of a cone with radius

2

and slant height

6

. It crawls to the vertex of the cone

X

in an infinite series of steps. In each step, if the ant is at a point

P

, it crawls along the shortest path on the exterior of the cone to a point

Q

on the opposite side of the cone such that

2QX = PX

. What is the total distance that the ant travels along the exterior of the cone?
P8. There is an infinite checkerboard with each square having side length

2

. If a circle with radius

1

is dropped randomly on the checkerboard, what is the probability that the circle lies inside of exactly

3

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

Problem Statement