MathDB

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names
B1. If the values of two angles in a triangle are

60

and

75

degrees respectively, what is the measure of the third angle?
B2. Square

ABCD

has side length

1

. What is the area of triangle

ABC

?
B3 / G1. An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is

8

, what is the square’s side length?
B4 / G2. What is the maximum possible number of sides and diagonals of equal length in a quadrilateral?
B5. A square of side length

4

is put within a circle such that all

4

corners lie on the circle. What is the diameter of the circle?
B6 / G3. Patrick is rafting directly across a river

20

meters across at a speed of

5

m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of

12

m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled?
B7 / G4. Quadrilateral

ABCD

has side lengths

AB = 7

BC = 15

CD = 20

, and

DA = 24

. It has a diagonal length of

BD = 25

. Find the measure, in degrees, of the sum of angles

ABC

and

ADC

.
B8 / G5. What is the largest

P

such that any rectangle inscribed in an equilateral triangle of side length

1

has a perimeter of at least

P

?
G6. A circle is inscribed in an equilateral triangle with side length

s

. Points

A

B

C

D

E

F

lie on the triangle such that line segments

AB

CD

, and

EF

are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon

ABCDEF = \frac{9\sqrt3}{2}

, find

s

.
G7. Let

\vartriangle ABC

be such that

\angle A = 105^o

\angle B = 45^o

\angle C = 30^o

. Let

M

be the midpoint of

AC

. What is

\angle MBC

?
G8. Points

A

B

, and

C

lie on a circle centered at

O

with radius

10

. Let the circumcenter of

\vartriangle AOC

P

. If

AB = 16

, find the minimum value of

PB

. 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