MathDB

[hide=E stands for Euclid , L stands for Lobachevsky]they had two problem sets under those two names
E1. How many positive divisors does

72

have?
E2 / L2. Raymond wants to travel in a car with

3

other (distinguishable) people. The car has

5

seats: a driver’s seat, a passenger seat, and a row of

3

seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car?
E3 / L3. Peter wants to make fruit punch. He has orange juice (

100\%

orange juice), tropical mix (

25\%

orange juice,

75\%

pineapple juice), and cherry juice (

100\%

cherry juice). If he wants his final mix to have

50\%

orange juice,

10\%

cherry juice, and

40\%

pineapple juice, in what ratios should he mix the

3

juices? Please write your answer in the form (orange):(tropical):(cherry), where the three integers are relatively prime.
E4 / L4. Points

A, B, C

, and

D

are chosen on a circle such that

m \angle ACD = 85^o

m\angle ADC = 40^o

,and

m\angle BCD = 60^o

. What is

m\angle CBD

?
E5.

a, b

, and

c

are positive real numbers. If

abc = 6

and

a + b = 2

, what is the minimum possible value of

a + b + c

?
E6 / L5. Circles

A

and

B

are drawn on a plane such that they intersect at two points. The centers of the two circles and the two intersection points lie on another circle, circle

C

. If the distance between the centers of circles

A

and

B

20

and the radius of circle

A

16

, what is the radius of circle

B

?
E7. Point

P

is inside rectangle

ABCD

. If

AP = 5

BP = 6

, and

CP = 7

, what is the length of

DP

?
E8 / L6. For how many integers

n

n^2 + 4

divisible by

n + 2

?
E9. How many of the perfect squares between

1

and

10000

, inclusive, can be written as the sum of two triangular numbers? We define the

n

th triangular number to be

1 + 2 + 3 + ... + n

, where

n

is a positive integer.
E10 / L7. A small sphere of radius

1

is sitting on the ground externally tangent to a larger sphere, also sitting on the ground. If the line connecting the spheres’ centers makes a

60^o

angle with the ground, what is the radius of the larger sphere?
E11 / L8. A classroom has

12

chairs in a row and

5

distinguishable students. The teacher wants to position the students in the seats in such a way that there is at least one empty chair between any two students. In how many ways can the teacher do this?
E12 / L9. Let there be real numbers

a

and

b

such that

a/b^2 + b/a^2 = 72

and

ab = 3

. Find the value of

a^2 + b^2

.
E13 / L10. Find the number of ordered pairs of positive integers

(x, y)

such that

gcd \, (x, y)+lcm \, (x, y) =x + y + 8

.
E14 / L11. Evaluate

\sum_{i=1}^{\infty}\frac{i}{4^i}=\frac{1}{4} +\frac{2}{16} +\frac{3}{64} +...

E15 / L12. Xavier and Olivia are playing tic-tac-toe. Xavier goes first. How many ways can the game play out such that Olivia wins on her third move? The order of the moves matters.
L1. What is the sum of the positive divisors of

100

?
L13. Let

ABCD

be a convex quadrilateral with

AC = 20

. Furthermore, let

M, N, P

, and

Q

be the midpoints of

DA, AB, BC

, and

CD

, respectively. Let

X

be the intersection of the diagonals of quadrilateral

MNPQ

. Given that

NX = 12

and

XP = 10

, compute the area of

ABCD

.
L14. Evaluate

(\sqrt3 + \sqrt5)^6

to the nearest integer.
L15. In Hatland, each citizen wears either a green hat or a blue hat. Furthermore, each citizen belongs to exactly one neighborhood. On average, a green-hatted citizen has

65\%

of his neighbors wearing green hats, and a blue-hatted citizen has

80\%

of his neighbors wearing blue hats. Each neighborhood has a different number of total citizens. What is the ratio of green-hatted to blue-hatted citizens in Hatland? (A citizen is his own neighbor.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement