MathDB

Set 1
p1. Farmer John has

4000

gallons of milk in a bucket. On the first day, he withdraws

10\%

of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is

10\%

more than the percentage he withdrew on the previous day. For example, he withdraws

20\%

of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day?
p2. Will multiplies the first four positive composite numbers to get an answer of

w

. Jeremy multiplies the first four positive prime numbers to get an answer of

j

. What is the positive difference between

w

and

j

?
p3. In Nathan’s math class of

60

students,

75\%

of the students like dogs and

60\%

of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats?
Set 2
p4. For how many integers

x

x^4 - 1

prime?
p5. Right triangle

\vartriangle ABC

satisfies

\angle BAC = 90^o

. Let

D

be the foot of the altitude from

A

BC

. If

AD = 60

and

AB = 65

, find the area of

\vartriangle ABC

.
p6. Define

n! = n \times (n - 1) \times ... \times 1

. Given that

3! + 4! + 5! = a^2 + b^2 + c^2

for distinct positive integers

a, b, c

, find

a + b + c

.
Set 3
p7. Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least

2

vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let

V

be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least

2

vertices. Find the nonnegative difference between

M

and

V

.
p8. Let a be the answer to this question, and suppose

a > 0

. Find

\sqrt{a +\sqrt{a +\sqrt{a +...}}}

.
p9. How many ordered pairs of integers

(x, y)

are there such that

x^2 - y^2 = 2019

?
Set 4
p10. Compute

\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}

where

p = 17

q = 7

, and

r = 8

.
p11. The unit squares of a

3 \times 3

grid are colored black and white. Call a coloring good if in each of the four

2 \times 2

squares in the

3 \times 3

grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings.
p12. Define a

k

-respecting string as a sequence of

k

consecutive positive integers

a_1

a_2

...

a_k

such that

a_i

is divisible by

i

for each

1 \le i \le k

. For example,

7

8

9

is a

3

-respecting string because

7

is divisible by

1

8

is divisible by

2

, and

9

is divisible by

3

. Let

S_7

be the set of the first terms of all

7

-respecting strings. Find the sum of the three smallest elements in

S_7

.
Set 5
p13. A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points

I

. Find the sum of all possible values of

I

.
p14. Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer

N

with probability

2^{-N}

, and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins?
p15. If

a, b, c, d

are all positive integers less than

5

, not necessarily distinct, find the number of ordered quadruples

(a, b, c, d)

such that

a^b - c^d

is divisible by

5

.

PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement