MathDB

p1. Farmer John wants to bring some cows to a pasture with grass that grows at a constant rate. Initially, the pasture has some nonzero amount of grass and it will stop growing if there is no grass left. The pasture sustains

100

cows for ten days. The pasture can also sustain

100

cows for five days, and then

120

cows for three more days. If cows eat at a constant rate, fund the maximum number of cows Farmer John can bring to the pasture so that they can be sustained indefinitely.
p2. Sam is learning basic arithmetic. He may place either the operation

+

-

in each of the blank spots between the numbers below:

5\,\, \_ \,\, 8\,\, \_ \,\,9\,\, \_ \,\,7\,\,\_ \,\,2\,\,\_ \,\,3

In how many ways can he place the operations so the result is divisible by

3

?
p3. Will loves the color blue, but he despises the color red. In the

5\times 6

rectangular grid below, how many rectangles are there containing at most one red square and with sides contained in the gridlines? https://cdn.artofproblemsolving.com/attachments/1/7/7ce55bdc9e05c7c514dddc7f8194f3031b93c4.png
p4. Let

r_1, r_2, r_3

be the three roots of a cubic polynomial

P(x)

. Suppose that

\frac{P(2) + P(-2)}{P(0)}= 200.

\frac{1}{r_1r_2}+ \frac{1}{r_2r_3}+\frac{1}{r_3r_1}= \frac{m}{n}

for relatively prime positive integers

m

and

n

, compute

m + n

.
p5. Consider a rectangle

ABCD

with

AB = 3

and

BC = 1

. Let

O

be the intersection of diagonals

AC

and

BD

. Suppose that the circumcircle of

\vartriangle ADO

intersects line

AB

again at

E \ne A

. Then, the length

BE

can be written as

\frac{m}{n}

for relatively prime positive integers

m

and

n

. Find

m + n

.
p6. Let

ABCD

be a square with side length

100

and

M

be the midpoint of side

AB

. The circle with center

M

and radius

50

intersects the circle with center

D

and radius

100

at point

E

CE

intersects

AB

F

. If

AF = \frac{m}{n}

for relatively prime positive integers

m

and

n

, find

m + n

.
p7. How many pairs of real numbers

(x, y)

, with

0 < x, y < 1

satisfy the property that both

3x + 5y

and

5x + 2y

are integers?
p8. Sebastian is coloring a circular spinner with

4

congruent sections. He randomly chooses one of four colors for each of the sections. If two or more adjacent sections have the same color, he fuses them and considers them as one section. (Sections meeting at only one point are not adjacent.) Suppose that the expected number of sections in the final colored spinner is equal to

\frac{m}{n}

for relatively prime positive integers

m

and

n

. Compute

m + n

.
p9. Let

ABC

be a triangle and

D

be a point on the extension of segment

BC

past

C

. Let the line through

A

perpendicular to

BC

\ell

. The line through

B

perpendicular to

AD

and the line through

C

perpendicular to

AD

intersect

\ell

H_1

and

H_2

, respectively. If

AB = 13

BC = 14

CA = 15

, and

H_1H_2 = 1001

, find

CD

.
p10. Find the sum of all positive integers

k

such that \frac21 -\frac{3}{2 \times 1}+\frac{4}{3\times 2\times 1} + ...+ (-1)^{k+1} \frac{k+1}{k\times (k - 1)\times ... \times 2\times 1} \ge 1 + \frac{1}{700^3}
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement