MathDB

Set 1
B1. Evaluate

2 + 0 - 2 \times 0

.
B2. It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses?
B3. Let

a

be the answer to this question. What is

\frac{1}{2-a}

?
Set 2
B4. Every day at Andover is either sunny or rainy. If today is sunny, there is a

60\%

chance that tomorrow is sunny and a

40\%

chance that tomorrow is rainy. On the other hand, if today is rainy, there is a

60\%

chance that tomorrow is rainy and a

40\%

chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is

n

?
B5. In the diagram below, what is the value of

\angle DD'Y

in degrees? https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png
B6. Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan?
Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid.
Set 3
B7. Let

P

be a point on side

AB

of square

ABCD

with side length

8

such that

PA = 3

. Let

Q

be a point on side

AD

such that

P Q \perp P C

. The area of quadrilateral

PQDB

can be expressed in the form

m/n

for relatively prime positive integers

m

and

n

. Compute

m + n

.
B8. Jessica and Jeffrey each pick a number uniformly at random from the set

\{1, 2, 3, 4, 5\}

(they could pick the same number). If Jessica’s number is

x

and Jeffrey’s number is

y

, the probability that

x^y

has a units digit of

1

can be expressed as

m/n

, where

m

and

n

are relatively prime positive integers. Find

m + n

.
B9. For two points

(x_1, y_1)

and

(x_2, y_2)

in the plane, we define the taxicab distance between them as

|x_1 - x_2| + |y_1 - y_2|

. For example, the taxicab distance between

(-1, 2)

and

(3,\sqrt2)

6-\sqrt2

. What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same?
Set 4
B10. Will wants to insert some × symbols between the following numbers:

1\,\,\,2\,\,\,3\,\,\,4\,\,\,6

to see what kinds of answers he can get. For example, here is one way he can insert

\times

symbols:

1 \times 23 \times 4 \times 6 = 552.

Will discovers that he can obtain the number

276

. What is the sum of the numbers that he multiplied together to get

276

?
B11. Let

ABCD

be a parallelogram with

AB = 5

BC = 3

, and

\angle BAD = 60^o

. Let the angle bisector of

\angle ADC

meet

AC

E

and

AB

F

. The length

EF

can be expressed as

m/n

, where

m

and

n

are relatively prime positive integers. What is

m + n

?
B12. Find the sum of all positive integers

n

such that

\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n

.
Note:

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

.
Set 5
B13. This year, February

29

fell on a Saturday. What is the next year in which February

29

will be a Saturday?
B14. Let

f(x) = \frac{1}{x} - 1

. Evaluate

f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) .

B15. Square

WXYZ

is inscribed in square

ABCD

with side length

1

such that

W

is on

AB

X

is on

BC

Y

is on

CD

, and

Z

is on

DA

. Line

W Y

hits

AD

and

BC

at points

P

and

R

respectively, and line

XZ

hits

AB

and

CD

at points

Q

and

S

respectively. If the area of

WXYZ

\frac{13}{18}

, then the area of

PQRS

can be expressed as

m/n

for relatively prime positive integers

m

and

n

. What is

m + n

?

PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777424p24371574]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement