MathDB

p1. David is taking a

50

-question test, and he needs to answer at least

70\%

of the questions correctly in order to pass the test. What is the minimum number of questions he must answer correctly in order to pass the test?
p2. You decide to flip a coin some number of times, and record each of the results. You stop flipping the coin once you have recorded either

20

heads, or

16

tails. What is the maximum number of times that you could have flipped the coin?
p3. The width of a rectangle is half of its length. Its area is

98

square meters. What is the length of the rectangle, in meters?
p4. Carol is twice as old as her younger brother, and Carol's mother is

4

times as old as Carol is. The total age of all three of them is

55

. How old is Carol's mother?
p5. What is the sum of all two-digit multiples of

9

?
p6. The number

2016

is divisible by its last two digits, meaning that

2016

is divisible by

16

. What is the smallest integer larger than

2016

that is also divisible by its last two digits?
p7. Let

Q

and

R

both be squares whose perimeters add to

80

. The area of

Q

to the area of

R

is in a ratio of

16 : 1

. Find the side length of

Q

.
p8. How many

8

-digit positive integers have the property that the digits are strictly increasing from left to right? For instance,

12356789

is an example of such a number, while

12337889

is not.
p9. During a game, Steve Korry attempts

20

free throws, making 16 of them. How many more free throws does he have to attempt to finish the game with

84\%

accuracy, assuming he makes them all?
p10. How many dierent ways are there to arrange the letters

MILKTEA

such that

TEA

is a contiguous substring? For reference, the term "contiguous substring" means that the letters

TEA

appear in that order, all next to one another. For example,

MITEALK

would be such a string, while

TMIELKA

would not be.
p11. Suppose you roll two fair

20

-sided dice. What is the probability that their sum is divisible by

10

?
p12. Suppose that two of the three sides of an acute triangle have lengths

20

and

16

, respectively. How many possible integer values are there for the length of the third side?
p13. Suppose that between Beijing and Shanghai, an airplane travels

500

miles per hour, while a train travels at

300

miles per hour. You must leave for the airport

2

hours before your flight, and must leave for the train station

30

minutes before your train. Suppose that the two methods of transportation will take the same amount of time in total. What is the distance, in miles, between the two cities?
p14. How many nondegenerate triangles (triangles where the three vertices are not collinear) with integer side lengths have a perimeter of

16

? Two triangles are considered distinct if they are not congruent.
p15. John can drive

100

miles per hour on a paved road and

30

miles per hour on a gravel road. If it takes John

100

minutes to drive a road that is

100

miles long, what fraction of the time does John spend on the paved road?
p16. Alice rolls one pair of

6

-sided dice, and Bob rolls another pair of

6

-sided dice. What is the probability that at least one of Alice's dice shows the same number as at least one of Bob's dice?
p17. When

20^{16}

is divided by

16^{20}

and expressed in decimal form, what is the number of digits to the right of the decimal point? Trailing zeroes should not be included.
p18. Suppose you have a

20 \times 16

bar of chocolate squares. You want to break the bar into smaller chunks, so that after some sequence of breaks, no piece has an area of more than

5

. What is the minimum possible number of times that you must break the bar? For an example of how breaking the chocolate works, suppose we have a

2\times 2

bar and wish to break it entirely into

1\times 1

bars. We can break it once to get two

2\times 1

bars. Then, we would have to break each of these individual bars in half in order to get all the bars to be size

1\times 1

, and we end up using

3

breaks in total.
p19. A class of

10

students decides to form two distinguishable committees, each with

3

students. In how many ways can they do this, if the two committees can have no more than one student in common?
p20. You have been told that you are allowed to draw a convex polygon in the Cartesian plane, with the requirements that each of the vertices has integer coordinates whose values range from

0

10

inclusive, and that no pair of vertices can share the same

x

y

coordinate value (so for example, you could not use both

(1, 2)

and

(1, 4)

in your polygon, but

(1, 2)

and

(2, 1)

is fine). What is the largest possible area that your polygon can have?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement