MathDB

p1. Compute

2020 \cdot \left( 2^{(0\cdot1)} + 9 - \frac{(20^1)}{8}\right)

.
p2. Nathan has five distinct shirts, three distinct pairs of pants, and four distinct pairs of shoes. If an “outfit” has a shirt, pair of pants, and a pair of shoes, how many distinct outfits can Nathan make?
p3. Let

ABCD

be a rhombus such that

\vartriangle ABD

and

\vartriangle BCD

are equilateral triangles. Find the angle measure of

\angle ACD

in degrees.
p4. Find the units digit of

2019^{2019}

.
p5. Determine the number of ways to color the four vertices of a square red, white, or blue if two colorings that can be turned into each other by rotations and reflections are considered the same.
p6. Kathy rolls two fair dice numbered from

1

6

. At least one of them comes up as a

4

5

. Compute the probability that the sumof the numbers of the two dice is at least

10

.
p7. Find the number of ordered pairs of positive integers

(x, y)

such that

20x +19y = 2019

.
p8. Let

p

be a prime number such that both

2p -1

and

10p -1

are prime numbers. Find the sum of all possible values of

p

.
p9. In a square

ABCD

with side length

10

, let

E

be the intersection of

AC

and

BD

. There is a circle inscribed in triangle

ABE

with radius

r

and a circle circumscribed around triangle

ABE

with radius

R

. Compute

R -r

.
p10. The fraction

\frac{13}{37 \cdot 77}

can be written as a repeating decimal

0.a_1a_2...a_{n-1}a_n

with

n

digits in its shortest repeating decimal representation. Find

a_1 +a_2 +...+a_{n-1}+a_n

.
p11. Let point

E

be the midpoint of segment

AB

of length

12

. Linda the ant is sitting at

A

. If there is a circle

O

of radius

3

centered at

E

, compute the length of the shortest path Linda can take from

A

B

if she can’t cross the circumference of

O

.
p12. Euhan and Minjune are playing tennis. The first one to reach

25

points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a

\frac34

chance of making the right call when the ball is in, meaning that he has a

\frac14

chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan.
p13. Find the number of subsets of

\{1, 2, 3, 4, 5, 6,7\}

which contain four consecutive numbers.
p14. Ezra and Richard are playing a game which consists of a series of rounds. In each round, one of either Ezra or Richard receives a point. When one of either Ezra or Richard has three more points than the other, he is declared the winner. Find the number of games which last eleven rounds. Two games are considered distinct if there exists a round in which the two games had different outcomes.
p15. There are

10

distinct subway lines in Boston, each of which consists of a path of stations. Using any

9

lines, any pair of stations are connected. However, among any

8

lines there exists a pair of stations that cannot be reached from one another. It happens that the number of stations is minimized so this property is satisfied. What is the average number of stations that each line passes through?
p16. There exist positive integers

k

and

3\nmid m

for which

1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times 54\times 55}.

Find the value

k

.
p17. Geronimo the giraffe is removing pellets from a box without replacement. There are

5

red pellets,

10

blue pellets, and

15

white pellets. Determine the probability that all of the red pellets are removed before all the blue pellets and before all of the white pellets are removed.
p18. Find the remainder when

70! \left( \frac{1}{4 \times 67}+ \frac{1}{5 \times 66}+...+ \frac{1}{66\times 5}+ \frac{1}{67\times 4} \right)

is divided by

71

.
p19. Let

A_1A_2...A_{12}

be the regular dodecagon. Let

X

be the intersection of

A_1A_2

and

A_5A_{11}

. Given that

X A_2 \cdot A_1A_2 = 10

, find the area of dodecagon.
p20. Evaluate the following infinite series:

\sum^{\infty}_{n=1}\sum^{\infty}_{m=1} \frac{n \sec^2m -m \tan^2 n}{3^{m+n}(m+n)}

.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

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