MathDB

Round 1
p1. How many ways are there to arrange the letters in the word

NEVERLAND

such that the

2

N

’s are adjacent and the two

E

’s are adjacent? Assume that letters that appear the same are not distinct.
p2. In rectangle

ABCD

E

and

F

are on

AB

and

CD

, respectively such that

DE = EF = FB

and

\angle CDE = 45^o

. Find

AB + AD

given that

AB

and

AD

are relatively prime positive integers.
p3. Maisy Airlines sees

n

takeoffs per day. Find the minimum value of

n

such that theremust exist two planes that take off within aminute of each other.
Round 2
p4. Nick is mixing two solutions. He has

100

mL of a solution that is

30\%

X

and

400

mL of a solution that is

10\%

X

. If he combines the two, what percent

X

is the final solution?
p5. Find the number of ordered pairs

(a,b)

, where

a

and

b

are positive integers, such that

\frac{1}{a}+\frac{2}{b}=\frac{1}{12}.

p6.

25

balls are arranged in a

5

5

square. Four of the balls are randomly removed from the square. Given that the probability that the square can be rotated

180^o

and still maintain the same configuration can be expressed as

\frac{m}{n}

, where

m

and

n

are relatively prime, find

m+n

.
Round 3
p7. Maisy the ant is on corner

A

of a

13\times 13\times 13

box. She needs to get to the opposite corner called

B

. Maisy can only walk along the surface of the cube and takes the path that covers the least distance. Let

C

and

D

be the possible points where she turns on her path. Find

AC^2 + AD^2 +BC^2 +BD^2 - AB^2 -CD^2

.
p8. Maisyton has recently built

5

intersections. Some intersections will get a park and some of those that get a park will also get a chess school. Find how many different ways this can happen.
p9. Let

f (x) = 2x -1

. Find the value of

x

that minimizes

| f ( f ( f ( f ( f (x)))))-2020|

.
Round 4
p10. Triangle

ABC

is isosceles, with

AB = BC > AC

. Let the angle bisector of

\angle A

intersect side

\overline{BC}

at point

D

, and let the altitude from

A

intersect side

\overline{BC}

at point

E

. If

\angle A = \angle C= x^o

, then the measure of

\angle DAE

can be expressed as

(ax -b)^o

, for some constants

a

and

b

. Find

ab

.
p11. Maisy randomly chooses

4

integers

w

x

y

, and

z

, where

w, x, y, z \in \{1,2,3, ... ,2019,2020\}

. Given that the probability that

w^2 + x^2 + y^2 + z^2

is not divisible by

4

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers, find

m+n

.
p12. Evaluate

-\log_4 \left(\log_2 \left(\sqrt{\sqrt{\sqrt{...\sqrt{16}}}} \right)\right),

where there are

100

square root signs.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement