MathDB

p1. What is the area of a triangle with side lengths

6

8

, and

10

?
p2. Let

f(n) = \sqrt{n}

. If

f(f(f(n))) = 2

, compute

n

.
p3. Anton is buying AguaFina water bottles. Each bottle costs

14

dollars, and Anton buys at least one water bottle. The number of dollars that Anton spends on AguaFina water bottles is a multiple of

10

. What is the least number of water bottles he can buy?
p4. Alex flips

3

fair coins in a row. The probability that the first and last flips are the same can be expressed in the form

m/n

for relatively prime positive integers

m

and

n

. Compute

m + n

.
p5. How many prime numbers

p

satisfy the property that

p^2 - 1

is not a multiple of

6

?
p6. In right triangle

\vartriangle ABC

with

AB = 5

BC = 12

, and

CA = 13

, point

D

lies on

\overline{CA}

such that

AD = BD

. The length of

CD

can then be expressed in the form

m/n

for relatively prime positive integers

m

and

n

. Compute

m + n

.
p7. Vivienne is deciding on what courses to take for Spring

2021

, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Vivienne have?
p8. Square

ABCD

has side length

2

. Square

ACEF

is drawn such that

B

lies inside square

ACEF

. Compute the area of pentagon

AFECD

.
p9. At the Boba Math Tournament, the Blackberry Milk Team has answered

4

out of the first

10

questions on the Boba Round correctly. If they answer all

p

remaining questions correctly, they will have answered exactly

\frac{9p}{5}\%

of the questions correctly in total. How many questions are on the Boba Round?
p10. The sum of two positive integers is

2021

less than their product. If one of them is a perfect square, compute the sum of the two numbers.
p11. Points

E

and

F

lie on edges

\overline{BC}

and

\overline{DA}

of unit square

ABCD

, respectively, such that

BE =\frac13

and

DF =\frac13

. Line segments

\overline{AE}

and

\overline{BF}

intersect at point

G

. The area of triangle

EFG

can be written in the form

m/n

, where

m

and

n

are relatively prime positive integers. Compute

m+n

.
p12. Compute the number of positive integers

n \le 2020

for which

n^{k+1}

is a factor of

(1+2+3+· · ·+n)^k

for some positive integer

k

.
p13. How many permutations of

123456

are divisible by their last digit? For instance,

123456

is divisible by

6

, but

561234

is not divisible by

4

.
p14. Compute the sum of all possible integer values for

n

such that

n^2 - 2n - 120

is a positive prime number.
p15. Triangle

\vartriangle ABC

has

AB =\sqrt{10}

BC =\sqrt{17}

, and

CA =\sqrt{41}

. The area of

\vartriangle ABC

can be expressed in the form

m/n

for relatively prime positive integers

m

and

n

. Compute

m + n

.
p16. Let

f(x) = \frac{1 + x^3 + x^{10}}{1 + x^{10}}

. Compute

f(-20) + f(-19) + f(-18) + ...+ f(20)

.
p17. Leanne and Jing Jing are walking around the

xy

-plane. In one step, Leanne can move from any point

(x, y)

(x + 1, y)

(x, y + 1)

and Jing Jing can move from

(x, y)

(x - 2, y + 5)

(x + 3, y - 1)

. The number of ways that Leanne can move from

(0, 0)

(20, 20)

is equal to the number of ways that Jing Jing can move from

(0, 0)

(a, b)

, where a and b are positive integers. Compute the minimum possible value of

a + b

.
p18. Compute the number positive integers

1 < k < 2021

such that the equation

x +\sqrt{kx} = kx +\sqrt{x}

has a positive rational solution for

x

.
p19. In triangle

\vartriangle ABC

, point

D

lies on

\overline{BC}

with

\overline{AD} \perp \overline{BC}

. If

BD = 3AD

, and the area of

\vartriangle ABC

15

, then the minimum value of

AC^2

is of the form

p\sqrt{q} - r

, where

p, q

, and

r

are positive integers and

q

is not divisible by the square of any prime number. Compute

p + q + r

.
p20. Suppose the decimal representation of

\frac{1}{n}

is in the form

0.p_1p_2...p_j\overline{d_1d_2...d_k}

, where

p_1, ... , p_j

d_1,... , d_k

are decimal digits, and

j

and

k

are the smallest possible nonnegative integers (i.e. it’s possible for

j = 0

k = 0

). We define the preperiod of

\frac{1}{n}

to be

j

and the period of

\frac{1}{n}

to be

k

. For example,

\frac16 = 0.16666...

has preperiod

1

and period

1

\frac17 = 0.\overline{142857}

has preperiod

0

and period

6

, and

\frac14 = 0.25

has preperiod

2

and period

0

. What is the smallest positive integer

n

such that the sum of the preperiod and period of

\frac{1}{n}

8

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

Problem Statement