MathDB

p1. If

3a + 1 = b

and

3b + 1 = 2020

, what is a?
p2. Tracy draws two triangles: one with vertices at

(0, 0)

(2, 0)

, and

(1, 8)

and another with vertices at

(1, 0)

(3, 0)

, and

(2, 8)

. What is the area of overlap of the two triangles?
p3. If

p

q

, and

r

are prime numbers such that

p + q + r = 50

, what is the maximum possible value of

pqr

?
p4. Points

A,B,C,D

lie on a circle of radius

4

such that

BC = 8

BD = 4

, and

m \angle ABC = 27^o

. If segments

\overline{AB}

and

\overline{CD}

do not intersect, what is the value of

m \angle ACD

? (Give your answer in degrees.)
p5. Express

\sqrt{14-5 \sqrt{52}}-\sqrt{14+5 \sqrt{52}}

as a rational number.
p6. Let

a_0 = 1

, and let

a_n = 1 + \frac{1}{a_{n-1}}

for every integer

n \ge 1

. Find the value of the product

a_1a_2...a_9

.
p7. Miki wants to distribute

75

identical candies to the students in her class such that each students gets at least

1

candy. For what number of students does Miki have the greatest number of possible ways to distribute the candies?
p8. Let

ABCD

be a rectangle. Let points

E

F

G

, and

H

lie on the segments

\overline{AB}

\overline{AD}

\overline{BC}

, and

\overline{CD}

(respectively) such that both

\overline{EF}

and

\overline{GH}

are parallel to

\overline{BD}

. If

\vartriangle AFE

is congruent to

\vartriangle BEG

and

\frac{AE}{HC} = \frac12

, what is

\frac{AB}{BC}

?
p9. If

a_1, a_2, a_3,...

is a geometric sequence satisfying

\frac{19a_{2019} + 19a_{2021}}{a_{2020}}= 25 +\frac{6a_{2006} + 6a_{2010}}{a_{2008}}

and

0 < a_1 < a_2

, what is the value of

\frac{a_2}{a_1}

?
p10. Elizabeth has an infinite grid of squares. (Each square is next to the four squares directly above it, below it, to its left, and to its right.) She colors in some of the squares such that the following two conditions are met: (1) no two colored squares are next to each other; (2) each uncolored square is next to exactly one colored square. In a

20 \times 20

subgrid of this infinite grid, how many colored squares are there?
p11. Find the smallest whole number

N \ge 2020

such that

N

has twice as many even divisors as odd divisors and

N^2

has a remainder of

1

when it is divided by

15

.
p12. We say that the sets A, B, and C form a "sunflower" if

A \cap B = A \cap C = B \cap C

. (

A \cap B

denotes the intersection of the sets

A

and

B

.) If

A

B

, and

C

are independently randomly chosen

4

-element subsets of the set

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

, what is the probability that

A

B

, and

C

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

Problem Statement