MathDB

p1. Square

ABCD

has side length

2

. Let the midpoint of

BC

E

. What is the area of the overlapping region between the circle centered at

E

with radius

1

and the circle centered at

D

with radius

2

? (You may express your answer using inverse trigonometry functions of noncommon values.)
p2. Find the number of times

f(x) = 2

occurs when

0 \le x \le 2022 \pi

for the function

f(x) = 2^x(cos(x) + 1)

.
p3. Stanford is building a new dorm for students, and they are looking to offer

2

room configurations:

\bullet

Configuration

A

: a one-room double, which is a square with side length of

x

\bullet

Configuration

B

: a two-room double, which is two connected rooms, each of them squares with a side length of

y

. To make things fair for everyone, Stanford wants a one-room double (rooms of configuration

A

) to be exactly

1

^2

larger than the total area of a two-room double. Find the number of possible pairs of side lengths

(x, y)

, where

x \in N

y \in N

, such that

x - y < 2022

.
p4. The island nation of Ur is comprised of

6

islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands

A

and

B

if each island chooses the other). Then, all islands connected by bridges together form an island-state. What is the expected number of island-states Ur is divided into?
p5. Let

a, b,

and

c

be the roots of the polynomial

x^3 - 3x^2 - 4x + 5

. Compute

\frac{a^4 + b^4}{a + b}+\frac{b^4 + c^4}{b + c}+\frac{c^4 + a^4}{c + a}

.
p6. Carol writes a program that finds all paths on an 10 by 2 grid from cell (1, 1) to cell (10, 2) subject to the conditions that a path does not visit any cell more than once and at each step the path can go up, down, left, or right from the current cell, excluding moves that would make the path leave the grid. What is the total length of all such paths? (The length of a path is the number of cells it passes through, including the starting and ending cells.)
p7. Consider the sequence of integers an defined by

a_1 = 1

a_p = p

for prime

p

and

a_{mn} = ma_n + na_m

for

m, n > 1

. Find the smallest

n

such that

\frac{a_n^2}{2022}

is a perfect power of

3

.
p8. Let

\vartriangle ABC

be a triangle whose

A

-excircle,

B

-excircle, and

C

-excircle have radii

R_A

R_B

, and

R_C

, respectively. If

R_AR_BR_C = 384

and the perimeter of

\vartriangle ABC

32

, what is the area of

\vartriangle ABC

?
p9. Consider the set

S

of functions

f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 243\}

satisfying: (a)

f(1) = 1

(b)

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

, (c)

n |f(n)

, (d)

f(lcm(m, n))f(gcd(m, n)) = f(m)f(n)

. If

|S|

can be written as

p^{\ell_1}_1 \cdot p^{\ell_2}_2 \cdot ... \cdot p^{\ell_k}_k

where

p_i

are distinct primes, compute

p_1\ell_1+p_2\ell_2+. . .+p_k\ell_k

.
p10. You are given that

\log_{10}2 \approx 0.3010

and that the first (leftmost) two digits of

2^{1000}

are 10. Compute the number of integers

n

with

1000 \le n \le 2000

such that

2^n

starts with either the digit

8

9

(in base

10

).
p11. Let

O

be the circumcenter of

\vartriangle ABC

. Let

M

be the midpoint of

BC

, and let

E

and

F

be the feet of the altitudes from

B

and

C

, respectively, onto the opposite sides.

EF

intersects

BC

P

. The line passing through

O

and perpendicular to

BC

intersects the circumcircle of

\vartriangle ABC

L

(on the major arc

BC

) and

N

, and intersects

BC

M

. Point

Q

lies on the line

LA

such that

OQ

is perpendicular to

AP

. Given that

\angle BAC = 60^o

and

\angle AMC = 60^o

, compute

OQ/AP

.
p12. Let

T

be the isosceles triangle with side lengths

5, 5, 6

. Arpit and Katherine simultaneously choose points

A

and

K

within this triangle, and compute

d(A, K)

, the squared distance between the two points. Suppose that Arpit chooses a random point

A

within

T

. Katherine plays the (possibly randomized) strategy which given Arpit’s strategy minimizes the expected value of

d(A, K)

. Compute this value.
p13. For a regular polygon

S

with

n

sides, let

f(S)

denote the regular polygon with

2n

sides such that the vertices of

S

are the midpoints of every other side of

f(S)

. Let

f^{(k)}(S)

denote the polygon that results after applying f a total of k times. The area of

\lim_{k \to \infty} f^{(k)}(P)

where

P

is a pentagon of side length

1

, can be expressed as

\frac{a+b\sqrt{c}}{d}\pi^m

for some positive integers

a, b, c, d, m

where

d

is not divisible by the square of any prime and

d

does not share any positive divisors with

a

and

b

. Find

a + b + c + d + m

.
p14. Consider the function

f(m) = \sum_{n=0}^{\infty}\frac{(n - m)^2}{(2n)!}

. This function can be expressed in the form

f(m) = \frac{a_m}{e} +\frac{b_m}{4}e

for sequences of integers

\{a_m\}_{m\ge 1}

\{b_m\}_{m\ge 1}

. Determine

\lim_{n \to \infty}\frac{2022b_m}{a_m}

.
p15. In

\vartriangle ABC

, let

G

be the centroid and let the circumcenters of

\vartriangle BCG

\vartriangle CAG

, and

\vartriangle ABG

I, J

, and

K

, respectively. The line passing through

I

and the midpoint of

BC

intersects

KJ

Y

. If the radius of circle

K

5

, the radius of circle

J

8

, and

AG = 6

, what is the length of

KY

?

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

Problem Statement