MathDB

p1. Let

x =\frac{15213}{15 - 213}.

Find the integer nearest to x.
p2. A grocery store sells oranges for either

\$1

each or five for

\$4

. If Theo wants to buy

40

oranges, they would save

\$k

by buying all five-packs instead of all single oranges. What is

k

?
p3. Let

ABCD

be a square. If

AB

and

CD

were increased in length by

20\%

and

AD

and

BC

were decreased in length by

20\%

while keeping

ABCD

a rectangle, the area of

ABCD

would change by

k\%

. Find

k

.
p4. Polly writes down all nonnegative integers that contain at most one

0

, at most three

2

s, and no other digits. What is the median of all numbers that Polly writes down?
p5. Let

P

be a point.

7

circles of distinct radii all pass through

P

. Let

n

be the total number of intersection points, including

P

. What is the ratio of the maximum possible value of

n

to the minimum possible value of

n

?
p6. Define the sequence

\{a_n\}

recursively with

a_0 = 2

a_1 = 3

, and

a_n = a_0 +...+ a_{n-1}

for

n \ge 2

. What is

a_{2022}

?
p7. Define the sequence

\{a_n\}

recursively with

a_0 = 1

a_1 = 0

, and

a_n = 2a_{n-1} + 9a_{n-2}

for all

n \ge 2

. What is the units digit of

a_{2022}

?
p8. Suppose that x satisfies

|2x - 2| -2 \ge x

. Find the sum of the minimum and maximum possible value of

x

.
p9. Clarabelle has

5000

cards numbered

1

5000

. They pick five at random and then place them face down such that their numbers are increasing from left to right. They then turn over the third card to reveal the number

2022

. What is the probability that the first card is a

1

?
p10. Rays

r_1

and

r_2

share a common endpoint. Three squares have sides on one of the rays and vertices on the other, as shown in the diagram. If the side lengths of the smallest two squares are

20

and

22

, find the side length of the largest square. https://cdn.artofproblemsolving.com/attachments/1/2/3717aa86a65e20b94a0d8161f89bea603411e9.png
p11. There exists a rectangle

ABCD

and a point

P

inside

ABCD

such that

AP = 20

BP = 21

, and

CP = 22

. In such a setup, find the square of the length

DP

.
p12. Compute the smallest integer

N

such that

5^6 = 15625

appears as the last five digits of

5^N

, where

N > 6

.
p13. There exist two complex numbers

z_1

z_2

such that

|z_1 + z_2| ^2 + |z_1 - z_2| ^2 = 338.

Find the length of the hypotenuse of the right triangle formed with legs of length

|z_1|

|z_2|

.
p14. Blahaj has two rays with a common endpoint A0 that form an angle of

1^o

. They construct a sequence of points

A_0

. . .

A_n

such that for all

1 \le i \le n

|A_{i-1}A_i | = 1

, and

|A_iA_0| > |A_{i-1}A_0|

. Find the largest possible value of

n

. https://cdn.artofproblemsolving.com/attachments/d/b/99c1adbdcf18e7b62ebfc1786cd6ad4bc2253e.png
p15. Consider the sequence

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, . .

. Find the sum of the first

100

terms of the sequence.
p16. Suppose Annie the Ant is walking on a regular icosahedron (as shown). She starts on point

A

and will randomly create a path to go to point

Z

which is the point directly opposite to

A

. Every move she makes never moves further from Z, and she has equal probability to go down every valid move. What is the expected number of moves she can make? https://cdn.artofproblemsolving.com/attachments/6/1/38b92c54a5f01cdb948ff565843cb08407e6db.png
p17. Suppose that

z

is a complex number, where the expression

\frac{z -2i}{z + 1}

is real. Find

min |z - 1|

.
p18. Scotty has a circular sheet of paper with radius

1

. They split this paper into

n

congruent sectors, and with each sector, tape the two straight edges together to form a cone. Let

V

be the combined volume of all n cones. What is the maximum value of

V

?
p19. Let

P(x) = (x- 3)^m \left(x -\frac13\right)^n

where

m, n

are positive integers. How many ordered pairs

(m, n)

for

m, n \le 100

result in

P(x)

having integer coefficients for its first three terms and last term? Assume

P(x)

is depicted from greatest to least exponent of

x

.
p20. Let

f(x) = |x| - 1

and

g(x) = |x - 1|

. Define

f^n(x) = \underbrace{f(f(f(...f}{n\,\, times}(x))),

and define

g^n(x)

similarly. Let the number of solutions to

f^{20}(x) = 0

and

g^{20}(x) = 0

a, b

,respectively. Find

a - b

.
p21. (Estimation) Let

M

be the mean absolute deviation of all submissions to this question. In other words, if the submissions to this question are

x_1

x-2

. . .

x_n

, with mean

x

, then

M =\frac{1}{n} \sum^n_{i=1} |x_i - x|.

Estimate

M

. Your answer must an integer between

0

and

999

, inclusive.
PS. You should use hide for answers.

Problem Statement