MathDB

p1. There exist real numbers

B

M

, and

T

such that

B + M + T = 23

and

B - M - T = 20

. Compute

M + T

.
p2. Kaity has a rectangular garden that measures

10

yards by

12

yards. Austin’s triangular garden has side lengths

6

yards,

8

yards, and

10

yards. Compute the ratio of the area of Kaity’s garden to the area of Austin’s garden.
p3. Nikhil’s mom and brother both have ages under

100

years that are perfect squares. His mom is

33

years older than his brother. Compute the sum of their ages.
p4. Madison wants to arrange

3

identical blue books and

2

identical pink books on a shelf so that each book is next to at least one book of the other color. In how many ways can Madison arrange the books?
p5. Two friends, Anna and Bruno, are biking together at the same initial speed from school to the mall, which is

6

miles away. Suddenly,

1

mile in, Anna realizes that she forgot her calculator at school. If she bikes

4

miles per hour faster than her initial speed, she could head back to school and still reach the mall at the same time as Bruno, assuming Bruno continues biking towards the mall at their initial speed. In miles per hour, what is Anna and Bruno’s initial speed, before Anna has changed her speed? (Assume that the rate at which Anna and Bruno bike is constant.)
p6. Let a number be “almost-perfect” if the sum of its digits is

28

. Compute the sum of the third smallest and third largest almost-perfect

4

-digit positive integers.
p7. Regular hexagon

ABCDEF

is contained in rectangle

PQRS

such that line

\overline{AB}

lies on line

\overline{PQ}

, point

C

lies on line

\overline{QR}

, line

\overline{DE}

lies on line

\overline{RS}

, and point

F

lies on line

\overline{SP}

. Given that

PQ = 4

, compute the perimeter of

AQCDSF

. https://cdn.artofproblemsolving.com/attachments/6/7/5db3d5806eaefa00d7fc90fb786a41c0466a90.png
p8. Compute the number of ordered pairs

(m, n)

, where

m

and

n

are relatively prime positive integers and

mn = 2520

. (Note that positive integers

x

and

y

are relatively prime if they share no common divisors other than

1

. For example, this means that

1

is relatively prime to every positive integer.)
p9. A geometric sequence with more than two terms has first term

x

, last term

2023

, and common ratio

y

, where

x

and

y

are both positive integers greater than

1

. An arithmetic sequence with a finite number of terms has first term

x

and common difference

y

. Also, of all arithmetic sequences with first term

x

, common difference

y

, and no terms exceeding

2023

, this sequence is the longest. What is the last term of the arithmetic sequence?
p10. Andrew is playing a game where he must choose three slips, uniformly at random and without replacement, from a jar that has nine slips labeled

1

through

9

. He wins if the sum of the three chosen numbers is divisible by

3

and one of the numbers is

1

. What is the probability Andrew wins?
p11. Circle

O

is inscribed in square

ABCD

. Let

E

be the point where

O

meets line segment

\overline{AB}

. Line segments

\overline{EC}

and

\overline{ED}

intersect

O

at points

P

and

Q

, respectively. Compute the ratio of the area of triangle

\vartriangle EPQ

to the area of triangle

\vartriangle ECD

.
p12. Define a recursive sequence by

a_1 = \frac12

and

a_2 = 1

, and

a_n =\frac{1 + a_{n-1}}{a_{n-2}}

for n ≥ 3. The product

a_1a_2a_3 ... a_{2023}

can be expressed in the form

a^b \cdot c^d \cdot e^f

, where

a

b

c

d

e

, and

f

are positive (not necessarily distinct) integers, and a, c, and e are prime. Compute

a + b + c + d + e + f

.
p13. An increasing sequence of

3

-digit positive integers satisfies the following properties:

\bullet

Each number is a multiple of

2

3

, or

5

\bullet

Adjacent numbers differ by only one digit and are relatively prime. (Note that positive integers x and y are relatively prime if they share no common divisors other than

1

.) What is the maximum possible length of the sequence?
p14. Circles

O_A

and

O_B

with centers

A

and

B

, respectively, have radii

3

and

8

, respectively, and are internally tangent to each other at point

P

. Point

C

is on circle

O_A

such that line

\overline{BC}

is tangent to circle

OA

. Extend line

\overline{PC}

to intersect circle

O_B

at point

D \ne P

. Compute

CD

.
p15. Compute the product of all real solutions

x

to the equation

x^2 + 20x - 23 = 2 \sqrt{x^2 + 20x + 1}

.
p16. Compute the number of divisors of

729, 000, 000

that are perfect powers. (A perfect power is an integer that can be written in the form

a^b

, where

a

and

b

are positive integers and

b > 1

.)
p17. The arithmetic mean of two positive integers

x

and

y

, each less than

100

, is

4

more than their geometric mean. Given

x > y

, compute the sum of all possible values for

x + y

. (Note that the geometric mean of

x

and

y

is defined to be

\sqrt{xy}

.)
p18. Ankit and Richard are playing a game. Ankit repeatedly writes the digits

2

0

2

3

, in that order, from left to right on a board until Richard tells him to stop. Richard wins if the resulting number, interpreted as a base-

10

integer, is divisible by as many positive integers less than or equal to

12

as possible. For example, if Richard stops Ankit after

7

digits have been written, the number would be

2023202

, which is divisible by

1

and

2

. Richard wants to win the game as early as possible. Assuming Ankit must write at least one digit, after how many digits should Richard stop Ankit?
p19. Eight chairs are set around a circular table. Among these chairs, two are red, two are blue, two are green, and two are yellow. Chairs that are the same color are identical. If rotations and reflections of arrangements of chairs are considered distinct, how many arrangements of chairs satisfy the property that each pair of adjacent chairs are different colors?
p20. Four congruent spheres are placed inside a right-circular cone such that they are all tangent to the base and the lateral face of the cone, and each sphere is tangent to exactly two other spheres. If the radius of the cone is

1

and the height of the cone is

2\sqrt2

, what is the radius of one of the spheres?
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