MathDB

p1. If

x

20\%

23

and

y

23\%

20

, compute

xy

.
p2. Pablo wants to eat a banana, a mango, and a tangerine, one at a time. How many ways can he choose the order to eat the three fruits?
p3. Let

a

b

, and

c

3

positive integers. If

a + \frac{b}{c} = \frac{11}{6}

, what is the minimum value of

a + b + c

?
p4. A rectangle has an area of

12

. If all of its sidelengths are increased by

2

, its area becomes

32

. What is the perimeter of the original rectangle?
p5. Rohit is trying to build a

3

-dimensional model by using several cubes of the same size. The model’s front view and top view are shown below. Suppose that every cube on the upper layer is directly above a cube on the lower layer and the rotations are considered distinct. Compute the total number of different ways to form this model. https://cdn.artofproblemsolving.com/attachments/b/b/40615b956f3d18313717259b12fcd6efb74cf8.png
p6. Priscilla has three octagonal prisms and two cubes, none of which are touching each other. If she chooses a face from these five objects in an independent and uniformly random manner, what is the probability the chosen face belongs to a cube? (One octagonal prism and cube are shown below.) https://cdn.artofproblemsolving.com/attachments/0/0/b4f56a381c400cae715e70acde2cdb315ee0e0.png
p7. Let triangle

\vartriangle ABC

and triangle

\vartriangle DEF

be two congruent isosceles right triangles where line segments

\overline{AC}

and

\overline{DF}

are their respective hypotenuses. Connecting a line segment

\overline{CF}

gives us a square

ACFD

but with missing line segments

\overline{AC}

\overline{AD}

, and

\overline{DF}

. Instead,

A

and

D

are connected by an arc defined by the semicircle with endpoints

A

and

D

. If

CF = 1

, what is the perimeter of the whole shape

ABCFED

? https://cdn.artofproblemsolving.com/attachments/2/5/098d4f58fee1b3041df23ba16557ed93ee9f5b.png
p8. There are two moles that live underground, and there are five circular holes that the moles can hop out of. The five holes are positioned as shown in the diagram below, where

A

B

C

D

, and

E

are the centers of the circles,

AE = 30

cm, and congruent triangles

\vartriangle ABC

\vartriangle CBD

, and

\vartriangle CDE

are equilateral. The two moles randomly choose exactly two of the five holes, hop out of the two chosen holes, and hop back in. What is the probability that the holes that the two moles hop out of have centers that are exactly

15

cm apart? https://cdn.artofproblemsolving.com/attachments/c/e/b46ba87b954a1904043020d7a211477caf321d.png
p9. Carson is planning a trip for

n

people. Let

x

be the number of cars that will be used and

y

be the number of people per car. What is the smallest value of

n

such that there are exactly

3

possibilities for

x

and

y

so that

y

is an integer,

x < y

, and exactly one person is left without a car?
p10. Iris is eating an ice cream cone, which consists of a hemisphere of ice cream with radius

r > 0

on top of a cone with height

12

and also radius

r

. Iris is a slow eater, so after eating one-third of the ice cream, she notices that the rest of the ice cream has melted and completely filled the cone. Assuming the ice cream did not change volume after it melted, what is the value of

r

?
p11. As Natasha begins eating brunch between

11:30

AM and

12

PM, she notes that the smaller angle between the minute and hour hand of the clock is

27

degrees. What is the number of degrees in the smaller angle between the minute and hour hand when Natasha finishes eating brunch

20

minutes later?
p12. On a regular hexagon

ABCDEF

, Luke the frog starts at point

A

, there is food on points

C

and

E

and there are crocodiles on points

B

and

D

. When Luke is on a point, he hops to any of the five other vertices with equal probability. What is the probability that Luke will visit both of the points with food before visiting any of the crocodiles?
p13.

2023

regular unit hexagons are arranged in a tessellating lattice, as follows. The first hexagon

ABCDEF

(with vertices in clockwise order) has leftmost vertex

A

at the origin, and hexagons

H_2

and

H_3

share edges

\overline{CD}

and

\overline{DE}

with hexagon

H_1

, respectively. Hexagon

H_4

shares edges with both hexagons

H_2

and

H_3

, and hexagons

H_5

and

H_6

are constructed similarly to hexagons H_2 and

H_3

. Hexagons

H_7

H_{2022}

are constructed following the pattern of hexagons

H_4

H_5

H_6

. Finally, hexagon H_{2023} is constructed, sharing an edge with both hexagons H2021 and H2022. Compute the perimeter of the resulting figure. https://cdn.artofproblemsolving.com/attachments/1/d/eaf0d04676bac3e3c197b4686dcddd08fce9ac.png
p14. Aditya’s favorite number is a positive two-digit integer. Aditya sums the integers from

5

to his favorite number, inclusive. Then, he sums the next

12

consecutive integers starting after his favorite number. If the two sums are consecutive integers and the second sum is greater than the first sum, what is Aditya’s favorite number?
p15. The

100^{th}

anniversary of BMT will fall in the year

2112

, which is a palindromic year. Compute the sum of all years from

0000

9999

, inclusive, that are palindromic when written out as four-digit numbers (including leading zeros). Examples include

2002

1991

, and

0110

.
p16. Points

A

B

C

D

, and

E

lie on line

r

, in that order, such that

DE = 2DC

and

AB = 2BC

. Let

M

be the midpoint of segment

\overline{AC}

. Finally, let point

P

lie on

r

such that

PE = x

. If

AB = 8x

ME = 9x

, and

AP = 112

, compute the sum of the two possible values of

CD

.
p17. A parabola

y = x^2

in the xy-plane is rotated

180^o

about a point

(a, b)

. The resulting parabola has roots at

x = 40

and

x = 48

. Compute

a + b

.
p18. Susan has a standard die with values

1

6

. She plays a game where every time she rolls the die, she permanently increases the value on the top face by

1

. What is the probability that, after she rolls her die 3 times, there is a face on it with a value of at least

7

?
p19. Let

N

be a

6

-digit number satisfying the property that the average value of the digits of

N^4

5

. Compute the sum of the digits of

N^4

.
p20. Let

O_1

O_2

...

O_8

be circles of radius

1

such that

O_1

is externally tangent to

O_8

and

O_2

but no other circles,

O_2

is externally tangent to

O_1

and

O_3

but no other circles, and so on. Let

C

be a circle that is externally tangent to each of

O_1

O_2

...

O_8

. Compute the radius of

C

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

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

2023 BmMT

Subcontests

2023 BmMT Individual Round - Berkley mini Math Tournament

2023 BmMT Team Round - Berkley mini Math Tournament Fall