MathDB

2023

1

2023 MBMT Team Round - Montgomery Blair Math Tournament

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names
B1 What is the sum of the first

5

positive integers?
B2 Bread picks a number

n

. He finds out that if he multiplies

n

by

23

and then subtracts

20

, he gets

46279

. What is

n

?
B3 A Harshad Number is a number that is divisible by the sum of its digits. For example,

27

is divisible by

2 + 7 = 9

. Only one two-digit multiple of

9

is not a Harshad Number. What is this number?
B4 / G1 There are

5

red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen?
B5 Let

a

be a

1

-digit positive integer and

b

be a

3

-digit positive integer. If the product of

a

and

b

is a

4

-digit integer, what is the minimum possible value of the sum of

a

and

b

?
B6 / G2 A circle has radius

6

. A smaller circle with the same center has radius

5

. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle?
B7 Call a two-digit integer “sus” if its digits sum to

10

. How many two-digit primes are sus?
B8 / G3 Alex and Jeff are playing against Max and Alan in a game of tractor with

2

standard decks of

52

cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the

5

s are worth

5

points, the

10

s are worth

10

points, and the kings are worth 10 points. Given that a team needs

50

percent more points than the other to win, what is the minimal score Alan and Max need to win?
B9 / G4 Bob has a sandwich in the shape of a rectangular prism. It has side lengths

10

,

5

, and

5

. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece?
B10 / G5 Aven makes a rectangular fence of area

96

with side lengths

x

and

y

. John makesva larger rectangular fence of area 186 with side lengths

x + 3

and

y + 3

. What is the value of

x + y

?
B11 / G6 A number is prime if it is only divisible by itself and

1

. What is the largest prime number

n

smaller than

1000

such that

n + 2

and

n - 2

are also prime? Note:

1

is not prime.
B12 / G7 Sally has

3

red socks,

1

green sock,

2

blue socks, and

4

purple socks. What is the probability she will choose a pair of matching socks when only choosing

2

socks without replacement?
B13 / G8 A triangle with vertices at

(0, 0)

,

(3, 0)

,

(0, 6)

is filled with as many

1 \times 1

lattice squares as possible. How much of the triangle’s area is not filled in by the squares?
B14 / G10 A series of concentric circles

w_1, w_2, w_3, ...

satisfy that the radius of

w_1 = 1

and the radius of

w_n =\frac34

times the radius of

w_{n-1}

. The regions enclosed in

w_{2n-1}

but not in

w_{2n}

are shaded for all integers

n > 0

. What is the total area of the shaded regions?
B15 / G12

10

cards labeled 1 through

10

lie on a table. Kevin randomly takes

3

cards and Patrick randomly takes 2 of the remaining

7

cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card?
G9 Let

A

and

B

be digits. If

125A^2 + B161^2 = 11566946

. What is

A + B

?
G11 How many ordered pairs of integers

(x, y)

satisfy

y^2 - xy + x = 0

?
G13

N

consecutive integers add to

27

. How many possible values are there for

N

?
G14 A circle with center O and radius

7

is tangent to a pair of parallel lines

\ell_1

and

\ell_2

. Let a third line tangent to circle

O

intersect

\ell_1

and

\ell_2

at points

A

and

B

. If

AB = 18

, find

OA + OB

.
G15 Let

M =\prod ^{42}_{i=0}(i^2 - 5).

Given that

43

doesn’t divide

M

, what is the remainder when M is divided by

43

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

2022

1

2022 MBMT Team Round - Montgomery Blair Math Tournament

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
D1. The product of two positive integers is

5

. What is their sum?
D2. Gavin is

4

feet tall. He walks

5

feet before falling forward onto a cushion. How many feet is the top of Gavin’s head from his starting point?
D3. How many times must Nathan roll a fair

6

-sided die until he can guarantee that the sum of his rolls is greater than

6

?
D4 / Z1. What percent of the first

20

positive integers are divisible by

3

?
D5. Let

a

be a positive integer such that

a^2 + 2a + 1 = 36

. Find

a

.
D6 / Z2. It is said that a sheet of printer paper can only be folded in half

7

times. A sheet of paper is

8.5

inches by

11

inches. What is the ratio of the paper’s area after it has been folded in half

7

times to its original area?
D7 / Z3. Boba has an integer. They multiply the number by

8

, which results in a two digit integer. Bubbles multiplies the same original number by 9 and gets a three digit integer. What was the original number?
D8. The average number of letters in the first names of students in your class of

24

is

7

. If your teacher, whose first name is Blair, is also included, what is the new class average?
D9 / Z4. For how many integers

x

is

9x^2

greater than

x^4

?
D10 / Z5. How many two digit numbers are the product of two distinct prime numbers ending in the same digit?
D11 / Z6. A triangle’s area is twice its perimeter. Each side length of the triangle is doubled,and the new triangle has area

60

. What is the perimeter of the new triangle?
D12 / Z7. Let

F

be a point inside regular pentagon

ABCDE

such that

\vartriangle FDC

is equilateral. Find

\angle BEF

.
D13 / Z8. Carl, Max, Zach, and Amelia sit in a row with

5

seats. If Amelia insists on sitting next to the empty seat, how many ways can they be seated?
D14 / Z9. The numbers

1, 2, ..., 29, 30

are written on a whiteboard. Gumbo circles a bunch of numbers such that for any two numbers he circles, the greatest common divisor of the two numbers is the same as the greatest common divisor of all the numbers he circled. Gabi then does the same. After this, what is the least possible number of uncircled numbers?
D15 / Z10. Via has a bag of veggie straws, which come in three colors: yellow, orange, and green. The bag contains

8

veggie straws of each color. If she eats

22

veggie straws without considering their color, what is the probability she eats all of the yellow veggie straws?
Z11. We call a string of letters purple if it is in the form

CVCCCV

, where

C

s are placeholders for (not necessarily distinct) consonants and

V

s are placeholders for (not necessarily distinct) vowels. If

n

is the number of purple strings, what is the remainder when

n

is divided by

35

? The letter

y

is counted as a vowel.
Z12. Let

a, b, c

, and d be integers such that

a+b+c+d = 0

and

(a+b)(c+d)(ab+cd) = 28

. Find

abcd

.
Z13. Griffith is playing cards. A

13

-card hand with Aces of all

4

suits is known as a godhand. If Griffith and

3

other players are dealt

13

-card hands from a standard

52

-card deck, then the probability that Griffith is dealt a godhand can be expressed in simplest form as

\frac{a}{b}

. Find

a

.
Z14. For some positive integer

m

, the quadratic

x^2 + 202200x + 2022m

has two (not necessarily distinct) integer roots. How many possible values of

m

are there?
Z15. Triangle

ABC

with altitudes of length

5

,

6

, and

7

is similar to triangle

DEF

. If

\vartriangle DEF

has integer side lengths, find the least possible value of its perimeter.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2019

1

2019 MBMT Team Round - Montgomery Blair Math Tournament

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
D1. What is the solution to the equation

3 \cdot x \cdot 5 = 4 \cdot 5 \cdot 6

?
D2. Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make?
D3. What fraction of the multiples of

5

between

1

and

100

inclusive are also multiples of

20

?
D4. What is the maximum number of times a circle can intersect a triangle?
D5 / L1. At an interesting supermarket, the nth apple you purchase costs

n

dollars, while pears are

3

dollars each. Given that Layla has exactly enough money to purchase either

k

apples or

2k

pears for

k > 0

, how much money does Layla have?
D6 / L3. For how many positive integers

1 \le n \le 10

does there exist a prime

p

such that the sum of the digits of

p

is

n

?
D7 / L2. Real numbers

a, b, c

are selected uniformly and independently at random between

0

and

1

. What is the probability that

a \ge b \le c

?
D8. How many ordered pairs of positive integers

(x, y)

satisfy

lcm(x, y) = 500

?
D9 / L4. There are

50

dogs in the local animal shelter. Each dog is enemies with at least

2

other dogs. Steven wants to adopt as many dogs as possible, but he doesn’t want to adopt any pair of enemies, since they will cause a ruckus. Considering all possible enemy networks among the dogs, find the maximum number of dogs that Steven can possibly adopt.
D10 / L7. Unit circles

a, b, c

satisfy

d(a, b) = 1

,

d(b, c) = 2

, and

d(c, a) = 3,

where

d(x, y)

is defined to be the minimum distance between any two points on circles

x

and

y

. Find the radius of the smallest circle entirely containing

a

,

b

, and

c

.
D11 / L8. The numbers

1

through

5

are written on a chalkboard. Every second, Sara erases two numbers

a

and

b

such that

a \ge b

and writes

\sqrt{a^2 - b^2}

on the board. Let M and m be the maximum and minimum possible values on the board when there is only one number left, respectively. Find the ordered pair

(M, m)

.
D12 / L9.

N

people stand in a line. Bella says, “There exists an assignment of nonnegative numbers to the

N

people so that the sum of all the numbers is

1

and the sum of any three consecutive people’s numbers does not exceed

1/2019

.” If Bella is right, find the minimum value of

N

possible.
D13 / L10. In triangle

\vartriangle ABC

,

D

is on

AC

such that

BD

is an altitude, and

E

is on

AB

such that

CE

is an altitude. Let F be the intersection of

BD

and

CE

. If

EF = 2FC

,

BF = 8DF

, and

DC = 3

, then find the area of

\vartriangle CDF

.
D14 / L11. Consider nonnegative real numbers

a_1, ..., a_6

such that

a_1 +... + a_6 = 20

. Find the minimum possible value of

\sqrt{a^2_1 + 1^2} +\sqrt{a^2_2 + 2^2} +\sqrt{a^2_3 + 3^2} +\sqrt{a^2_4 + 4^2} +\sqrt{a^2_5 + 5^2} +\sqrt{a^2_6 + 6^2}.

D15 / L13. Find an

a < 1000000

so that both

a

and

101a

are triangular numbers. (A triangular number is a number that can be written as

1 + 2 +... + n

for some

n \ge 1

.)
Note: There are multiple possible answers to this problem. You only need to find one.
L6. How many ordered pairs of positive integers

(x, y)

, where

x

is a perfect square and

y

is a perfect cube, satisfy

lcm(x, y) = 81000000

?
L12. Given two points

A

and

B

in the plane with

AB = 1

, define

f(C)

to be the incenter of triangle

ABC

, if it exists. Find the area of the region of points

f(f(X))

where

X

is arbitrary.
L14. Leptina and Zandar play a game. At the four corners of a square, the numbers

1, 2, 3

, and

4

are written in clockwise order. On Leptina’s turn, she must swap a pair of adjacent numbers. On Zandar’s turn, he must choose two adjacent numbers

a

and

b

with

a \ge b

and replace

a

with

a - b

. Zandar wants to reduce the sum of the numbers at the four corners of the square to

2

in as few turns as possible, and Leptina wants to delay this as long as possible. If Leptina goes first and both players play optimally, find the minimum number of turns Zandar can take after which Zandar is guaranteed to have reduced the sum of the numbers to

2

.
L15. There exist polynomials

P, Q

and real numbers

c_0, c_1, c_2, ... , c_{10}

so that the three polynomials

P, Q

, and

c_0P^{10} + c_1P^9Q + c_2P^8Q^2 + ... + c_{10}Q^{10}

are all polynomials of degree 2019. Suppose that

c_0 = 1

,

c_1 = -7

,

c_2 = 22

. Find all possible values of

c_{10}

.
Note: The answer(s) are rational numbers. It suffices to give the prime factorization(s) of the numerator(s) and denominator(s).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2020.45

1

senators' committee in Flatland Congress 2020 MBMT p45

In the Flatland Congress there are senators who are on committees. Each senator is on at least one committee, and each committee has at least one senator. The rules for forming committees are as follows:

\bullet

For any pair of senators, there is exactly one committee which contains both senators.

\bullet

For any two committees, there is exactly one senator who is on both committees.

\bullet

There exist a set of four senators, no three of whom are all on the same committee.

\bullet

There exists a committee with exactly

6

senators.
If there are at least

25

senators in this Congress, compute the minimum possible number of senators

s

and minimum number of committees

c

in this Congress. Express your answer in the form

(s, c)

.

2020.44

1

a_n =sum_{d|n} 1/2^{d+ n/d} 2020 MBMT p44

Let

a_n =\sum_{d|n} \frac{1}{2^{d+ \frac{n}{d}}}

. In other words,

a_n

is the sum of

\frac{1}{2^{d+ \frac{n}{d}}}

over all divisors

d

of

n

. Find

\frac{\sum_{k=1} ^{\infty}ka_k}{\sum_{k=1}^{\infty} a_k} =\frac{a_1 + 2a_2 + 3a_3 + ....}{a_1 + a_2 + a_3 +....}

2020.43

1

\sigma_k(n)/ n^{k+2} (2020n + 2019)^2 > m 2020 MBMT p43

Let

\sigma_k(n)

be the sum of the

k^{th}

powers of the divisors of

n

. For all

k \ge 2

and all

n \ge 3

, we have that

\frac{\sigma_k(n)}{n^{k+2}} (2020n + 2019)^2 > m.

Find the largest possible value of

m

.

2020.42

1

integer lenght of angle bisector 2020 MBMT p42

\vartriangle ABC

has side lengths

AB = 4

and

AC = 9

. Angle bisector

AD

bisects angle

A

and intersects

BC

at

D

. Let

k

be the ratio

\frac{BD}{AB}

. Given that the length

AD

is an integer, find the sum of all possible

k^2

.

2020.41

1

last two digits of $$2^{3^{4^{...^{2019}}}} 2020 MBMT p41

What are the last two digits of

2^{3^{4^{...^{2019}}}} ?

2018

1

2018 MBMT Team Round - Montgomery Blair Math Tournament

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
C1. Mr. Pham flips

2018

coins. What is the difference between the maximum and minimum number of heads that can appear?
C2 / G1. Brandon wants to maximize

\frac{\Box}{\Box} +\Box

by placing the numbers

1

,

2

, and

3

in the boxes. If each number may only be used once, what is the maximum value attainable?
C3. Guang has

10

cents consisting of pennies, nickels, and dimes. What are all the possible numbers of pennies he could have?
C4. The ninth edition of Campbell Biology has

1464

pages. If Chris reads from the beginning of page

426

to the end of page

449

, what fraction of the book has he read?
C5 / G2. The planet Vriky is a sphere with radius

50

meters. Kyerk starts at the North Pole, walks straight along the surface of the sphere towards the equator, runs one full circle around the equator, and returns to the North Pole. How many meters did Kyerk travel in total throughout his journey?
C6 / G3. Mr. Pham is lazy and decides Stan’s quarter grade by randomly choosing an integer from

0

to

100

inclusive. However, according to school policy, if the quarter grade is less than or equal to

50

, then it is bumped up to

50

. What is the probability that Stan’s final quarter grade is

50

?
C7 / G5. What is the maximum (finite) number of points of intersection between the boundaries of a equilateral triangle of side length

1

and a square of side length

20

?
C8. You enter the MBMT lottery, where contestants select three different integers from

1

to

5

(inclusive). The lottery randomly selects two winning numbers, and tickets that contain both of the winning numbers win. What is the probability that your ticket will win?
C9 / G7. Find a possible solution

(B, E, T)

to the equation

THE + MBMT = 2018

, where

T, H, E, M, B

represent distinct digits from

0

to

9

.
C10.

ABCD

is a unit square. Let

E

be the midpoint of

AB

and

F

be the midpoint of

AD

.

DE

and

CF

meet at

G

. Find the area of

\vartriangle EFG

.
C11. The eight numbers

2015

,

2016

,

2017

,

2018

,

2019

,

2020

,

2021

, and

2022

are split into four groups of two such that the two numbers in each pair differ by a power of

2

. In how many different ways can this be done?
C12 / G4. We define a function f such that for all integers

n, k, x

, we have that

f(n, kx) = k^n f(n, x) and f(n + 1, x) = xf(n, x).

If

f(1, k) = 2k

for all integers

k

, then what is

f(3, 7)

?
C13 / G8. A sequence of positive integers is constructed such that each term is greater than the previous term, no term is a multiple of another term, and no digit is repeated in the entire sequence. An example of such a sequence would be

4

,

79

,

1035

. How long is the longest possible sequence that satisfies these rules?
C14 / G11.

ABC

is an equilateral triangle of side length

8

.

P

is a point on side AB. If

AC +CP = 5 \cdot AP

, find

AP

.
C15. What is the value of

(1) + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + 49 + 50)

?
G6. An ant is on a coordinate plane. It starts at

(0, 0)

and takes one step each second in the North, South, East, or West direction. After

5

steps, what is the probability that the ant is at the point

(2, 1)

?
G10. Find the set of real numbers

S

so that

\prod_{c\in S}(x^2 + cxy + y^2) = (x^2 - y^2)(x^{12} - y^{12}).

G12. Given a function

f(x)

such that

f(a + b) = f(a) + f(b) + 2ab

and

f(3) = 0

, find

f\left( \frac12 \right)

.
G13. Badville is a city on the infinite Cartesian plane. It has

24

roads emanating from the origin, with an angle of

15

degrees between each road. It also has beltways, which are circles centered at the origin with any integer radius. There are no other roads in Badville. Steven wants to get from

(10, 0)

to

(3, 3)

. What is the minimum distance he can take, only going on roads?
G14. Team

A

and Team

B

are playing basketball. Team A starts with the ball, and the ball alternates between the two teams. When a team has the ball, they have a

50\%

chance of scoring

1

point. Regardless of whether or not they score, the ball is given to the other team after they attempt to score. What is the probability that Team

A

will score

5

points before Team

B

scores any?
G15. The twelve-digit integer

\overline{A58B3602C91D},

where

A, B, C, D

are digits with

A > 0

, is divisible by

10101

. Find

\overline{ABCD}

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

2017

1

2017 MBMT Team Round - Montgomery Blair Math Tournament

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names

R1. What is

11^2 - 9^2

?
R2. Write

\frac{9}{15}

as a decimal.
R3. A

90^o

sector of a circle is shaded, as shown below. What percent of the circle is shaded?
R4. A fair coin is flipped twice. What is the probability that the results of the two flips are different?
R5. Wayne Dodson has

55

pounds of tungsten. If each ounce of tungsten is worth

75

cents, and there are

16

ounces in a pound, how much money, in dollars, is Wayne Dodson’s tungsten worth?
R6. Tenley Towne has a collection of

28

sticks. With these

28

sticks he can build a tower that has

1

stick in the top row,

2

in the next row, and so on. Let

n

be the largest number of rows that Tenley Towne’s tower can have. What is n?
R7. What is the sum of the four smallest primes?
R8 / P1. Let

ABC

be an isosceles triangle such that

\angle B = 42^o

. What is the sum of all possible degree measures of angle

A

?
R9. Consider a line passing through

(0, 0)

and

(4, 8)

. This line passes through the point

(2, a)

. What is the value of

a

?
R10 / P2. Brian and Stan are playing a game. In this game, Brian rolls a fair six-sided die, while Stan rolls a fair four-sided die. Neither person shows the other what number they rolled. Brian tells Stan, “The number I rolled is guaranteed to be higher than the number you rolled.” Stan now has to guess Brian’s number. If Stan plays optimally, what is the probability that Stan correctly guesses the number that Brian rolled?
R11. Guang chooses

4

distinct integers between

0

and

9

, inclusive. How many ways can he choose the integers such that every pair of chosen integers sums up to an even number?
R12 / P4. David is trying to write a problem for MBMT. He assigns degree measures to every interior angle in a convex

n

-gon, and it so happens that every angle he assigned is less than

144

degrees. He tells Pratik the value of

n

and the degree measures in the

n

-gon, and to David’s dismay, Pratik claims that such an

n

-gon does not exist. What is the smallest value of

n \ge 3

such that Pratik’s claim is necessarily true?
R13 / P3. Consider a triangle

ABC

with side lengths of

5

,

5

, and

2\sqrt5

. There exists a triangle with side lengths of

5, 5

, and

x

(

x \ne 2\sqrt5

) which has the same area as

ABC

. What is the value of

x

?
R14 / P5. A mother has

11

identical apples and

9

identical bananas to distribute among her

3

kids. In how many ways can the fruits be allocated so that each child gets at least one apple and one banana?
R15 / P7. Find the sum of the five smallest positive integers that cannot be represented as the sum of two not necessarily distinct primes.
P6. Srinivasa Ramanujan has the polynomial

P(x) = x^5 - 3x^4 - 5x^3 + 15x^2 + 4x - 12

. His friend Hardy tells him that

3

is one of the roots of

P(x)

. What is the sum of the other roots of

P(x)

?
P8.

ABC

is an equilateral triangle with side length

10

. Let

P

be a point which lies on ray

\overrightarrow{BC}

such that

PB = 20

. Compute the ratio

\frac{PA}{PC}

.
P9. Let

ABC

be a triangle such that

AB = 10

,

BC = 14

, and

AC = 6

. The median

CD

and angle bisector

CE

are both drawn to side

AB

. What is the ratio of the area of triangle

CDE

to the area of triangle

ABC

?
P10. Find all integer values of

x

between

0

and

2017

inclusive, which satisfy

2016x^{2017} + 990x^{2016} + 2x + 17 \equiv 0 \,\,\, (mod \,\,\, 2017).

P11. Let

x^2 + ax + b

be a quadratic polynomial with positive integer roots such that

a^2 - 2b = 97

. Compute

a + b

.
P12. Let

S

be the set

\{2, 3, ... , 14\}

. We assign a distinct number from

S

to each side of a six-sided die. We say a numbering is predictable if prime numbers are always opposite prime numbers and composite numbers are always opposite composite numbers. How many predictable numberings are there? (Rotations of a die are not distinct)
P13. In triangle

ABC

,

AB = 10

,

BC = 21

, and

AC = 17

.

D

is the foot of the altitude from

A

to

BC

,

E

is the foot of the altitude from

D

to

AB

, and

F

is the foot of the altitude from

D

to

AC

. Find the area of the smallest circle that contains the quadrilateral

AEDF

.
P14. What is the greatest distance between any two points on the graph of

3x^2 + 4y^2 + z^2 - 12x + 8y + 6z = -11

?
P15. For a positive integer

n

,

\tau (n)

is defined to be the number of positive divisors of

n

. Given this information, find the largest positive integer

n

less than

1000

such that

\sum_{d|n} \tau (d) = 108.

In other words, we take the sum of

\tau (d)

for every positive divisor

d

of

n

, which has to be

108

.

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

2016

1

2016 MBMT Team Round - Montgomery Blair Math Tournament

[hide=E stands for Euclid , L stands for Lobachevsky]they had two problem sets under those two names
E1. How many positive divisors does

72

have?
E2 / L2. Raymond wants to travel in a car with

3

other (distinguishable) people. The car has

5

seats: a driver’s seat, a passenger seat, and a row of

3

seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car?
E3 / L3. Peter wants to make fruit punch. He has orange juice (

100\%

orange juice), tropical mix (

25\%

orange juice,

75\%

pineapple juice), and cherry juice (

100\%

cherry juice). If he wants his final mix to have

50\%

orange juice,

10\%

cherry juice, and

40\%

pineapple juice, in what ratios should he mix the

3

juices? Please write your answer in the form (orange):(tropical):(cherry), where the three integers are relatively prime.
E4 / L4. Points

A, B, C

, and

D

are chosen on a circle such that

m \angle ACD = 85^o

,

m\angle ADC = 40^o

,and

m\angle BCD = 60^o

. What is

m\angle CBD

?
E5.

a, b

, and

c

are positive real numbers. If

abc = 6

and

a + b = 2

, what is the minimum possible value of

a + b + c

?
E6 / L5. Circles

A

and

B

are drawn on a plane such that they intersect at two points. The centers of the two circles and the two intersection points lie on another circle, circle

C

. If the distance between the centers of circles

A

and

B

is

20

and the radius of circle

A

is

16

, what is the radius of circle

B

?
E7. Point

P

is inside rectangle

ABCD

. If

AP = 5

,

BP = 6

, and

CP = 7

, what is the length of

DP

?
E8 / L6. For how many integers

n

is

n^2 + 4

divisible by

n + 2

?
E9. How many of the perfect squares between

1

and

10000

, inclusive, can be written as the sum of two triangular numbers? We define the

n

th triangular number to be

1 + 2 + 3 + ... + n

, where

n

is a positive integer.
E10 / L7. A small sphere of radius

1

is sitting on the ground externally tangent to a larger sphere, also sitting on the ground. If the line connecting the spheres’ centers makes a

60^o

angle with the ground, what is the radius of the larger sphere?
E11 / L8. A classroom has

12

chairs in a row and

5

distinguishable students. The teacher wants to position the students in the seats in such a way that there is at least one empty chair between any two students. In how many ways can the teacher do this?
E12 / L9. Let there be real numbers

a

and

b

such that

a/b^2 + b/a^2 = 72

and

ab = 3

. Find the value of

a^2 + b^2

.
E13 / L10. Find the number of ordered pairs of positive integers

(x, y)

such that

gcd \, (x, y)+lcm \, (x, y) =x + y + 8

.
E14 / L11. Evaluate

\sum_{i=1}^{\infty}\frac{i}{4^i}=\frac{1}{4} +\frac{2}{16} +\frac{3}{64} +...

E15 / L12. Xavier and Olivia are playing tic-tac-toe. Xavier goes first. How many ways can the game play out such that Olivia wins on her third move? The order of the moves matters.
L1. What is the sum of the positive divisors of

100

?
L13. Let

ABCD

be a convex quadrilateral with

AC = 20

. Furthermore, let

M, N, P

, and

Q

be the midpoints of

DA, AB, BC

, and

CD

, respectively. Let

X

be the intersection of the diagonals of quadrilateral

MNPQ

. Given that

NX = 12

and

XP = 10

, compute the area of

ABCD

.
L14. Evaluate

(\sqrt3 + \sqrt5)^6

to the nearest integer.
L15. In Hatland, each citizen wears either a green hat or a blue hat. Furthermore, each citizen belongs to exactly one neighborhood. On average, a green-hatted citizen has

65\%

of his neighbors wearing green hats, and a blue-hatted citizen has

80\%

of his neighbors wearing blue hats. Each neighborhood has a different number of total citizens. What is the ratio of green-hatted to blue-hatted citizens in Hatland? (A citizen is his own neighbor.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2015 E15

1

MBMT Team -- Euler #15

In quadrilateral

ABCD

, diagonals

AC

and

BD

intersect at

O

. If the area of triangle

DOC

is

4

and the area of triangle

AOB

is

36

, compute the minimum possible value of the area of

ABCD

.

2015 E14

1

MBMT Team -- Euler #14

Jane tells you that she is thinking of a three-digit number that is greater than

500

that has exactly

20

positive divisors. If Jane tells you the sum of the positive divisors of her number, you would not be able to figure out her number. If, instead, Jane had told you the sum of the prime divisors of her number, then you also would not have been able to figure out her number. What is Jane's number? (Note: the sum of the prime divisors of

12

is

2 + 3 = 5

, not

2 + 2 + 3 = 7

.)

2015 E13

1

MBMT Team -- Euler #13

A blind ant is walking on the coordinate plane. It is trying to reach an anthill, placed at all points where both the

x

-coordinate and

y

-coordinate are odd. The ant starts at the origin, and each minute it moves one unit either up, down, to the right, or to the left, each with probability

\frac{1}{4}

. The ant moves

3

times and doesn't reach an anthill during this time. On average, how many additional moves will the ant need to reach an anthill? (Compute the expected number of additional moves needed.)

2015 E8

1

MBMT Team -- Euler #8

You are trying to maximize a function of the form

f(x, y, z) = ax + by + cz

, where

a

,

b

, and

c

are constants. You know that

f(3, 1, 1) > f(2, 1, 1)

,

f(2, 2, 3) > f(2, 3, 4)

, and

f(3, 3, 4) > f(3, 3, 3)

. For

-5 \le x,y,z \le 5

, what value of

(x,y,z)

maximizes the value of

f(x, y, z)

? Give your answer as an ordered triple.

2015 F15 E12

1

MBMT Team -- Fermat #15/Euler #12

Adam, Bendeguz, Cathy, and Dennis all see a positive integer

n

. Adam says, "

n

leaves a remainder of

2

when divided by

3

." Bendeguz says, "For some

k

,

n

is the sum of the first

k

positive integers." Cathy says, "Let

s

be the largest perfect square that is less than

2n

. Then

2n - s = 20

." Dennis says, "For some

m

, if I have

m

marbles, there are

n

ways to choose two of them." If exactly one of them is lying, what is

n

?

2015 F14

1

MBMT Team -- Fermat #14

Let

ABCD

be a square with side length

1

. If point

E

is on

BC

, point

F

is on

DC

, and triangle

AEF

is equilateral, compute the side length of triangle

AEF

. (Note: if your answer has a square root inside a square root, you have not fully simplified your answer.)

2015 F13 E11

1

MBMT Team -- Fermat #13/Euler #11

Two (not necessarily different) integers between

1

and

60

, inclusive, are chosen independently and at random. What is the probability that their product is a multiple of

60

?

2015 F12 E10

1

MBMT Team -- Fermat #12/Euler #10

Let

ABC

be a triangle with

AB = 13

,

BC = 14

, and

AC = 15

. Let

D

and

E

be the midpoints of

BC

and

AB

, respectively. If

AD

and

CE

intersect at

G

, compute the area of quadrilateral

BEGD

.

2015 F11 E9

1

MBMT Team -- Fermat #11/Euler #9

The right triangle below has legs of length

1

and

2

. Find the sum of the areas of the shaded regions (of which there are infinitely many), given that the regions into which the triangle has been divided are all right triangles.

2015 F10 E7

1

MBMT Team -- Fermat #10/Euler #7

Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in

6

minutes, Mr. Rose in

10

minutes, and Mr. Schwartz in

18

minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?

2018 F9 E6

1

MBMT Team -- Fermat #9/Euler #6

If

x + y = 3

and

x^2 + y^2 = 7

, compute

x^3 + y^3 + x^4 + y^4

.

2015 F8 E5

1

MBMT Team -- Fermat #8/Euler #5

Victor has

3

piles of

3

cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the

3

piles.) In how many orders can Victor draw the cards?

2015 F7 E4

1

MBMT Team -- Fermat #7/Euler #4

Compute

\frac{x^2 + 8x + 7}{x^2 + 9x + 14}

, if

x = 2015

.

2015 F6 E3

1

MBMT Team -- Fermat #6/Euler #3

If

a

,

b

and

c

are real numbers such that

ab = 31

,

ac = 13

, and

bc = 5

, compute the product of all possible values of

abc

.

2015 F5 E2

1

MBMT Team -- Fermat #5/Euler #2

An unfair

6

-sided die has faces labeled

1

,

2

,

3

,

4

,

5

, and

6

. The probability that a die lands with a certain face up is proportional to the number on the face. What is the probability that at least one of the first three rolls is a

1

or a

2

?

2015 F4

1

MBMT Team -- Fermat #4

For a list of

9

positive integers that are not necessarily all different, the mean, median, and (unique) mode are all

9

. What is largest possible positive difference between the largest element and the smallest element of the set?

2015 F3 E1

1

MBMT Team -- Fermat #3/Euler #1

Compute

1 - 2 + 3 - 4 + \dots + 2013 - 2014 + 2015

.

2015 F2

1

MBMT Team -- Fermat #2

Mr. Rose gave a test to his two calculus classes. His first period class has

20

students, and their average score on the test was

80

. His second period class has

30

students, and their average score was

90

. What was the average score of all

50

of his calculus students?

2015 F1

1

MBMT Team -- Fermat #1

Point

A

is located at

(0,0)

. Point

B

is located at

(2,3)

and is the midpoint of

AC

. Point

D

is located at

(10,0)

. What are the coordinates of the midpoint of segment

CD

?

MBMT Team Rounds

Subcontests

2023 MBMT Team Round - Montgomery Blair Math Tournament

2022 MBMT Team Round - Montgomery Blair Math Tournament

2019 MBMT Team Round - Montgomery Blair Math Tournament

senators' committee in Flatland Congress 2020 MBMT p45

a_n =sum_{d|n} 1/2^{d+ n/d} 2020 MBMT p44

\sigma_k(n)/ n^{k+2} (2020n + 2019)^2 > m 2020 MBMT p43

integer lenght of angle bisector 2020 MBMT p42

last two digits of $$2^{3^{4^{...^{2019}}}} 2020 MBMT p41

2018 MBMT Team Round - Montgomery Blair Math Tournament

2017 MBMT Team Round - Montgomery Blair Math Tournament

2016 MBMT Team Round - Montgomery Blair Math Tournament

MBMT Team -- Euler #15

MBMT Team -- Euler #14

MBMT Team -- Euler #13

MBMT Team -- Euler #8

MBMT Team -- Fermat #15/Euler #12

MBMT Team -- Fermat #14

MBMT Team -- Fermat #13/Euler #11

MBMT Team -- Fermat #12/Euler #10

MBMT Team -- Fermat #11/Euler #9

MBMT Team -- Fermat #10/Euler #7

MBMT Team -- Fermat #9/Euler #6

MBMT Team -- Fermat #8/Euler #5

MBMT Team -- Fermat #7/Euler #4

MBMT Team -- Fermat #6/Euler #3

MBMT Team -- Fermat #5/Euler #2

MBMT Team -- Fermat #4

MBMT Team -- Fermat #3/Euler #1

MBMT Team -- Fermat #2

MBMT Team -- Fermat #1

MBMT Team Rounds

Subcontests

2023 MBMT Team Round - Montgomery Blair Math Tournament

2022 MBMT Team Round - Montgomery Blair Math Tournament

2019 MBMT Team Round - Montgomery Blair Math Tournament

senators' committee in Flatland Congress 2020 MBMT p45

a_n =sum_{d|n} 1/2^{d+ n/d} 2020 MBMT p44

\sigma_k(n)/ n^{k+2} (2020n + 2019)^2 &gt; m 2020 MBMT p43

integer lenght of angle bisector 2020 MBMT p42

last two digits of $$2^{3^{4^{...^{2019}}}} 2020 MBMT p41

2018 MBMT Team Round - Montgomery Blair Math Tournament

2017 MBMT Team Round - Montgomery Blair Math Tournament

2016 MBMT Team Round - Montgomery Blair Math Tournament

MBMT Team -- Euler #15

MBMT Team -- Euler #14

MBMT Team -- Euler #13

MBMT Team -- Euler #8

MBMT Team -- Fermat #15/Euler #12

MBMT Team -- Fermat #14

MBMT Team -- Fermat #13/Euler #11

MBMT Team -- Fermat #12/Euler #10

MBMT Team -- Fermat #11/Euler #9

MBMT Team -- Fermat #10/Euler #7

MBMT Team -- Fermat #9/Euler #6

MBMT Team -- Fermat #8/Euler #5

MBMT Team -- Fermat #7/Euler #4

MBMT Team -- Fermat #6/Euler #3

MBMT Team -- Fermat #5/Euler #2

MBMT Team -- Fermat #4

MBMT Team -- Fermat #3/Euler #1

MBMT Team -- Fermat #2

MBMT Team -- Fermat #1

\sigma_k(n)/ n^{k+2} (2020n + 2019)^2 > m 2020 MBMT p43