MathDB

[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

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.

2023

Problems(1)