MathDB

p1. Quarters are worth

25

cents and weigh

5.7

grams each. Dimes are worth

10

cents and weigh

2.3

grams each. A pile of quarters weighs

342

grams, and a pile of dimes is equal in value to the pile of quarters. What is the weight in grams of the pile of dimes?
p2. Three-digit palindrome

\overline{ABA}

with

A \ne 0

has the property that the sum of the squares of its digits equals the two-digit number

\overline{AB}

. What is the sum of all possible values of

\overline{ABA}

?
p3. Let

N

be the number

N = 01020304 . . . 979899

, formed by concatenating the integers from

01

99

inclusive, with leading zeros. How many times does the digit

7

appear in

7 \times N

?
p4. Winry and Riza are each writing the integers

1, 2, . . . , N

on a chalkboard. However, Winry erases all occurrences of the digit

1

, and Riza erases all occurrences of the digit

3

. They notice that at

N = 13

, the sum of the digits written on the chalkboard on Winry’s list is the same as the sum of the digits written on the chalkboard on Riza’s list. What is the next smallest

N

where this property holds?
p5. Define the Hexicab distance between two points on the Cartesian plane to be the length of the shortest path connecting the points that consists of line segments parallel to the

x

-axis, the line

y =\frac{\sqrt3}{3}x

, or the line

y =\sqrt3 x

. The region

R

consists of all points that are a Hexicab distance at most

1

from the origin. If the area of R can be expressed in the form

\frac{a+b\sqrt{c}}{d}

where

a, b, c, d

are positive integers such that c is square-free and

gcd(a, b, d) = 1

, find

a + b + c + d

.
p6.

\vartriangle A_0B_0C_0

has

\angle A_0 = 40^o

\angle B_0 = 60^o

, and angle

\angle C_0 = 80^o

. It is rotated about vertex

A_0

clockwise by some angle

\alpha

to get

\vartriangle A_1B_1C_1

. Then,

\vartriangle A_1B_1C_1

is rotated clockwise about

B_1

by some angle

\beta

to get

\vartriangle A_2B_2C_2

. After that,

\vartriangle A_2B_2C_2

is rotated clockwise about vertex

C_2

by some angle

\gamma

to get

\vartriangle A_3B_3C_3

. If

A_0 = A_3

B_0 = B_3

C_0 = C_3

, and

0^o < \alpha\le 180^o

, then what is

\alpha

in degrees?
p7. Ally, Bella, Aria, Barbara, Ava, Brianna, and Cindy are playing a game, with turns taken in the order listed above. Ally starts with

1

muffin, and has the choice to either “double it and give it to the next person” or “square it and give it to the next person”. Ally then passes the resulting amount to Bella, and this process continues, with each person given the same choice and passing it to the next person. This process ends with Cindy receiving C muffins. Ally, Aria, and Ava are trying to maximize

|2023 - C|

, while Bella, Barbara, and Brianna are trying to minimize

|2023 - C|

. If both sides play optimally, what is

|2023 - C|

?
p8. A set of integers has a coprime cycle if it can be listed in a cyclic list of the form

(a_1, a_2, a_3, . . . , a_n)

, where each element of the set appears exactly once and

gcd(a_i, a_{i+1}) = 1

for

i = 1, . . . , n

where

a_{n+1} = a_1

. For example,

(7, 8, 9, 10, 11, 14, 13, 12)

is a coprime cycle for

k = 7

, but

(7, 8, 9, 10, 11, 12, 13, 14)

is not because

gcd(14, 7) \ne 1

. What is the smallest positive integer k such that the set

\{k, k + 1, . . . , k + 7\}

does not possess a coprime cycle?
p9.

ABCD

is a regular tetrahedron with side length

1

. Spheres

O_1,

O_2

O_3

, and

O_4

have equal radii, and are positioned inside

ABCD

such that they are internally tangent to three of the faces at a time, and all four spheres intersect at a single point. If the radius of

O_1

can be expressed as

\frac{a \sqrt{b}}{c}

, where

a, b, c

are positive integers such that

b

is square-free and

gcd(a, c) = 1

, find

a + b + c

.
p10. Danielle has

2023^3

boxes in which she can place non-negative integers. Each box is assigned a unique ordered triple

(i, j, k)

with

i, j, k \in \{1, 2, 3, · · · , 2023\}

. In the box with ordered triple

(1, 1, 1)

, Danielle places

a_0

. Then, for a box labeled

(a, b, c)

, she places the smallest non-negative integer that has not been placed in any of

(a_0, b, c)

for

0 \le a_0 < a

(a, b_0, c)

for

0 \le b_0 < b

, or

(a, b, c_0)

for

0 \le c_0 < c

. Find the number Danielle places in the box labeled

(2023, 203, 20)

.
p11. For a fixed integer

n

, let

f(a, b, c, d) = (4a - 6b + 7c - 3d + 1)^n

. The

k

th degree terms of a polynomial are the terms whose exponents sum to

k

, e.g.

a^2 b^3 c

has degree

6

. Let

A

and

B

be the sum of the coefficients of all

7

th degree terms and of all

9

th degree terms of

f

, respectively. If

B = 224A,

find the total number of positive divisors of

A

.
p12. If

n

is the smallest positive integer such that

10^{3n^3+10n+1 } \equiv -1

(mod

14641

), then find the last three digits of

3^n

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

Problem Statement