MathDB

p1. Find the largest

4

digit integer that is divisible by

2

and

5

, but not

3

.
p2. The diagram below shows the eight vertices of a regular octagon of side length

2

. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure

270^o

. Compute the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png
p3. Consider the numbers formed by writing full copies of

2023

next to each other, like so:

2023202320232023...

How many copies of

2023

are next to each other in the smallest multiple of

11

that can be written in this way?
p4. A positive integer

n

with base-

10

representation

n = a_1a_2 ...a_k

is called powerful if the digits

a_i

are nonzero for all

1 \le i \le k

and

n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k .

What is the unique four-digit positive integer that is powerful?
p5. Six

(6)

chess players, whose names are Alice, Bob, Crystal, Daniel, Esmeralda, and Felix, are sitting in a circle to discuss future content pieces for a show. However, due to fights they’ve had, Bob can’t sit beside Alice or Crystal, and Esmeralda can’t sit beside Felix. Determine the amount of arrangements the chess players can sit in. Two arrangements are the same if they only differ by a rotation.
p6. Given that the infinite sum

\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +...

is equal to

\frac{\pi^4}{90}

, compute the value of

\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}

p7. Triangle

ABC

is equilateral. There are

3

distinct points,

X

Y

Z

inside

\vartriangle ABC

that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio

1 : 1 : 2

in some order. Find the ratio of the area of

\vartriangle ABC

to that of

\vartriangle XY Z

.
p8. For a fixed prime

p

, a finite non-empty set

S = \{s_1,..., s_k\}

of integers is

p

-admissible if there exists an integer

n

for which the product

(s_1 + n)(s_2 + n) ... (s_k + n)

is not divisible by

p

. For example,

\{4, 6, 8\}

2

-admissible since

(4+1)(6+1)(8+1) = 315

is not divisible by

2

. Find the size of the largest subset of

\{1, 2,... , 360\}

that is two-,three-, and five-admissible.
p9. Kwu keeps score while repeatedly rolling a fair

6

-sided die. On his first roll he records the number on the top of the die. For each roll, if the number was prime, the following roll is tripled and added to the score, and if the number was composite, the following roll is doubled and added to the score. Once Kwu rolls a

1

, he stops rolling. For example, if the first roll is

1

, he gets a score of

1

, and if he rolls the sequence

(3, 4, 1)

, he gets a score of

3 + 3 \cdot 4 + 2 \cdot 1 = 17

. What is his expected score?
p10. Let

\{a_1, a_2, a_3, ...\}

be a geometric sequence with

a_1 = 4

and

a_{2023} = \frac14

. Let

f(x) = \frac{1}{7(1+x^2)}

. Find

f(a_1) + f(a_2) + ... + f(a_{2023}).

p11. Let

S

be the set of quadratics

x^2 + ax + b

, with

a

and

b

real, that are factors of

x^{14} - 1

. Let

f(x)

be the sum of the quadratics in

S

. Find

f(11)

.
p12. Find the largest integer

0 < n < 100

such that

n^2 + 2n

divides

4(n- 1)! + n + 4

.
p13. Let

\omega

be a unit circle with center

O

and radius

OQ

. Suppose

P

is a point on the radius

OQ

distinct from

Q

such that there exists a unique chord of

\omega

through

P

whose midpoint when rotated

120^o

counterclockwise about

Q

lies on

\omega

. Find

OP

.
p14. A sequence of real numbers

\{a_i\}

satisfies

n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n

for each integer

n \ge 1

. Find the value of

a_{2023}

.
p15. In

\vartriangle ABC

, let

\angle ABC = 90^o

and let

I

be its incenter. Let line

BI

intersect

AC

at point

D

, and let line

CI

intersect

AB

at point

E

. If

ID = IE = 1

, find

BI

.
p16. For a positive integer

n

, let

S_n

be the set of permutations of the first

n

positive integers. If

p = (a_1, ..., a_n) \in S_n

, then define the bijective function

\sigma_p : \{1,..., n\} \to \{1, ..., n\}

such that

\sigma_p (i) = a_i

for all integers

1 \le i \le n

. For any two permutations

p, q \in S_n

, we say

p

and

q

are friends if there exists a third permutation

r \in S_n

such that for all integers

1 \le i \le n

\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)).

Find the number of friends, including itself, that the permutation

(4, 5, 6, 7, 8, 9, 10, 2, 3, 1)

has in

S_{10}

.
PS. You had better use hide for answers.

Problem Statement