MathDB

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
Z15. Let

AOB

be a quarter circle with center

O

and radius

4

. Let

\omega_1

and

\omega_2

be semicircles inside

AOB

with diameters

OA

and

OB

, respectively. Find the area of the region within

AOB

but outside of

\omega_1

and

\omega_2

.
Set 4
Z16. Integers

a, b, c

form a geometric sequence with an integer common ratio. If

c = a + 56

, find

b

.
Z17 / D24. In parallelogram

ABCD

\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o

where all angles are in degrees. Find the value of

\angle C

.
Z18. Steven likes arranging his rocks. A mountain formation is where the sequence of rocks to the left of the tallest rock increase in height while the sequence of rocks to the right of the tallest rock decrease in height. If his rocks are

1, 2, . . . , 10

inches in height, how many mountain formations are possible? For example: the sequences

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

and

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

are considered mountain formations.
Z19. Find the smallest

5

-digit multiple of

11

whose sum of digits is

15

.
Z20. Two circles,

\omega_1

and

\omega_2

, have radii of

2

and

8

, respectively, and are externally tangent at point

P

. Line

\ell

is tangent to the two circles, intersecting

\omega_1

A

and

\omega_2

B

. Line

m

passes through

P

and is tangent to both circles. If line

m

intersects line

\ell

at point

Q

, calculate the length of

P Q

.
Set 5
Z21. Sen picks a random

1

million digit integer. Each digit of the integer is placed into a list. The probability that the last digit of the integer is strictly greater than twice the median of the digit list is closest to

\frac{1}{a}

, for some integer

a

. What is

a

?
Z22. Let

6

points be evenly spaced on a circle with center

O

, and let

S

be a set of

7

points: the

6

points on the circle and

O

. How many equilateral polygons (not self-intersecting and not necessarily convex) can be formed using some subset of

S

as vertices?
Z23. For a positive integer

n

, define

r_n

recursively as follows:

r_n = r^2_{n-1} + r^2_{n-2} + ... + r^2_0

,where

r_0 = 1

. Find the greatest integer less than

\frac{r_2}{r^2_1}+\frac{r_3}{r^2_2}+ ...+\frac{r_{2023}}{r^2_{2022}}.

Z24. Arnav starts at

21

on the number line. Every minute, if he was at

n

, he randomly teleports to

2n^2

n^2

, or

\frac{n^2}{4}

with equal chance. What is the probability that Arnav only ever steps on integers?
Z25. Let

ABCD

be a rectangle inscribed in circle

\omega

with

AB = 10

. If

P

is the intersection of the tangents to

\omega

C

and

D

, what is the minimum distance from

P

AB

?
PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here and D.16-30/Z.9-14, 17, 26-30 [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement