MathDB

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names
Set 4
D16. The cooking club at Blair creates

14

croissants and

21

danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes?
D17. Each digit in a

3

digit integer is either

1, 2

, or

4

with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit?
D18 / Z11. How many two digit numbers are there such that the product of their digits is prime?
D19 / Z9. In the coordinate plane, a point is selected in the rectangle defined by

-6 \le x \le 4

and

-2 \le y \le 8

. What is the largest possible distance between the point and the origin,

(0, 0)

?
D20 / Z10. The sum of two numbers is

6

and the sum of their squares is

32

. Find the product of the two numbers.
Set 5
D21 / Z12. Triangle

ABC

has area

4

and

\overline{AB} = 4

. What is the maximum possible value of

\angle ACB

?
D22 / Z13. Let

ABCD

be an iscoceles trapezoid with

AB = CD

and M be the midpoint of

AD

. If

\vartriangle ABM

and

\vartriangle MCD

are equilateral, and

BC = 4

, find the area of trapezoid

ABCD

.
D23 / Z14. Let

x

and

y

be positive real numbers that satisfy

(x^2 + y^2)^2 = y^2

. Find the maximum possible value of

x

.
D24 / Z17. 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

.
D25. The number

12ab9876543

is divisible by

101

, where

a, b

represent digits between

0

and

9

. What is

10a + b

?
Set 6
D26 / Z26. For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get

n

. Estimate the greatest integer

a

such that

2^a

evenly divides

n

.
D27 / Z27. Circles of radius

5

are centered at each corner of a square with side length

6

. If a random point

P

is chosen randomly inside the square, what is the probability that

P

lies within all four circles?
D28 / Z28. Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s

4

th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class?
D29 / Z29. Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are

10

meters west from a roast turkey. Beard, can turn exactly

0.7^o

and Bored can turn exactly

0.2^o

degrees. Driving at a consistent

2

meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey. Suppose Beard gets to the Turkey in about

818.5

seconds. Estimate the amount of time it will take Bored.
D30 / Z30. Let a be the probability that

4

randomly chosen positive integers have no common divisor except for

1

. Estimate

300a

. Note that the integers

1, 2, 3, 4

have no common divisor except for

1

.
Remark. This problem is asking you to find

300 \lim_{n\to \infty} a_n

, if

a_n

is defined to be the probability that

4

randomly chosen integers from

\{1, 2, ..., n\}

have greatest common divisor

1

.
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 Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement