MathDB

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
Set 4
D.16 / L.6 Alex has

100

Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most

1

spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them?
D.17 I start with the number

1

in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket?
D.18 / L.12 Point

P

inside square

ABCD

is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area

25

25

, and

15

, what are all the possible values for the area of the fourth triangle?
D.19 Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at

100

or more decibels, then they lose the game. Mr. Stein yells first, at

88

decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win?
D.20 / L.15 A semicircle of radius

1

has line

\ell

along its base and is tangent to line

m

. Let

r

be the radius of the largest circle tangent to

\ell

m

, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of

r

is the interval

[a, b]

. Find

b - a

.
Set 5
D.21 / L.14 Hungryman starts at the tile labeled “

S

”. On each move, he moves

1

unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats.
https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png
D.22 / L.11 How many triples of nonnegative integers

(x, y, z)

satisfy the equation

6x + 10y +15z = 300

?
D.23 / L.16 Anson, Billiam, and Connor are looking at a

3D

figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a

5 \times 5

square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure.
D.24 / L.13 Tse and Cho are playing a game. Cho chooses a number

x \in [0, 1]

uniformly at random, and Tse guesses the value of

x(1 - x)

. Tse wins if his guess is at most

\frac{1}{50}

away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins?
D.25 / L.20 Find the largest solution to the equation

2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.

Set 6
This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.
D.26 / L.26 What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were

47

volunteers; this is the fifth MBMT.
D.27 / L.27 William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than

10\%

of the entire chocolate bar.
D.28 / L.28 Let

x

be the positive solution to the equation

x^{x^{x^x}}= 1.1

. Estimate

\frac{1}{x-1}

.
D.29 / L.29 Estimate the number of dots in the following box: https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png It may be useful to know that this image was produced by plotting

(4\sqrt{x}, y)

some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval

[0, 1]

.
D.30 / L.30 For a positive integer

n

, let

f(n)

be the smallest prime greater than or equal to

n

. Estimate

(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).

For

26 \le i \le 30

, let

E_i

be your team’s answer to problem

i

and let

A_i

be the actual answer to problem

i

. Your score

S_i

for problem

i

is given by

S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5)

S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|)

S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|))

S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right)

S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)

PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement