MathDB

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
L.10 Given the following system of equations where

x, y, z

are nonzero, find

x^2 + y^2 + z^2

x + 2y = xy

3y + z = yz

3x + 2z = xz

Set 4
L.16 / D.23 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.
L.17 The repeating decimal

0.\overline{MBMT}

is equal to

\frac{p}{q}

, where

p

and

q

are relatively prime positive integers, and

M, B, T

are distinct digits. Find the minimum value of

q

.
L.18 Annie, Bob, and Claire have a bag containing the numbers

1, 2, 3, . . . , 9

. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so

123

213

, and

321

all count as arithmetic sequences.)
L.19 Consider a set

S

of positive integers. Define the operation

f(S)

to be the smallest integer

n > 1

such that the base

2^k

representation of

n

consists only of ones and zeros for all

k \in S

. Find the size of the largest set

S

such that

f(S) < 2^{2019}

.
L.20 / D.25 Find the largest solution to the equation

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

Set 5
L.21 Steven is concerned about his artistic abilities. To make himself feel better, he creates a

100 \times 100

square grid and randomly paints each square either white or black, each with probability

\frac12

. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png
L.22 Let x be chosen uniformly at random from

[0, 1]

. Let n be the smallest positive integer such that

3^n x

is at most

\frac14

away from an integer. What is the expected value of

n

?
L.23 Let

A

and

B

be two points in the plane with

AB = 1

. Let

\ell

be a variable line through

A

. Let

\ell'

be a line through

B

perpendicular to

\ell

. Let X be on

\ell

and

Y

be on

\ell'

with

AX = BY = 1

. Find the length of the locus of the midpoint of

XY

.
L.24 Each of the numbers

a_i

, where

1 \le i \le n

, is either

-1

1

. Also,

a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0.

Find the number of possible values for

n

between

4

and

100

, inclusive.
L.25 Let

S

be the set of positive integers less than

3^{2019}

that have only zeros and ones in their base

3

representation. Find the sum of the squares of the elements of

S

. Express your answer in the form

a^b(c^d - 1)(e^f - 1)

, where

a, b, c, d, e, f

are positive integers and

a, c, e

are not perfect powers.
PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here and D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement