MathDB

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
Set 4
C.16 / G.6 Let

a, b

, and

c

be real numbers. If

a^3 + b^3 + c^3 = 64

and

a + b = 0

, what is the value of

c

?
C.17 / G.8 Bender always turns

60

degrees clockwise. He walks

3

meters, turns, walks

2

meters, turns, walks

1

meter, turns, walks

4

meters, turns, walks

1

meter, and turns. How many meters does Bender have to walk to get back to his original position?
C.18 / G.13 Guang has

4

identical packs of gummies, and each pack has a red, a blue, and a green gummy. He eats all the gummies so that he finishes one pack before going on to the next pack, but he never eats two gummies of the same color in a row. How many different ways can Guang eat the gummies?
C.19 Find the sum of all digits

q

such that there exists a perfect square that ends in

q

.
C.20 / G.14 The numbers

5

and

7

are written on a whiteboard. Every minute Stev replaces the two numbers on the board with their sum and difference. After

2017

minutes the product of the numbers on the board is

m

. Find the number of factors of

m

.
Set 5
C.21 / G.10 On the planet Alletas,

\frac{32}{33}

of the people with silver hair have purple eyes and

\frac{8}{11}

of the people with purple eyes have silver hair. On Alletas, what is the ratio of the number of people with purple eyes to the number of people with silver hair?
C.22 / G.15 Let

P

be a point on

y = -1

. Let the clockwise rotation of

P

60^o

about

(0, 0)

P'

. Find the minimum possible distance between

P'

and

(0, -1)

.
C.23 / G.18 How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.
C.24 Jeremy and Kevin are arguing about how cool a sweater is on a scale of

1-5

. Jeremy says “

3

”, and Kevin says “

4

”. Jeremy angrily responds “

3.5

”, to which Kevin replies “

3.75

”. The two keep going at it, responding with the average of the previous two ratings. What rating will they converge to (and settle on as the coolness of the sweater)?
C.25 / G.20 An even positive integer

n

has an odd factorization if the largest odd divisor of

n

is also the smallest odd divisor of

n

greater than

1

. Compute the number of even integers

n

less than

50

with an odd factorization.
Set 6
C.26 / G.26 When

2018! = 2018 \times 2017 \times ... \times 1

is multiplied out and written as an integer, find the number of

4

’s.
If the correct answer is

A

and your answer is

E

, you will receive

12 \min\, \, (A/E, E/A)^3

points.
C.27 / G.27 A circle of radius

10

is cut into three pieces of equal area with two parallel cuts. Find the width of the center piece. https://cdn.artofproblemsolving.com/attachments/e/2/e0ab4a2d51052ee364dd14336677b053a40352.png If the correct answer is

A

and your answer is

E

, you will receive

\max \, \,(0, 12 - 6|A - E|)

points.
C.28 / G.28 An equilateral triangle of side length

1

is randomly thrown onto an infinite set of lines, spaced

1

apart. On average, how many times will the boundary of the triangle intersect one of the lines? https://cdn.artofproblemsolving.com/attachments/0/1/773c3d3e0dfc1df54945824e822feaa9c07eb7.png For example, in the above diagram, the boundary of the triangle intersects the lines in

2

places.
If the correct answer is

A

and your answer is

E

, you will receive

\max\, \,(0, 12-120|A-E|/A)

points.
C.29 / G.29 Call an ordered triple of integers

(a, b, c)

nice if there exists an integer

x

such that

ax^2 + bx + c = 0

. How many nice triples are there such that

-100 \le a, b, c \le 100

?
If the correct answer is

A

and your answer is

E

, you will receive

12 \min\, \,(A/E, E/A)

points.
C.30 / G.30 Let

f(i)

denote the number of MBMT volunteers to be born in the

i

th state to join the United States. Find the value of

1f(1) + 2f(2) + 3f(3) + ... + 50f(50)

.
Note 1: Maryland was the

7

th state to join the US. Note 2: Last year’s MBMT competition had

42

volunteers.
If the correct answer is

A

and your answer is

E

, you will receive

\max\, \,(0, 12 - 500(|A -E|/A)^2)

points.
PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement