MathDB

Round 1
p1. Alex is writing a sequence of

A

’s and

B

’s on a chalkboard. Any

20

consecutive letters must have an equal number of

A

’s and

B

’s, but any 22 consecutive letters must have a different number of

A

’s and

B

’s. What is the length of the longest sequence Alex can write?.
p2. A positive number is placed on each of the

10

circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all

10

numbers are different? https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png
p3. Pablo and Nina take turns entering integers into the cells of a

3 \times 3

table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to

0

. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does?
p4. All possible simplified fractions greater than

0

and less than

1

with denominators less than or equal to

100

are written in a row with a space before each number (including the first). Zeke and Qing play a game, taking turns choosing a blank space and writing a “

+

” or “

-

” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer. Can Zeke win no matter what Qing does? https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.png
p5. A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? https://cdn.artofproblemsolving.com/attachments/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png
Round 2
p6. Prove that among any

3^{2022}

integers, it is possible to find exactly

3^{2021}

of them whose sum is divisible by

3^{2021}

.
p7. Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is

(5, 7, 10)

and you zap

5

and

10

, whose average is

7.5

, the new list is

(-2.5, 7, 2.5)

. Is it possible to start with the list

(3, 1, 4)

and, through some sequence of zaps, end with a list in which the sum of the three numbers is

0

?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement