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National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2022 Bangladesh Mathematical Olympiad
7
7
Part of
2022 Bangladesh Mathematical Olympiad
Problems
(1)
A city where the names of people are either A or B and 10 letters long
Source: BdMO 2022 Secondary P7
4/12/2022
Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length
10
10
10
, where each letter is either
A
A
A
or
B
B
B
. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word
A
B
B
ABB
A
BB
is located consecutively, then these letters can be changed to
B
B
A
BBA
BB
A
or if
B
B
A
BBA
BB
A
is located somewhere in the word consecutively, then these letters can be changed to
A
B
B
ABB
A
BB
(if wanted, the word can be kept as it is, without making this change.) For example
A
A
B
B
A
AABBA
AA
BB
A
can be transformed into
A
A
A
B
B
AAABB
AAA
BB
(the opposite is also possible.) Now Sabbir made a team of
N
N
N
citizens where no one is friends with anyone. What is the highest value of
N
.
N.
N
.
combinatorics