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Problems
Contests
International Contests
Austrian-Polish
1992 Austrian-Polish Competition
9
9
Part of
1992 Austrian-Polish Competition
Problems
(1)
consider words composed of n letters A and n letters $B
Source: Austrian - Polish 1992 APMC
5/7/2020
Given an integer
n
>
1
n > 1
n
>
1
, consider words composed of
n
n
n
letters
A
A
A
and
n
n
n
letters
B
B
B
. A word
X
1
.
.
.
X
2
n
X_1...X_{2n}
X
1
...
X
2
n
is said to belong to set
R
(
n
)
R(n)
R
(
n
)
(respectively,
S
(
n
)
S(n)
S
(
n
)
) if no initial segment (respectively, exactly one initial segment)
X
1
.
.
.
X
k
X_1...X_k
X
1
...
X
k
with
1
≤
k
<
2
n
1 \le k < 2n
1
≤
k
<
2
n
consists of equally many letters
A
A
A
and
B
B
B
. If
r
(
n
)
r(n)
r
(
n
)
and
s
(
n
)
s(n)
s
(
n
)
denote the cardinalities of
R
(
n
)
R(n)
R
(
n
)
and
S
(
n
)
S(n)
S
(
n
)
respectively, compute
s
(
n
)
/
r
(
n
)
s(n)/r(n)
s
(
n
)
/
r
(
n
)
.
Words
combinatorics