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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1972 Kurschak Competition
2
2
Part of
1972 Kurschak Competition
Problems
(1)
f(X)$ be the number of ways of dividing the line of n boys and n girls
Source: 1972 Hungary - Kürschák Competition p2
10/15/2022
A class has
n
>
1
n > 1
n
>
1
boys and
n
n
n
girls. For each arrangement
X
X
X
of the class in a line let
f
(
X
)
f(X)
f
(
X
)
be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with
f
(
X
)
=
0
f(X) = 0
f
(
X
)
=
0
be
A
A
A
, and the number of arrangements with
f
(
X
)
=
1
f(X) = 1
f
(
X
)
=
1
be
B
B
B
. Show that
B
=
2
A
B = 2A
B
=
2
A
.
combinatorics