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National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2013 Bangladesh Mathematical Olympiad
10
10
Part of
2013 Bangladesh Mathematical Olympiad
Problems
(1)
Ordered sequence of elements
Source: Bangladesh National MO 2013
3/5/2019
Higher Secondary P10
X
X
X
is a set of
n
n
n
elements.
P
m
(
X
)
P_m(X)
P
m
(
X
)
is the set of all
m
m
m
element subsets (i.e. subsets that contain exactly
m
m
m
elements) of
X
X
X
. Suppose
P
m
(
X
)
P_m(X)
P
m
(
X
)
has
k
k
k
elements. Prove that the elements of
P
m
(
X
)
P_m(X)
P
m
(
X
)
can be ordered in a sequence
A
1
,
A
2
,
.
.
.
A
i
,
.
.
.
A
k
A_1, A_2,...A_i,...A_k
A
1
,
A
2
,
...
A
i
,
...
A
k
such that it satisfies the two conditions: (A) each element of
P
m
(
X
)
P_m(X)
P
m
(
X
)
occurs exactly once in the sequence, (B) for any
i
i
i
such that
0
<
i
<
k
0<i<k
0
<
i
<
k
, the size of the set
A
i
∩
A
i
+
1
A_i \cap A_{i+1}
A
i
∩
A
i
+
1
is
m
−
1
m-1
m
−
1
.
algebra