MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2004 India IMO Training Camp
2004 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(4)
1
5
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4
2
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A permutation
Given a permutation
σ
=
(
a
1
,
a
2
,
a
3
,
.
.
.
a
n
)
\sigma = (a_1,a_2,a_3,...a_n)
σ
=
(
a
1
,
a
2
,
a
3
,
...
a
n
)
of
(
1
,
2
,
3
,
.
.
.
n
)
(1,2,3,...n)
(
1
,
2
,
3
,
...
n
)
, an ordered pair
(
a
j
,
a
k
)
(a_j,a_k)
(
a
j
,
a
k
)
is called an inversion of
σ
\sigma
σ
if
a
≤
j
<
k
≤
n
a \leq j < k \leq n
a
≤
j
<
k
≤
n
and
a
j
>
a
k
a_j > a_k
a
j
>
a
k
. Let
m
(
σ
)
m(\sigma)
m
(
σ
)
denote the no. of inversions of the permutation
σ
\sigma
σ
. Find the average of
m
(
σ
)
m(\sigma)
m
(
σ
)
as
σ
\sigma
σ
varies over all permutations.
A functional ineq
Let
f
f
f
be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers
a
<
a
+
d
<
a
+
2
d
a < a+d < a+ 2d
a
<
a
+
d
<
a
+
2
d
such that
f
(
a
)
<
f
(
a
+
d
)
<
f
(
a
+
2
d
)
f(a) < f(a+d) <f(a+2d)
f
(
a
)
<
f
(
a
+
d
)
<
f
(
a
+
2
d
)
3
5
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2
7
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