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ISI B.Stat Entrance Exam
2010 ISI B.Stat Entrance Exam
1
Rearrangement inequality
Rearrangement inequality
Source: ISI(BS) 2010 #1
May 16, 2012
inequalities
rearrangement inequality
inequalities proposed
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots, a_n
a
1
,
a
2
,
⋯
,
a
n
and
b
1
,
b
2
,
⋯
,
b
n
b_1,b_2,\cdots, b_n
b
1
,
b
2
,
⋯
,
b
n
be two permutations of the numbers
1
,
2
,
⋯
,
n
1,2,\cdots, n
1
,
2
,
⋯
,
n
. Show that
∑
i
=
1
n
i
(
n
+
1
−
i
)
≤
∑
i
=
1
n
a
i
b
i
≤
∑
i
=
1
n
i
2
\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2
i
=
1
∑
n
i
(
n
+
1
−
i
)
≤
i
=
1
∑
n
a
i
b
i
≤
i
=
1
∑
n
i
2
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