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Contests
International Contests
IMO Longlists
1977 IMO Longlists
38
38
Part of
1977 IMO Longlists
Problems
(1)
Inequality for three sequences a_i, b_i and c_i - [ILL 1977]
Source:
1/11/2011
Let
m
j
>
0
m_j > 0
m
j
>
0
for
j
=
1
,
2
,
…
,
n
j = 1, 2,\ldots, n
j
=
1
,
2
,
…
,
n
and
a
1
≤
⋯
≤
a
n
<
b
1
≤
⋯
≤
b
n
<
c
1
≤
⋯
≤
c
n
a_1 \leq \cdots \leq a_n < b_1 \leq \cdots \leq b_n < c_1 \leq \cdots \leq c_n
a
1
≤
⋯
≤
a
n
<
b
1
≤
⋯
≤
b
n
<
c
1
≤
⋯
≤
c
n
be real numbers. Prove that
(
∑
j
=
1
n
m
j
(
a
j
+
b
j
+
c
j
)
)
2
>
3
(
∑
j
=
1
n
m
j
)
(
∑
j
=
1
n
m
j
(
a
j
b
j
+
b
j
c
j
+
c
j
a
j
)
)
.
\Biggl( \sum_{j=1}^{n} m_j(a_j+b_j+c_j) \Biggr)^2 > 3 \Biggl( \sum_{j=1}^{n} m_j \Biggr) \Biggl( \sum_{j=1}^{n} m_j(a_jb_j+b_jc_j+c_ja_j) \Biggr).
(
j
=
1
∑
n
m
j
(
a
j
+
b
j
+
c
j
)
)
2
>
3
(
j
=
1
∑
n
m
j
)
(
j
=
1
∑
n
m
j
(
a
j
b
j
+
b
j
c
j
+
c
j
a
j
)
)
.
inequalities
inequalities proposed