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National and Regional Contests
Russia Contests
239 Open Math Olympiad
2019 239 Open Mathematical Olympiad
7
Inequality with 4n numbers
Inequality with 4n numbers
Source: 239 2019 S7
July 31, 2020
inequalities
algebra
Problem Statement
Given positive numbers
a
1
,
…
,
a
n
a_1, \ldots , a_n
a
1
,
…
,
a
n
,
b
1
,
…
,
b
n
b_1, \ldots , b_n
b
1
,
…
,
b
n
,
c
1
,
…
,
c
n
c_1, \ldots , c_n
c
1
,
…
,
c
n
. Let
m
k
m_k
m
k
be the maximum of the products
a
i
b
j
c
l
a_ib_jc_l
a
i
b
j
c
l
over the sets
(
i
,
j
,
l
)
(i, j, l)
(
i
,
j
,
l
)
for which
m
a
x
(
i
,
j
,
l
)
=
k
max(i, j, l) = k
ma
x
(
i
,
j
,
l
)
=
k
. Prove that
(
a
1
+
…
+
a
n
)
(
b
1
+
…
+
b
n
)
(
c
1
+
…
+
c
n
)
≤
n
2
(
m
1
+
…
+
m
n
)
.
(a_1 + \ldots + a_n) (b_1 +\ldots + b_n) (c_1 +\ldots + c_n) \leq n^2 (m_1 + \ldots + m_n).
(
a
1
+
…
+
a
n
)
(
b
1
+
…
+
b
n
)
(
c
1
+
…
+
c
n
)
≤
n
2
(
m
1
+
…
+
m
n
)
.
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