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National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2000 Iran MO (3rd Round)
2
A,b,c
A,b,c
Source: 17-th Iranian Mathematical Olympiad 1999/2000
December 14, 2005
vector
inequalities proposed
inequalities
Problem Statement
Suppose that
a
,
b
,
c
a, b, c
a
,
b
,
c
are real numbers such that for all positive numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
we have
(
1
n
∑
i
=
1
n
x
i
)
a
(
1
n
∑
i
=
1
n
x
i
2
)
b
(
1
n
∑
i
=
1
n
x
i
3
)
c
≥
1
(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1
(
n
1
∑
i
=
1
n
x
i
)
a
(
n
1
∑
i
=
1
n
x
i
2
)
b
(
n
1
∑
i
=
1
n
x
i
3
)
c
≥
1
Prove that vector
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is a nonnegative linear combination of vectors
(
−
2
,
1
,
0
)
(-2,1,0)
(
−
2
,
1
,
0
)
and
(
−
1
,
2
,
−
1
)
(-1,2,-1)
(
−
1
,
2
,
−
1
)
.
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