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IMC
2013 IMC
4
IMC 2013/1/4
IMC 2013/1/4
Source: Imc 2013 Problem 4
August 8, 2013
inequalities
function
IMC
college contests
Problem Statement
Let
n
⩾
3
\displaystyle{n \geqslant 3}
n
⩾
3
and let
x
1
,
x
2
,
.
.
.
,
x
n
\displaystyle{{x_1},{x_2},...,{x_n}}
x
1
,
x
2
,
...
,
x
n
be nonnegative real numbers. Define
A
=
∑
i
=
1
n
x
i
,
B
=
∑
i
=
1
n
x
i
2
,
C
=
∑
i
=
1
n
x
i
3
\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }
A
=
i
=
1
∑
n
x
i
,
B
=
i
=
1
∑
n
x
i
2
,
C
=
i
=
1
∑
n
x
i
3
. Prove that:
(
n
+
1
)
A
2
B
+
(
n
−
2
)
B
2
⩾
A
4
+
(
2
n
−
2
)
A
C
.
\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.
(
n
+
1
)
A
2
B
+
(
n
−
2
)
B
2
⩾
A
4
+
(
2
n
−
2
)
A
C
.
Proposed by Géza Kós, Eötvös University, Budapest.
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