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International Contests
Balkan MO Shortlist
2018 Balkan MO Shortlist
A6
A inequality
A inequality
Source: Shortlist BMO 2018, A6
May 3, 2019
inequalities
Problem Statement
Let
x
1
,
x
2
,
⋯
,
x
n
x_1, x_2, \cdots, x_n
x
1
,
x
2
,
⋯
,
x
n
be positive real numbers . Prove that:
∑
i
=
1
n
x
i
2
≥
1
n
+
1
(
∑
i
=
1
n
x
i
)
2
+
12
(
∑
i
=
1
n
i
x
i
)
2
n
(
n
+
1
)
(
n
+
2
)
(
3
n
+
1
)
.
\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}.
i
=
1
∑
n
x
i
2
≥
n
+
1
1
(
i
=
1
∑
n
x
i
)
2
+
n
(
n
+
1
)
(
n
+
2
)
(
3
n
+
1
)
12
(
∑
i
=
1
n
i
x
i
)
2
.
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