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2023 South East Mathematical Olympiad
6
China South East Mathematical Olympiad 2023 Grade10 Q6
China South East Mathematical Olympiad 2023 Grade10 Q6
Source: Wenzhou
July 31, 2023
inequalities proposed
algebra
inequalities
Problem Statement
Let
a
1
≥
a
2
≥
⋯
≥
a
n
>
0.
a_1\geq a_2\geq \cdots \geq a_n >0 .
a
1
≥
a
2
≥
⋯
≥
a
n
>
0.
Prove that
(
1
a
1
+
1
a
2
+
.
.
.
+
1
a
n
)
2
≥
∑
k
=
1
n
k
(
2
k
−
1
)
a
1
2
+
a
2
2
+
⋯
+
a
k
2
\left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}
(
a
1
1
+
a
2
1
+
...
+
a
n
1
)
2
≥
k
=
1
∑
n
a
1
2
+
a
2
2
+
⋯
+
a
k
2
k
(
2
k
−
1
)
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