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National and Regional Contests
Nepal Contests
Nepal National Olympiad
2024 Nepal Mathematics Olympiad (Pre-TST)
Problem 2
Problem 2
Part of
2024 Nepal Mathematics Olympiad (Pre-TST)
Problems
(1)
NMO(Nepal) Problem 2
Source:
3/17/2024
Let,
S
=
∑
i
=
1
k
n
i
2
\displaystyle{S =\sum_{i=1}^{k} {n_i}^2}
S
=
i
=
1
∑
k
n
i
2
. Prove that for
n
i
∈
R
+
n_i \in \mathbb{R}^+
n
i
∈
R
+
∑
i
=
1
k
n
i
S
−
n
i
2
≥
4
n
1
+
n
2
+
⋯
+
n
k
\sum_{i=1}^{k} \frac{n_i}{S-n_i^2} \geq \frac{4}{n_1+n_2+ \cdots+ n_k}
i
=
1
∑
k
S
−
n
i
2
n
i
≥
n
1
+
n
2
+
⋯
+
n
k
4
Proposed by Kang Taeyoung, South Korea
algebra
Inequality