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2020 IMO Shortlist
A7
n variable inequalities where all variables are bounded
n variable inequalities where all variables are bounded
Source: ISL 2020 A7
July 20, 2021
inequalities
n-variable inequality
IMO Shortlist
IMO Shortlist 2020
Problem Statement
Let
n
n
n
and
k
k
k
be positive integers. Prove that for
a
1
,
…
,
a
n
∈
[
1
,
2
k
]
a_1, \dots, a_n \in [1,2^k]
a
1
,
…
,
a
n
∈
[
1
,
2
k
]
one has
∑
i
=
1
n
a
i
a
1
2
+
⋯
+
a
i
2
≤
4
k
n
.
\sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}.
i
=
1
∑
n
a
1
2
+
⋯
+
a
i
2
a
i
≤
4
kn
.
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