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Balkan MO Shortlist
2016 Balkan MO Shortlist
A7
BMO Shortlist 2016
BMO Shortlist 2016
Source: Romania
May 3, 2019
inequalities
number theory
Problem Statement
Find all integers
n
≥
2
n\geq 2
n
≥
2
for which there exist the real numbers
a
k
,
1
≤
k
≤
n
a_k, 1\leq k \leq n
a
k
,
1
≤
k
≤
n
, which are satisfying the following conditions:
∑
k
=
1
n
a
k
=
0
,
∑
k
=
1
n
a
k
2
=
1
and
n
⋅
(
∑
k
=
1
n
a
k
3
)
=
2
(
b
n
−
1
)
,
where
b
=
max
1
≤
k
≤
n
{
a
k
}
.
\sum_{k=1}^n a_k=0, \sum_{k=1}^n a_k^2=1 \text{ and } \sqrt{n}\cdot \Bigr(\sum_{k=1}^n a_k^3\Bigr)=2(b\sqrt{n}-1), \text{ where } b=\max_{1\leq k\leq n} \{a_k\}.
k
=
1
∑
n
a
k
=
0
,
k
=
1
∑
n
a
k
2
=
1
and
n
⋅
(
k
=
1
∑
n
a
k
3
)
=
2
(
b
n
−
1
)
,
where
b
=
1
≤
k
≤
n
max
{
a
k
}
.
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