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National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2007 Hanoi Open Mathematics Competitions
9
Prove that $ a_k\in[2006;2007]$
Prove that $ a_k\in[2006;2007]$
Source: HOMC
February 15, 2017
algebra
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
2007
a_1,a_2,...,a_{2007}
a
1
,
a
2
,
...
,
a
2007
be real number such that
a
1
+
a
2
+
.
.
.
+
a
2007
≥
200
7
2
a_1+a_2+...+a_{2007}\geq 2007^{2}
a
1
+
a
2
+
...
+
a
2007
≥
200
7
2
and
a
1
2
+
a
2
2
+
.
.
.
+
a
2007
2
≤
200
7
3
−
1
a_1^{2}+a_2^{2}+...+a_{2007}^{2}\leq 2007^{3}-1
a
1
2
+
a
2
2
+
...
+
a
2007
2
≤
200
7
3
−
1
. Prove that
a
k
∈
[
2006
;
2008
]
a_k\in[2006;2008]
a
k
∈
[
2006
;
2008
]
for all
k
∈
{
1
,
2
,
.
.
.
,
2007
}
k\in\left \{ 1,2,...,2007 \right \}
k
∈
{
1
,
2
,
...
,
2007
}
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