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Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2019 Korea Junior Math Olympiad.
4
4
Part of
2019 Korea Junior Math Olympiad.
Problems
(1)
Prove that the sequence is constant
Source: KJMO 2019 p4
1/8/2021
{
a
n
}
\{a_{n}\}
{
a
n
}
is a sequence of natural numbers satisfying the following inequality for all natural number
n
n
n
:
(
a
1
+
⋯
+
a
n
)
(
1
a
1
+
⋯
+
1
a
n
)
≤
n
2
+
2019
(a_{1}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\cdots+\frac{1}{a_{n}}\right)\le{n^{2}}+2019
(
a
1
+
⋯
+
a
n
)
(
a
1
1
+
⋯
+
a
n
1
)
≤
n
2
+
2019
Prove that
{
a
n
}
\{a_{n}\}
{
a
n
}
is constant.
Sequence
inequalities
KJMO
constant