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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2019 Korea National Olympiad
1
1
Part of
2019 Korea National Olympiad
Problems
(1)
Inequality in sequence
Source: Korea National Olympiad 2019 P1
11/16/2019
The sequence
a
1
,
a
2
,
.
.
.
,
a
2019
{a_1, a_2, ..., a_{2019}}
a
1
,
a
2
,
...
,
a
2019
satisfies the following condition.
a
1
=
1
,
a
n
+
1
=
2019
a
n
+
1
a_1=1, a_{n+1}=2019a_{n}+1
a
1
=
1
,
a
n
+
1
=
2019
a
n
+
1
Now let
x
1
,
x
2
,
.
.
.
,
x
2019
x_1, x_2, ..., x_{2019}
x
1
,
x
2
,
...
,
x
2019
real numbers such that
x
1
=
a
2019
,
x
2019
=
a
1
x_1=a_{2019}, x_{2019}=a_1
x
1
=
a
2019
,
x
2019
=
a
1
(The others are arbitary.) Prove that
∑
k
=
1
2018
(
x
k
+
1
−
2019
x
k
−
1
)
2
≥
∑
k
=
1
2018
(
a
2019
−
k
−
2019
a
2020
−
k
−
1
)
2
\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2
∑
k
=
1
2018
(
x
k
+
1
−
2019
x
k
−
1
)
2
≥
∑
k
=
1
2018
(
a
2019
−
k
−
2019
a
2020
−
k
−
1
)
2
inequalities