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China Northern MO
2020 China Northern MO
P1
P1
Part of
2020 China Northern MO
Problems
(1)
Unbounded sum of sequence
Source: 2020 China North Mathematical Olympiad Advanced Level P1
8/4/2020
The function
f
(
x
)
=
x
2
+
sin
x
f(x)=x^2+ \sin x
f
(
x
)
=
x
2
+
sin
x
and the sequence of positive numbers
{
a
n
}
\{ a_n \}
{
a
n
}
satisfy
a
1
=
1
a_1=1
a
1
=
1
,
f
(
a
n
)
=
a
n
−
1
f(a_n)=a_{n-1}
f
(
a
n
)
=
a
n
−
1
, where
n
≥
2
n \geq 2
n
≥
2
. Prove that there exists a positive integer
n
n
n
such that
a
1
+
a
2
+
⋯
+
a
n
>
2020
a_1+a_2+ \dots + a_n > 2020
a
1
+
a
2
+
⋯
+
a
n
>
2020
.
Sequences
unbounded
recursion
algebra