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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2006 China Second Round Olympiad
15
2006 China Second Round Olympiad Test 1 #15
2006 China Second Round Olympiad Test 1 #15
Source:
September 28, 2014
Problem Statement
Suppose
f
(
x
)
=
x
2
+
a
f(x)=x^2+a
f
(
x
)
=
x
2
+
a
. Define
f
1
(
x
)
=
f
(
x
)
f^1(x)=f(x)
f
1
(
x
)
=
f
(
x
)
,
f
n
(
x
)
=
f
(
f
n
−
1
(
x
)
)
f^n(x)=f(f^{n-1}(x))
f
n
(
x
)
=
f
(
f
n
−
1
(
x
))
,
n
=
2
,
3
,
⋯
n=2, 3, \cdots
n
=
2
,
3
,
⋯
, and let
M
=
{
a
∈
R
∣
∣
f
n
(
0
)
∣
≤
2
,
for any
n
∈
N
}
M=\{a\in\mathbb{R}| |f^n(0)|\le 2, \text{for any } n\in\mathbb{N}\}
M
=
{
a
∈
R
∣∣
f
n
(
0
)
∣
≤
2
,
for any
n
∈
N
}
. Prove that
M
=
[
−
2
,
1
4
]
M=[-2, \frac{1}{4}]
M
=
[
−
2
,
4
1
]
.
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