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National and Regional Contests
Brazil Contests
Brazil National Olympiad
2004 Brazil National Olympiad
6
6
Part of
2004 Brazil National Olympiad
Problems
(1)
Periodic points
Source: Brazilian Math Olympiad, 2004
10/19/2004
Let
a
a
a
and
b
b
b
be real numbers. Define
f
a
,
b
:
R
2
→
R
2
f_{a,b}\colon R^2\to R^2
f
a
,
b
:
R
2
→
R
2
by
f
a
,
b
(
x
;
y
)
=
(
a
−
b
y
−
x
2
;
x
)
f_{a,b}(x;y)=(a-by-x^2;x)
f
a
,
b
(
x
;
y
)
=
(
a
−
b
y
−
x
2
;
x
)
. If
P
=
(
x
;
y
)
∈
R
2
P=(x;y)\in R^2
P
=
(
x
;
y
)
∈
R
2
, define
f
a
,
b
0
(
P
)
=
P
f^0_{a,b}(P) = P
f
a
,
b
0
(
P
)
=
P
and
f
a
,
b
k
+
1
(
P
)
=
f
a
,
b
(
f
a
,
b
k
(
P
)
)
f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))
f
a
,
b
k
+
1
(
P
)
=
f
a
,
b
(
f
a
,
b
k
(
P
))
for all nonnegative integers
k
k
k
. The set
p
e
r
(
a
;
b
)
per(a;b)
p
er
(
a
;
b
)
of the periodic points of
f
a
,
b
f_{a,b}
f
a
,
b
is the set of points
P
∈
R
2
P\in R^2
P
∈
R
2
such that
f
a
,
b
n
(
P
)
=
P
f_{a,b}^n(P) = P
f
a
,
b
n
(
P
)
=
P
for some positive integer
n
n
n
. Fix
b
b
b
. Prove that the set
A
b
=
{
a
∈
R
∣
p
e
r
(
a
;
b
)
≠
∅
}
A_b=\{a\in R \mid per(a;b)\neq \emptyset\}
A
b
=
{
a
∈
R
∣
p
er
(
a
;
b
)
=
∅
}
admits a minimum. Find this minimum.
algebra unsolved
algebra