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Taiwan National Olympiad
1997 Taiwan National Olympiad
1
1
Part of
1997 Taiwan National Olympiad
Problems
(1)
$f$ is periodic
Source: 6-th Taiwanese Mathematical Olympiad 1997
1/18/2007
Let
a
a
a
be rational and
b
,
c
,
d
b,c,d
b
,
c
,
d
are real numbers, and let
f
:
R
→
[
−
1.1
]
f: \mathbb{R}\to [-1.1]
f
:
R
→
[
−
1.1
]
be a function satisfying
f
(
x
+
a
+
b
)
−
f
(
x
+
b
)
=
c
[
x
+
2
a
+
[
x
]
−
2
[
x
+
a
]
−
]
+
d
f(x+a+b)-f(x+b)=c[x+2a+[x]-2[x+a]-]+d
f
(
x
+
a
+
b
)
−
f
(
x
+
b
)
=
c
[
x
+
2
a
+
[
x
]
−
2
[
x
+
a
]
−
]
+
d
for all
x
x
x
. Show that
f
f
f
is periodic.
function
algebra unsolved
algebra