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China National Olympiad
1991 China National Olympiad
2
China Mathematical Olympiad 1991 problem2
China Mathematical Olympiad 1991 problem2
Source: China Mathematical Olympiad 1991 problem2
October 8, 2013
function
algebra unsolved
algebra
Problem Statement
Given
I
=
[
0
,
1
]
I=[0,1]
I
=
[
0
,
1
]
and
G
=
{
(
x
,
y
)
∣
x
,
y
∈
I
}
G=\{(x,y)|x,y \in I\}
G
=
{(
x
,
y
)
∣
x
,
y
∈
I
}
, find all functions
f
:
G
→
I
f:G\rightarrow I
f
:
G
→
I
, such that
∀
x
,
y
,
z
∈
I
\forall x,y,z \in I
∀
x
,
y
,
z
∈
I
we have: i.
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
)
)
f(f(x,y),z)=f(x,f(y,z))
f
(
f
(
x
,
y
)
,
z
)
=
f
(
x
,
f
(
y
,
z
))
; ii.
f
(
x
,
1
)
=
x
,
f
(
1
,
y
)
=
y
f(x,1)=x, f(1,y)=y
f
(
x
,
1
)
=
x
,
f
(
1
,
y
)
=
y
; iii.
f
(
z
x
,
z
y
)
=
z
k
f
(
x
,
y
)
f(zx,zy)=z^kf(x,y)
f
(
z
x
,
zy
)
=
z
k
f
(
x
,
y
)
. (
k
k
k
is a positive real number irrelevant to
x
,
y
,
z
x,y,z
x
,
y
,
z
.)
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