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National and Regional Contests
Korea Contests
Korea National Olympiad
2008 Korean National Olympiad
7
2008 KMO P7
2008 KMO P7
Source:
August 9, 2015
functional equation
function
algebra
Problem Statement
Prove that the only function
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
satisfying the following is
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
. (i)
∀
x
≠
0
\forall x \not= 0
∀
x
=
0
,
f
(
x
)
=
x
2
f
(
1
x
)
f(x) = x^2f(\frac{1}{x})
f
(
x
)
=
x
2
f
(
x
1
)
. (ii)
∀
x
,
y
\forall x, y
∀
x
,
y
,
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f(x+y) = f(x)+f(y)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
. (iii)
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
.
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