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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2013 Vietnam National Olympiad
1
function equation
function equation
Source: Problem 5 VMO 2013 day 2
January 12, 2013
function
algebra proposed
algebra
Problem Statement
Find all
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
that satisfies
f
(
0
)
=
0
,
f
(
1
)
=
2013
f(0)=0,f(1)=2013
f
(
0
)
=
0
,
f
(
1
)
=
2013
and
(
x
−
y
)
(
f
(
f
2
(
x
)
)
−
f
(
f
2
(
y
)
)
)
=
(
f
(
x
)
−
f
(
y
)
)
(
f
2
(
x
)
−
f
2
(
y
)
)
(x-y)(f(f^2(x))-f(f^2(y)))=(f(x)-f(y))(f^2(x)-f^2(y))
(
x
−
y
)
(
f
(
f
2
(
x
))
−
f
(
f
2
(
y
)))
=
(
f
(
x
)
−
f
(
y
))
(
f
2
(
x
)
−
f
2
(
y
))
Note:
f
2
(
x
)
=
(
f
(
x
)
)
2
f^2(x)=(f(x))^2
f
2
(
x
)
=
(
f
(
x
)
)
2
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