MathDB
Problems
Contests
International Contests
Baltic Way
1993 Baltic Way
6
Prove that f(3)=g(3)
Prove that f(3)=g(3)
Source: Baltic Way 1993
June 15, 2012
function
algebra proposed
algebra
Problem Statement
Suppose two functions
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
are defined for all
x
x
x
with
2
<
x
<
4
2<x<4
2
<
x
<
4
and satisfy:
2
<
f
(
x
)
<
4
,
2
<
g
(
x
)
<
4
,
f
(
g
(
x
)
)
=
g
(
f
(
x
)
)
=
x
,
f
(
x
)
ā
g
(
x
)
=
x
2
2<f(x)<4,2<g(x)<4,f(g(x))=g(f(x))=x,f(x)\cdot g(x)=x^2
2
<
f
(
x
)
<
4
,
2
<
g
(
x
)
<
4
,
f
(
g
(
x
))
=
g
(
f
(
x
))
=
x
,
f
(
x
)
ā
g
(
x
)
=
x
2
for all
2
<
x
<
4
2<x<4
2
<
x
<
4
. Prove that
f
(
3
)
=
g
(
3
)
f(3)=g(3)
f
(
3
)
=
g
(
3
)
.
Back to Problems
View on AoPS