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Asian Pacific Mathematics Olympiad
›
2011
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Problem 5
Problem 5
Find all functions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
that are bounded above and satisfy
f
(
x
f
(
y
)
)
+
y
f
(
x
)
=
x
f
(
y
)
+
f
(
x
y
)
f(xf(y))+yf(x)=xf(y)+f(xy)
f
(
x
f
(
y
))
+
y
f
(
x
)
=
x
f
(
y
)
+
f
(
x
y
)
for all real
x
,
y
x,y
x
,
y
.
Step 1 of 7: Start at
(
1
,
1
)
(1,1)
(
1
,
1
)
Previous step
Next step
f
(
f
(
1
)
)
=
f
(
1
)
f(f(1))=f(1)
f
(
f
(
1
))
=
f
(
1
)
Detailed analysis
Putting
x
=
y
=
1
x=y=1
x
=
y
=
1
in the equation gives
f
(
f
(
1
)
)
=
f
(
1
)
f(f(1))=f(1)
f
(
f
(
1
))
=
f
(
1
)
.
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