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Iran MO (3rd Round)
2017 Iran MO (3rd round)
3
Functional equation
Functional equation
Source: Iran 3rd round 2017 first Algebra exam
August 7, 2017
algebra
functional equation
Problem Statement
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R^+}\rightarrow\mathbb{R^+}
f
:
R
+
→
R
+
such that
x
+
f
(
y
)
x
f
(
y
)
=
f
(
1
y
+
f
(
1
x
)
)
\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))
x
f
(
y
)
x
+
f
(
y
)
=
f
(
y
1
+
f
(
x
1
))
for all positive real numbers
x
x
x
and
y
y
y
.
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