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Mathematical Excellence Olympiad
2020 IMEO
Problem 3
Yet another functional equation
Yet another functional equation
Source: IMEO 2020 Problem 3
July 15, 2020
IMEO
functional equation
algebra
Problem Statement
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R^+} \to \mathbb{R^+}
f
:
R
+
→
R
+
such that for all positive real
x
,
y
x, y
x
,
y
holds
x
f
(
x
)
+
y
f
(
y
)
=
(
x
+
y
)
f
(
x
2
+
y
2
x
+
y
)
xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)
x
f
(
x
)
+
y
f
(
y
)
=
(
x
+
y
)
f
(
x
+
y
x
2
+
y
2
)
. Fedir Yudin
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