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Mongolian Mathematical Olympiad
2023 Mongolian Mathematical Olympiad
1
FE with two functions
FE with two functions
Source: Mongolian Mathematical Olympiad P4
January 30, 2024
algebra
functional equation
function
Problem Statement
Find all functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
and
h
:
R
2
→
R
h : \mathbb{R}^2 \to \mathbb{R}
h
:
R
2
→
R
such that
f
(
x
+
y
−
z
)
2
=
f
(
x
y
)
+
h
(
x
+
y
+
z
,
x
y
+
y
z
+
z
x
)
f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)
f
(
x
+
y
−
z
)
2
=
f
(
x
y
)
+
h
(
x
+
y
+
z
,
x
y
+
yz
+
z
x
)
for all real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
.
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