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2022 District Olympiad
P1
(x^3+3x^2f(y))/(x+f(y))+(y^3+3y^2f(x))/(y+f(x))=(x+y)^3/f(x+y)
(x^3+3x^2f(y))/(x+f(y))+(y^3+3y^2f(x))/(y+f(x))=(x+y)^3/f(x+y)
Source: Romanian District Olympiad 2022 - Grade 9 - Problem 1
March 27, 2022
Arithmetic Function
functional equation
function
algebra
Problem Statement
Let
f
:
N
∗
→
N
∗
f:\mathbb{N}^*\rightarrow \mathbb{N}^*
f
:
N
∗
→
N
∗
be a function such that
x
3
+
3
x
2
f
(
y
)
x
+
f
(
y
)
+
y
3
+
3
y
2
f
(
x
)
y
+
f
(
x
)
=
(
x
+
y
)
3
f
(
x
+
y
)
,
(
∀
)
x
,
y
∈
N
∗
.
\frac{x^3+3x^2f(y)}{x+f(y)}+\frac{y^3+3y^2f(x)}{y+f(x)}=\frac{(x+y)^3}{f(x+y)},~(\forall)x,y\in\mathbb{N}^*.
x
+
f
(
y
)
x
3
+
3
x
2
f
(
y
)
+
y
+
f
(
x
)
y
3
+
3
y
2
f
(
x
)
=
f
(
x
+
y
)
(
x
+
y
)
3
,
(
∀
)
x
,
y
∈
N
∗
.
a
)
a)
a
)
Prove that
f
(
1
)
=
1.
f(1)=1.
f
(
1
)
=
1.
b
)
b)
b
)
Find function
f
.
f.
f
.
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