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2002 Romania National Olympiad
2
The real polynomials f and g
The real polynomials f and g
Source: Romanian MO 2002
December 8, 2010
algebra
polynomial
algebra proposed
Problem Statement
Find all real polynomials
f
f
f
and
g
g
g
, such that:
(
x
2
+
x
+
1
)
⋅
f
(
x
2
−
x
+
1
)
=
(
x
2
−
x
+
1
)
⋅
g
(
x
2
+
x
+
1
)
,
(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1),
(
x
2
+
x
+
1
)
⋅
f
(
x
2
−
x
+
1
)
=
(
x
2
−
x
+
1
)
⋅
g
(
x
2
+
x
+
1
)
,
for all
x
∈
R
x\in\mathbb{R}
x
∈
R
.
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