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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2006 Vietnam National Olympiad
5
Vmo 2006 a5
Vmo 2006 a5
Source:
February 28, 2006
algebra
polynomial
algebra unsolved
Problem Statement
Find all polynomyals
P
(
x
)
P(x)
P
(
x
)
with real coefficients which satisfy the following equality for all real numbers
x
x
x
:
P
(
x
2
)
+
x
(
3
P
(
x
)
+
P
(
ā
x
)
)
=
(
P
(
x
)
)
2
+
2
x
2
.
P(x^2)+x(3P(x)+P(-x))=(P(x))^2+2x^2 .
P
(
x
2
)
+
x
(
3
P
(
x
)
+
P
(
ā
x
))
=
(
P
(
x
)
)
2
+
2
x
2
.
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