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Vietnam National Olympiad
2024 Vietnam National Olympiad
2
2
Part of
2024 Vietnam National Olympiad
Problems
(1)
Polynomial is back at VMO!
Source: 2024 Vietnam National Olympiad - Problem 2
1/5/2024
Find all polynomials
P
(
x
)
,
Q
(
x
)
P(x), Q(x)
P
(
x
)
,
Q
(
x
)
with real coefficients such that for all real numbers
a
a
a
,
P
(
a
)
P(a)
P
(
a
)
is a root of the equation
x
2023
+
Q
(
a
)
x
2
+
(
a
2024
+
a
)
x
+
a
3
+
2025
a
=
0.
x^{2023}+Q(a)x^2+(a^{2024}+a)x+a^3+2025a=0.
x
2023
+
Q
(
a
)
x
2
+
(
a
2024
+
a
)
x
+
a
3
+
2025
a
=
0.
algebra
polynomial