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2003 All-Russian Olympiad Regional Round
11.5
P(Q(x)) = Q(P(x), trinomials - All-Russian MO 2003 Regional (R4) 11.5
P(Q(x)) = Q(P(x), trinomials - All-Russian MO 2003 Regional (R4) 11.5
Source:
September 17, 2024
algebra
trinomial
Problem Statement
Square trinomials
P
(
x
)
=
x
2
+
a
x
+
b
P(x) = x^2 + ax + b
P
(
x
)
=
x
2
+
a
x
+
b
and
Q
(
x
)
=
x
2
+
c
x
+
d
Q(x) = x^2 + cx + d
Q
(
x
)
=
x
2
+
c
x
+
d
are such that the equation
P
(
Q
(
x
)
)
=
Q
(
P
(
x
)
)
P(Q(x)) = Q(P(x))
P
(
Q
(
x
))
=
Q
(
P
(
x
))
has no real roots. Prove that
b
≠
d
b \ne d
b
=
d
.
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