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Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2017 India Regional Mathematical Olympiad
3
RMO 2017 P3
RMO 2017 P3
Source: RMO 2017 P3
October 8, 2017
algebra
Real Roots
Problem Statement
Let
P
(
x
)
=
x
2
+
x
2
+
b
P(x)=x^2+\dfrac x 2 +b
P
(
x
)
=
x
2
+
2
x
+
b
and
Q
(
x
)
=
x
2
+
c
x
+
d
Q(x)=x^2+cx+d
Q
(
x
)
=
x
2
+
c
x
+
d
be two polynomials with real coefficients such that
P
(
x
)
Q
(
x
)
=
Q
(
P
(
x
)
)
P(x)Q(x)=Q(P(x))
P
(
x
)
Q
(
x
)
=
Q
(
P
(
x
))
for all real
x
x
x
. Find all real roots of
P
(
Q
(
x
)
)
=
0
P(Q(x))=0
P
(
Q
(
x
))
=
0
.
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