MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
CMIMC Problems
2021 CMIMC
6
2021 Team P6
2021 Team P6
Source:
March 2, 2021
algebra
Problem Statement
Let
P
(
x
)
,
Q
(
x
)
,
P(x), Q(x),
P
(
x
)
,
Q
(
x
)
,
and
R
(
x
)
R(x)
R
(
x
)
be three monic quadratic polynomials with only real roots, satisfying
P
(
Q
(
x
)
)
=
(
x
−
1
)
(
x
−
3
)
(
x
−
5
)
(
x
−
7
)
P(Q(x))=(x-1)(x-3)(x-5)(x-7)
P
(
Q
(
x
))
=
(
x
−
1
)
(
x
−
3
)
(
x
−
5
)
(
x
−
7
)
Q
(
R
(
x
)
)
=
(
x
−
2
)
(
x
−
4
)
(
x
−
6
)
(
x
−
8
)
Q(R(x))=(x-2)(x-4)(x-6)(x-8)
Q
(
R
(
x
))
=
(
x
−
2
)
(
x
−
4
)
(
x
−
6
)
(
x
−
8
)
for all real numbers
x
.
x.
x
.
What is
P
(
0
)
+
Q
(
0
)
+
R
(
0
)
?
P(0)+Q(0)+R(0)?
P
(
0
)
+
Q
(
0
)
+
R
(
0
)?
Proposed by Kyle Lee
Back to Problems
View on AoPS