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Putnam
1981 Putnam
A5
A5
Part of
1981 Putnam
Problems
(1)
Putnam 1981 A5
Source: Putnam 1981
3/31/2022
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with real coefficients and form the polynomial
Q
(
x
)
=
(
x
2
+
1
)
P
(
x
)
P
′
(
x
)
+
x
(
P
(
x
)
2
+
P
′
(
x
)
2
)
.
Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).
Q
(
x
)
=
(
x
2
+
1
)
P
(
x
)
P
′
(
x
)
+
x
(
P
(
x
)
2
+
P
′
(
x
)
2
)
.
Given that the equation
P
(
x
)
=
0
P(x) = 0
P
(
x
)
=
0
has
n
n
n
distinct real roots exceeding
1
1
1
, prove or disprove that the equation
Q
(
x
)
=
0
Q(x)=0
Q
(
x
)
=
0
has at least
2
n
−
1
2n - 1
2
n
−
1
distinct real roots.
Putnam
polynomial
roots