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Undergraduate contests
IMC
2017 IMC
7
7
Part of
2017 IMC
Problems
(1)
IMC 2017 Problem 7
Source:
8/3/2017
Let
p
(
x
)
p(x)
p
(
x
)
be a nonconstant polynomial with real coefficients. For every positive integer~
n
n
n
, let
q
n
(
x
)
=
(
x
+
1
)
n
p
(
x
)
+
x
n
p
(
x
+
1
)
.
q_n(x) = (x+1)^np(x)+x^n p(x+1) .
q
n
(
x
)
=
(
x
+
1
)
n
p
(
x
)
+
x
n
p
(
x
+
1
)
.
Prove that there are only finitely many numbers
n
n
n
such that all roots of
q
n
(
x
)
q_n(x)
q
n
(
x
)
are real.
college contests
IMC
imc 2017