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Austrian-Polish
2000 Austrian-Polish Competition
1
1
Part of
2000 Austrian-Polish Competition
Problems
(1)
\sum_{k=1}^{2n+1}(-1)^k [ k/2 ] P(x + k)=0 for infinitely many real x
Source: Austrian - Polish 2000 APMC
5/4/2020
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients having the following property: There exists a positive integer n such that the equality
∑
k
=
1
2
n
+
1
(
−
1
)
k
[
k
2
]
P
(
x
+
k
)
=
0
\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0
k
=
1
∑
2
n
+
1
(
−
1
)
k
[
2
k
]
P
(
x
+
k
)
=
0
holds for infinitely many real numbers
x
x
x
.
floor function
Sum
polynomial
algebra