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International Contests
Tournament Of Towns
1997 Tournament Of Towns
(545) 6
(545) 6
Part of
1997 Tournament Of Towns
Problems
(1)
TOT 545 1997 Spring S A6 F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}
Source:
9/11/2024
Prove that if
F
(
x
)
F(x)
F
(
x
)
and
G
(
x
)
G(x)
G
(
x
)
are polynomials with coefficients
0
0
0
and
1
1
1
such that
F
(
x
)
G
(
x
)
=
1
+
x
+
x
2
+
.
.
.
+
x
n
−
1
F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}
F
(
x
)
G
(
x
)
=
1
+
x
+
x
2
+
...
+
x
n
−
1
holds for some
n
>
1
n > 1
n
>
1
, then one of them can be represented in the form
(
1
+
x
+
x
2
+
.
.
.
+
x
k
−
1
)
T
(
x
)
(1 +x + x^2 +...+ x^{k-1}) T(x)
(
1
+
x
+
x
2
+
...
+
x
k
−
1
)
T
(
x
)
for some
k
>
1
k > 1
k
>
1
where
T
(
x
)
T(x)
T
(
x
)
is a polynomial with coefficients
0
0
0
and
1
1
1
.(V Senderov, M Vialiy)
algebra
polynomial