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Balkan MO Shortlist
2013 Balkan MO Shortlist
A4
A4
Part of
2013 Balkan MO Shortlist
Problems
(1)
1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x), integer pol.
Source: Balkan MO Shortlist 2013 A4 BMO
3/9/2020
Find all positive integers
n
n
n
such that there exist non-constant polynomials with integer coefficients
f
1
(
x
)
,
.
.
.
,
f
n
(
x
)
f_1(x),...,f_n(x)
f
1
(
x
)
,
...
,
f
n
(
x
)
(not necessarily distinct) and
g
(
x
)
g(x)
g
(
x
)
such that
1
+
∏
k
=
1
n
(
f
k
2
(
x
)
−
1
)
=
(
x
2
+
2013
)
2
g
2
(
x
)
1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)
1
+
k
=
1
∏
n
(
f
k
2
(
x
)
−
1
)
=
(
x
2
+
2013
)
2
g
2
(
x
)
Integer Polynomial
polynomial
algebra