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Balkan MO Shortlist
2013 Balkan MO Shortlist
A5
A5
Part of
2013 Balkan MO Shortlist
Problems
(1)
x^{2n}+y^{2n}+z^{2n}-xy-yz-zx divides (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}
Source: Balkan MO Shortlist 2013 A5 BMO
3/9/2020
Determine all positive integers
n
n
n
such that
f
n
(
x
,
y
,
z
)
=
x
2
n
+
y
2
n
+
z
2
n
−
x
y
−
y
z
−
z
x
f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx
f
n
(
x
,
y
,
z
)
=
x
2
n
+
y
2
n
+
z
2
n
−
x
y
−
yz
−
z
x
divides
g
n
(
x
,
y
,
z
)
=
(
x
−
y
)
5
n
+
(
y
−
z
)
5
n
+
(
z
−
x
)
5
n
g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}
g
n
(
x
,
y
,
z
)
=
(
x
−
y
)
5
n
+
(
y
−
z
)
5
n
+
(
z
−
x
)
5
n
, as polynomials in
x
,
y
,
z
x, y, z
x
,
y
,
z
with integer coefficients.
divides
polynomial
algebra
Integer Polynomial