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Middle European Mathematical Olympiad
2018 Middle European Mathematical Olympiad
8
8
Part of
2018 Middle European Mathematical Olympiad
Problems
(1)
Silesian integers
Source: MEMO 2018 T8
9/2/2018
An integer
n
n
n
is called silesian if there exist positive integers
a
,
b
a,b
a
,
b
and
c
c
c
such that
n
=
a
2
+
b
2
+
c
2
a
b
+
b
c
+
c
a
.
n=\frac{a^2+b^2+c^2}{ab+bc+ca}.
n
=
ab
+
b
c
+
c
a
a
2
+
b
2
+
c
2
.
(
a
)
(a)
(
a
)
prove that there are infinitely many silesian integers.
(
b
)
(b)
(
b
)
prove that not every positive integer is silesian.
number theory