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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2020 Greece National Olympiad
4
4
Part of
2020 Greece National Olympiad
Problems
(1)
\frac{a+b}{a^2+k^2b^2-k^2ab} is a composite number for all a,b \in N+
Source: Greek National 2020 p4 (Archimedes)
2/23/2020
Find all values of the positive integer
k
k
k
that has the property: There are no positive integers
a
,
b
a,b
a
,
b
such that the expression
A
(
k
,
a
,
b
)
=
a
+
b
a
2
+
k
2
b
2
−
k
2
a
b
A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}
A
(
k
,
a
,
b
)
=
a
2
+
k
2
b
2
−
k
2
ab
a
+
b
is a composite positive number.
composite number
positive integers
number theory
Composite