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Junior Balkan MO
2013 Junior Balkan MO
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2013 Junior Balkan MO
Problems
(1)
JBMO 2013 Problem 1
Source: Proposed by Serbia
6/23/2013
Find all ordered pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers for which the numbers
a
3
b
−
1
a
+
1
\dfrac{a^3b-1}{a+1}
a
+
1
a
3
b
−
1
and
b
3
a
+
1
b
−
1
\dfrac{b^3a+1}{b-1}
b
−
1
b
3
a
+
1
are both positive integers.
algebra
polynomial
Vieta
modular arithmetic
number theory proposed
number theory
JBMO