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Balkan MO Shortlist
2013 Balkan MO Shortlist
N8
N8
Part of
2013 Balkan MO Shortlist
Problems
(1)
a+b+c+d=0 and ac+bd=0, iff a-b divides 2ab
Source: Balkan MO Shortlist 2013 N8 BMO
3/10/2020
Suppose that
a
a
a
and
b
b
b
are integers. Prove that there are integers
c
c
c
and
d
d
d
such that
a
+
b
+
c
+
d
=
0
a+b+c+d=0
a
+
b
+
c
+
d
=
0
and
a
c
+
b
d
=
0
ac+bd=0
a
c
+
b
d
=
0
, if and only if
a
ā
b
a-b
a
ā
b
divides
2
a
b
2ab
2
ab
.
number theory
divides
Integers