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Balkan MO Shortlist
2023 Balkan MO Shortlist
N1
N1
Part of
2023 Balkan MO Shortlist
Problems
(1)
A number that is sum of two squares
Source: BMO SL 2023 N1
5/3/2024
For positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
(not necessarily distinct), suppose that
a
+
b
c
,
b
+
a
c
,
c
+
a
b
a+bc, b+ac, c+ab
a
+
b
c
,
b
+
a
c
,
c
+
ab
are all perfect squares. Show that
a
2
(
b
+
c
)
+
b
2
(
a
+
c
)
+
c
2
(
a
+
b
)
+
2
a
b
c
a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc
a
2
(
b
+
c
)
+
b
2
(
a
+
c
)
+
c
2
(
a
+
b
)
+
2
ab
c
can be written as sum of two squares.
number theory
BMO Shortlist