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2010 Chile National Olympiad
1
1
Part of
2010 Chile National Olympiad
Problems
(1)
a- b, 2a + 2b + 1, 3a + 3b + 1 perfect sqaures if 2a^2 + a = 3b^2 + b
Source: Chile Finals 2010 L2 p1
10/5/2022
The integers
a
,
b
a, b
a
,
b
satisfy the following identity
2
a
2
+
a
=
3
b
2
+
b
.
2a^2 + a = 3b^2 + b.
2
a
2
+
a
=
3
b
2
+
b
.
Prove that
a
ā
b
a- b
a
ā
b
,
2
a
+
2
b
+
1
2a + 2b + 1
2
a
+
2
b
+
1
, and
3
a
+
3
b
+
1
3a + 3b + 1
3
a
+
3
b
+
1
are perfect squares.
number theory
Perfect Square