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Canada National Olympiad
2019 Canada National Olympiad
2
2
Part of
2019 Canada National Olympiad
Problems
(1)
Prove a^2+3ab+3b^2-1|a+b^3 implies c^3|a^2+3ab+3b^2-1
Source: 2019 Canadian Mathematical Olympiad Problem 2
3/28/2019
Let
a
,
b
a,b
a
,
b
be positive integers such that
a
+
b
3
a+b^3
a
+
b
3
is divisible by
a
2
+
3
a
b
+
3
b
2
−
1
a^2+3ab+3b^2-1
a
2
+
3
ab
+
3
b
2
−
1
. Prove that
a
2
+
3
a
b
+
3
b
2
−
1
a^2+3ab+3b^2-1
a
2
+
3
ab
+
3
b
2
−
1
is divisible by the cube of an integer greater than 1.
number theory
cube of a natural number.