MathDB
Problems
Contests
International Contests
Caucasus Mathematical Olympiad
2018 Caucasus Mathematical Olympiad
1
Easy algebra
Easy algebra
Source: III Caucasus Mathematical Olympiad
March 17, 2018
algebra
Problem Statement
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers, not all of them are equal. Prove that
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
if and only if
a
2
+
a
b
+
b
2
=
b
2
+
b
c
+
c
2
=
c
2
+
c
a
+
a
2
a^2+ab+b^2=b^2+bc+c^2=c^2+ca+a^2
a
2
+
ab
+
b
2
=
b
2
+
b
c
+
c
2
=
c
2
+
c
a
+
a
2
.
Back to Problems
View on AoPS