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239 Open Math Olympiad
2024 239 Open Mathematical Olympiad
5
3-variable inequality with a weird condition
3-variable inequality with a weird condition
Source: 239 MO 2024 J5
May 22, 2024
algebra
inequalities proposed
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be reals such that
a
2
(
c
2
−
2
b
−
1
)
+
b
2
(
a
2
−
2
c
−
1
)
+
c
2
(
b
2
−
2
a
−
1
)
=
0.
a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0.
a
2
(
c
2
−
2
b
−
1
)
+
b
2
(
a
2
−
2
c
−
1
)
+
c
2
(
b
2
−
2
a
−
1
)
=
0.
Show that
3
(
a
2
+
b
2
+
c
2
)
+
4
(
a
+
b
+
c
)
+
3
≥
6
a
b
c
.
3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.
3
(
a
2
+
b
2
+
c
2
)
+
4
(
a
+
b
+
c
)
+
3
≥
6
ab
c
.
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