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1990 IMO Longlists
8
8
Part of
1990 IMO Longlists
Problems
(1)
Inequality on the side lengths of a triangle - ILL 1990 YUG3
Source:
9/19/2010
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side lengths and
P
P
P
be area of a triangle, respectively. Prove that
(
a
2
+
b
2
+
c
2
−
4
3
P
)
(
a
2
+
b
2
+
c
2
)
≥
2
(
a
2
(
b
−
c
)
2
+
b
2
(
c
−
a
)
2
+
c
2
(
a
−
b
)
2
)
.
(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).
(
a
2
+
b
2
+
c
2
−
4
3
P
)
(
a
2
+
b
2
+
c
2
)
≥
2
(
a
2
(
b
−
c
)
2
+
b
2
(
c
−
a
)
2
+
c
2
(
a
−
b
)
2
)
.
inequalities
geometry
area of a triangle
inequalities unsolved