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2018 Irish Math Olympiad
7
7
Part of
2018 Irish Math Olympiad
Problems
(1)
(a^3+b^3+c^3)<(a+b+c)(a^2+b^2+c^2)<=3(a^3+b^3+c^3) where a,b,c sidelengths
Source: Irmo 2018 p2 q7
9/16/2018
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the side lengths of a triangle. Prove that
2
(
a
3
+
b
3
+
c
3
)
<
(
a
+
b
+
c
)
(
a
2
+
b
2
+
c
2
)
≤
3
(
a
3
+
b
3
+
c
3
)
2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)
2
(
a
3
+
b
3
+
c
3
)
<
(
a
+
b
+
c
)
(
a
2
+
b
2
+
c
2
)
≤
3
(
a
3
+
b
3
+
c
3
)
geometric inequality
inequalities
algebra