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Austrian-Polish
2001 Austrian-Polish Competition
3
Nice (triangle inequality)
Nice (triangle inequality)
Source: APMC 2001
February 5, 2006
inequalities
trigonometry
triangle inequality
inequalities proposed
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be sides of a triangle. Prove that
2
<
a
+
b
c
+
b
+
c
a
+
c
+
a
b
−
a
3
+
b
3
+
c
3
a
b
c
≤
3
2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3
2
<
c
a
+
b
+
a
b
+
c
+
b
c
+
a
−
ab
c
a
3
+
b
3
+
c
3
≤
3
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