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National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2015 JBMO TST - Turkey
4
Turkey 2015 JBMO TST P4 (Inequality)
Turkey 2015 JBMO TST P4 (Inequality)
Source:
March 31, 2016
Inequality
Problem Statement
Prove that
1
a
+
1
b
+
1
c
≥
a
b
+
b
c
+
c
a
+
2
(
a
+
b
+
c
)
\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)
a
1
+
b
1
+
c
1
≥
b
a
+
c
b
+
a
c
+
2
(
a
+
b
+
c
)
for the all
a
,
b
,
c
a,b,c
a
,
b
,
c
positive real numbers satisfying
a
2
+
b
2
+
c
2
+
2
a
b
c
≤
1
a^2+b^2+c^2+2abc \le 1
a
2
+
b
2
+
c
2
+
2
ab
c
≤
1
.
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