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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2016 Iran MO (2nd Round)
1
Iran inequality
Iran inequality
Source: Iran second round 2016, day 1, problem 1
April 28, 2016
inequalities
algebra
Problem Statement
If
0
<
a
≤
b
≤
c
0<a\leq b\leq c
0
<
a
≤
b
≤
c
prove that
(
c
−
a
)
2
6
c
≤
a
+
b
+
c
3
−
3
1
a
+
1
b
+
1
c
\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}
6
c
(
c
−
a
)
2
≤
3
a
+
b
+
c
−
a
1
+
b
1
+
c
1
3
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