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Mediterranean Mathematics Olympiad
2016 Mediterranean Mathematics Olympiad
2
Inequality: a + b + c = 3; sums of square roots
Inequality: a + b + c = 3; sums of square roots
Source: Mediterranean Mathematics Olympiad 2016 P2
June 4, 2016
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers with
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
. Prove that
b
a
2
+
3
+
c
b
2
+
3
+
a
c
2
+
3
≤
3
2
1
a
b
c
4
\sqrt{\frac{b}{a^2+3}}+ \sqrt{\frac{c}{b^2+3}}+ \sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}
a
2
+
3
b
+
b
2
+
3
c
+
c
2
+
3
a
≤
2
3
4
ab
c
1
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