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Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2011 Bosnia Herzegovina Team Selection Test
2
a+b+c=1, trivial
a+b+c=1, trivial
Source: Bosnia and Herzegovina 2011
May 16, 2011
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive reals such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that the inequality
a
1
+
b
−
c
3
+
b
1
+
c
−
a
3
+
c
1
+
a
−
b
3
≤
1
a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1
a
3
1
+
b
−
c
+
b
3
1
+
c
−
a
+
c
3
1
+
a
−
b
≤
1
holds.
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