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Regional Mathematical Olympiad
2019 India Regional Mathematical Olympiad
3
Cyclic Non-homogeneous in 3 variables
Cyclic Non-homogeneous in 3 variables
Source: RMO 2019 P3
October 20, 2019
Inequality
RMO
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove that
a
a
2
+
b
3
+
c
3
+
b
b
2
+
a
3
+
c
3
+
c
c
2
+
a
3
+
b
3
≤
1
5
a
b
c
\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}
a
2
+
b
3
+
c
3
a
+
b
2
+
a
3
+
c
3
b
+
c
2
+
a
3
+
b
3
c
≤
5
ab
c
1
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