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2016 India Regional Mathematical Olympiad
2
A bound on product of three numbers
A bound on product of three numbers
Source: RMO Mumbai 2016, P2
October 11, 2016
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
1
+
a
+
b
1
+
b
+
c
1
+
c
=
1.
\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.
1
+
a
a
+
1
+
b
b
+
1
+
c
c
=
1.
Prove that
a
b
c
≤
1
8
abc \le \frac{1}{8}
ab
c
≤
8
1
.
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