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Japan MO Finals
2004 Japan MO Finals
4
4
Part of
2004 Japan MO Finals
Problems
(1)
Proof of inequality
Source: Japan Mathematical Olympiad Finals 2004, Problem 4
3/30/2005
For positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying
a
+
b
+
c
=
1
,
a+b+c=1,
a
+
b
+
c
=
1
,
Prove that we have
1
+
a
1
−
a
+
1
+
b
1
−
b
+
1
+
c
1
−
c
≦
2
(
b
a
+
c
b
+
a
c
)
.
\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leqq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).
1
−
a
1
+
a
+
1
−
b
1
+
b
+
1
−
c
1
+
c
≦
2
(
a
b
+
b
c
+
c
a
)
.
Note that you don't need to state for the condition for which the equality holds.
inequalities
inequalities proposed