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Austrian-Polish
2000 Austrian-Polish Competition
9
a+b+c=1
a+b+c=1
Source: 23rd Austrian-Polish MO 2000
June 27, 2005
inequalities
inequalities unsolved
algebra
Problem Statement
If three nonnegative reals
a
a
a
,
b
b
b
,
c
c
c
satisfy
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
, prove that
2
≤
(
1
−
a
2
)
2
+
(
1
−
b
2
)
2
+
(
1
−
c
2
)
2
≤
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)
2
≤
(
1
−
a
2
)
2
+
(
1
−
b
2
)
2
+
(
1
−
c
2
)
2
≤
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
.
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