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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2009 Mexico National Olympiad
3
Cubes and 2s
Cubes and 2s
Source: OMM 2009 3
July 15, 2014
inequalities
inequalities unsolved
Problem Statement
Let
a
a
a
,
b
b
b
, and
c
c
c
be positive numbers satisfying
a
b
c
=
1
abc=1
ab
c
=
1
. Show that
a
3
a
3
+
2
+
b
3
b
3
+
2
+
c
3
c
3
+
2
≥
1
and
1
a
3
+
2
+
1
b
3
+
2
+
1
c
3
+
2
≤
1
\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1
a
3
+
2
a
3
+
b
3
+
2
b
3
+
c
3
+
2
c
3
≥
1
and
a
3
+
2
1
+
b
3
+
2
1
+
c
3
+
2
1
≤
1
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