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International Contests
Middle European Mathematical Olympiad
2018 Middle European Mathematical Olympiad
1
Simple inequality
Simple inequality
Source: MEMO 2018 T1
September 2, 2018
inequalities
am-gm inequality
easy problem
Problem Statement
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive real numbers satisfying
a
b
c
=
1.
abc=1.
ab
c
=
1.
Prove that
a
2
−
b
2
a
+
b
c
+
b
2
−
c
2
b
+
c
a
+
c
2
−
a
2
c
+
a
b
≤
a
+
b
+
c
−
3.
\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.
a
+
b
c
a
2
−
b
2
+
b
+
c
a
b
2
−
c
2
+
c
+
ab
c
2
−
a
2
≤
a
+
b
+
c
−
3.
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