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National and Regional Contests
Austria Contests
Austrian MO National Competition
2005 Federal Competition For Advanced Students, Part 2
2
Austrian MO 2005 inequality
Austrian MO 2005 inequality
Source: Austrian MO 2005 round 2
June 27, 2005
inequalities
inequalities unsolved
Problem Statement
Prove that for all positive reals
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
, we have
a
+
b
+
c
+
d
a
b
c
d
≤
1
a
3
+
1
b
3
+
1
c
3
+
1
d
3
\frac{a+b+c+d}{abcd}\leq \frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}+\frac{1}{d^{3}}
ab
c
d
a
+
b
+
c
+
d
≤
a
3
1
+
b
3
1
+
c
3
1
+
d
3
1
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