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Poland - Second Round
2007 Poland - Second Round
3
Polish MO 2007 Second Round Day 2, problem 3
Polish MO 2007 Second Round Day 2, problem 3
Source:
February 28, 2007
Problem Statement
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
are positive real numbers satisfying the following condition:
1
a
+
1
b
+
1
c
+
1
d
=
4
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=4
a
1
+
b
1
+
c
1
+
d
1
=
4
Prove that:
a
3
+
b
3
2
3
+
b
3
+
c
3
2
3
+
c
3
+
d
3
2
3
+
d
3
+
a
3
2
3
≤
2
(
a
+
b
+
c
+
d
)
−
4
\sqrt[3]{\frac{a^{3}+b^{3}}{2}}+\sqrt[3]{\frac{b^{3}+c^{3}}{2}}+\sqrt[3]{\frac{c^{3}+d^{3}}{2}}+\sqrt[3]{\frac{d^{3}+a^{3}}{2}}\leq 2(a+b+c+d)-4
3
2
a
3
+
b
3
+
3
2
b
3
+
c
3
+
3
2
c
3
+
d
3
+
3
2
d
3
+
a
3
≤
2
(
a
+
b
+
c
+
d
)
−
4
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