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A5
JBMO Shortlist 2019 A5
JBMO Shortlist 2019 A5
Source:
September 12, 2020
algebra
inequalities
Problem Statement
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive real numbers such that
a
b
c
d
=
1
abcd = 1
ab
c
d
=
1
. Prove the inequality
1
a
3
+
b
+
c
+
d
+
1
a
+
b
3
+
c
+
d
+
1
a
+
b
+
c
3
+
d
+
1
a
+
b
+
c
+
d
3
≤
a
+
b
+
c
+
d
4
\frac{1}{a^3 + b + c + d} +\frac{1}{a + b^3 + c + d}+\frac{1}{a + b + c^3 + d} +\frac{1}{a + b + c + d^3} \leq \frac{a+b+c+d}{4}
a
3
+
b
+
c
+
d
1
+
a
+
b
3
+
c
+
d
1
+
a
+
b
+
c
3
+
d
1
+
a
+
b
+
c
+
d
3
1
≤
4
a
+
b
+
c
+
d
Proposed by Romania
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