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2018 JBMO Shortlist
A1
JBMO 2018. Shortlist Algebra
JBMO 2018. Shortlist Algebra
Source:
July 11, 2019
algebra
Inequality
inequalities
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive real numbers . Prove:
x
y
4
+
z
4
+
y
z
4
+
x
4
+
z
x
4
+
y
4
≥
(
x
+
y
+
z
)
7
4
2
27
\frac{x}{\sqrt{\sqrt[4]{y}+\sqrt[4]{z}}}+\frac{y}{\sqrt{\sqrt[4]{z}+\sqrt[4]{x}}}+\frac{z}{\sqrt{\sqrt[4]{x}+\sqrt[4]{y}}}\geq \frac{\sqrt[4]{(\sqrt{x}+\sqrt{y}+\sqrt{z})^7}}{\sqrt{2\sqrt{27}}}
4
y
+
4
z
x
+
4
z
+
4
x
y
+
4
x
+
4
y
z
≥
2
27
4
(
x
+
y
+
z
)
7
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