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2007 JBMO Shortlist
5
ineqyality with Σ x\sqrt{yz(x + my)(x + nz)}
ineqyality with Σ x\sqrt{yz(x + my)(x + nz)}
Source: JBMO Shortlist 2007 A5
October 14, 2017
inequalities
JBMO
algebra
Problem Statement
The real numbers
x
,
y
,
z
,
m
,
n
x,y,z, m, n
x
,
y
,
z
,
m
,
n
are positive, such that
m
+
n
≥
2
m + n \ge 2
m
+
n
≥
2
. Prove that
x
y
z
(
x
+
m
y
)
(
x
+
n
z
)
+
y
x
z
(
y
+
m
x
)
(
y
+
n
z
)
+
z
x
y
(
z
+
m
x
)
(
x
+
n
y
)
≤
3
(
m
+
n
)
8
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + z\sqrt{xy(z + mx)(x + ny) }\le \frac{3(m + n)}{8} (x + y)(y + z)(z + x)
x
yz
(
x
+
m
y
)
(
x
+
n
z
)
+
y
x
z
(
y
+
m
x
)
(
y
+
n
z
)
+
z
x
y
(
z
+
m
x
)
(
x
+
n
y
)
≤
8
3
(
m
+
n
)
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
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