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2014 JBMO Shortlist
8
8
Part of
2014 JBMO Shortlist
Problems
(1)
1/x(ay+b)+1/y(az+b)+1/z(ax+b)>=3 if xyz=1, real and positive, under conditions
Source: JBMO Shortlist 2014 A8
4/24/2019
Let
x
,
y
,
z
\displaystyle {x, y, z}
x
,
y
,
z
be positive real numbers such that
x
y
z
=
1
\displaystyle {xyz = 1}
x
yz
=
1
. Prove the inequality:
1
x
(
a
y
+
b
)
+
1
y
(
a
z
+
b
)
+
1
z
(
a
x
+
b
)
≥
3
\displaystyle{\dfrac{1}{x\left(ay+b\right)}+\dfrac{1}{y\left(az+b\right)}+\dfrac{1}{z\left(ax+b\right)}\geq 3}
x
(
a
y
+
b
)
1
+
y
(
a
z
+
b
)
1
+
z
(
a
x
+
b
)
1
≥
3
if: (A)
a
=
0
,
b
=
1
\displaystyle {a = 0, b = 1}
a
=
0
,
b
=
1
(B)
a
=
1
,
b
=
0
\displaystyle {a = 1, b = 0}
a
=
1
,
b
=
0
(C)
a
+
b
=
1
,
a
,
b
>
0
\displaystyle {a + b = 1, \; a, b> 0}
a
+
b
=
1
,
a
,
b
>
0
When the equality holds?
algebra
inequalities
three variable inequality