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Prove that xyz+(1-x)(1-y)(1-z) ≤ 1 [Slovenia 2010]
Prove that xyz+(1-x)(1-y)(1-z) ≤ 1 [Slovenia 2010]
Source:
November 14, 2010
probability
function
inequalities
inequalities proposed
Problem Statement
Let
x
,
y
x,y
x
,
y
and
z
z
z
be real numbers such that
0
≤
x
,
y
,
z
≤
1.
0 \leq x,y,z \leq 1.
0
≤
x
,
y
,
z
≤
1.
Prove that
x
y
z
+
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
≤
1.
xyz+(1-x)(1-y)(1-z) \leq 1.
x
yz
+
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
≤
1.
When does equality hold?
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