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2020 Turkey Team Selection Test
8
Minimum Value
Minimum Value
Source: Turkey TST 2020 Day 3 P8
March 10, 2020
inequalities
algebra
Turkey
High school olympiad
Problem Statement
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers such that
0
<
x
,
y
,
z
<
1
0<x,y,z<1
0
<
x
,
y
,
z
<
1
. Find the minimum value of:
x
y
z
(
x
+
y
+
z
)
+
(
x
y
+
y
z
+
z
x
)
(
1
−
x
y
z
)
x
y
z
1
−
x
y
z
\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}
x
yz
1
−
x
yz
x
yz
(
x
+
y
+
z
)
+
(
x
y
+
yz
+
z
x
)
(
1
−
x
yz
)
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