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Turkey Team Selection Test
2019 Turkey Team SeIection Test
9
Non-symmetric Inequality
Non-symmetric Inequality
Source: Turkey Team Selection Test 2019 Day 3 Problem 9
March 25, 2019
inequalities
Problem Statement
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers such that
y
≥
2
z
≥
4
x
y\geq 2z \geq 4x
y
≥
2
z
≥
4
x
and
2
(
x
3
+
y
3
+
z
3
)
+
15
(
x
y
2
+
y
z
2
+
z
x
2
)
≥
16
(
x
2
y
+
y
2
z
+
z
2
x
)
+
2
x
y
z
.
2(x^3+y^3+z^3)+15(xy^2+yz^2+zx^2)\geq 16(x^2y+y^2z+z^2x)+2xyz.
2
(
x
3
+
y
3
+
z
3
)
+
15
(
x
y
2
+
y
z
2
+
z
x
2
)
≥
16
(
x
2
y
+
y
2
z
+
z
2
x
)
+
2
x
yz
.
Prove that:
4
x
+
y
≥
4
z
4x+y\geq 4z
4
x
+
y
≥
4
z
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