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0171 inequalities 1st edition Round 7 p1
0171 inequalities 1st edition Round 7 p1
Source:
May 9, 2021
inequalities
1st edition
Problem Statement
Prove that for every positive numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
the following inequality holds:
4
x
2
+
4
x
(
y
+
z
)
+
(
y
−
z
)
2
<
4
y
2
+
4
y
(
z
+
x
)
+
(
z
−
x
)
2
+
4
z
2
+
4
z
(
x
+
y
)
+
(
x
−
y
)
2
.
\sqrt{4x^2 + 4x(y + z) + (y - z)^2} <\sqrt{4y^2 + 4y(z + x) + (z - x)^2}+\sqrt{4z^2 + 4z(x + y) + (x - y)^2}.
4
x
2
+
4
x
(
y
+
z
)
+
(
y
−
z
)
2
<
4
y
2
+
4
y
(
z
+
x
)
+
(
z
−
x
)
2
+
4
z
2
+
4
z
(
x
+
y
)
+
(
x
−
y
)
2
.
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