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Turkey Contests
JBMO TST - Turkey
2018 JBMO TST-Turkey
8
8
Part of
2018 JBMO TST-Turkey
Problems
(1)
A hard problem with triangle inequality
Source: 2018 JBMO TST - Turkey, P8
3/27/2020
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive real numbers such that
x
,
y
,
z
\sqrt {x}, \sqrt {y}, \sqrt {z}
x
,
y
,
z
are sides of a triangle and
x
y
+
y
z
+
z
x
=
5
\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5
y
x
+
z
y
+
x
z
=
5
. Prove that
x
(
y
2
−
2
z
2
)
z
+
y
(
z
2
−
2
x
2
)
x
+
z
(
x
2
−
2
y
2
)
y
⩾
0
\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0
z
x
(
y
2
−
2
z
2
)
+
x
y
(
z
2
−
2
x
2
)
+
y
z
(
x
2
−
2
y
2
)
⩾
0
algebra
triangle inequality
positive real numbers
inequalities
Turkey