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Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2017 Macedonia JBMO TST
3
3
Part of
2017 Macedonia JBMO TST
Problems
(1)
cyc sum (x^2+y^2+z)/(x^2+2) \geq3 [Macedonia JBMO TST 2017, P3]
Source: Macedonia JBMO TST 2017, Problem 3
6/26/2018
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive reals such that
x
y
z
=
1
xyz=1
x
yz
=
1
. Show that
x
2
+
y
2
+
z
x
2
+
2
+
y
2
+
z
2
+
x
y
2
+
2
+
z
2
+
x
2
+
y
z
2
+
2
≥
3.
\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.
x
2
+
2
x
2
+
y
2
+
z
+
y
2
+
2
y
2
+
z
2
+
x
+
z
2
+
2
z
2
+
x
2
+
y
≥
3.
When does equality happen?
inequalities