MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2016 Greece JBMO TST
1
1
Part of
2016 Greece JBMO TST
Problems
(1)
x^2y/z+y^2z/x+z^2x/y+2(y/xz+z/xy+x/yz)>= 9 if x,y,z>0
Source: Greece JBMO TST 2016 p1
4/29/2019
a) Prove that, for any real
x
>
0
x>0
x
>
0
, it is true that
x
3
−
3
x
≥
−
2
x^3-3x\ge -2
x
3
−
3
x
≥
−
2
. b) Prove that, for any real
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
, it is true that
x
2
y
z
+
y
2
z
x
+
z
2
x
y
+
2
(
y
x
z
+
z
x
y
+
x
y
z
)
≥
9
\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}+2\left(\frac{y}{xz}+\frac{z}{xy}+\frac{x}{yz} \right)\ge 9
z
x
2
y
+
x
y
2
z
+
y
z
2
x
+
2
(
x
z
y
+
x
y
z
+
yz
x
)
≥
9
. When we have equality ?
algebra
inequalities
three variable inequality