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Balkan MO Shortlist
2016 Balkan MO Shortlist
A2
A2
Part of
2016 Balkan MO Shortlist
Problems
(1)
1/(x^2+y+z)+1/(y^2+z+x)+1/(z^2+x+y) <= 1 if x/yz+y/zx+z/xy<=1, for x,y,z>0
Source: Balkan BMO Shortlist 2016 A2
7/30/2019
For all
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
satisfying
x
y
z
+
y
z
x
+
z
x
y
≤
x
+
y
+
z
\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z
yz
x
+
z
x
y
+
x
y
z
≤
x
+
y
+
z
, prove that
1
x
2
+
y
+
z
+
1
y
2
+
z
+
x
+
1
z
2
+
x
+
y
≤
1
\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1
x
2
+
y
+
z
1
+
y
2
+
z
+
x
1
+
z
2
+
x
+
y
1
≤
1
inequalities
three variable inequality
algebra