MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2005 Greece JBMO TST
2
(x^2-y^2)/ (2x^2+1) + (y^2-z^2)/(2y^2+1)+(z^2-x^2)/ (2z^2+1)<=0
(x^2-y^2)/ (2x^2+1) + (y^2-z^2)/(2y^2+1)+(z^2-x^2)/ (2z^2+1)<=0
Source: Greece JBMO TST 2005 p2
June 16, 2019
algebra
inequalities
three variable inequality
Problem Statement
Prove that for each
x
,
y
,
z
∈
R
x,y,z \in R
x
,
y
,
z
∈
R
it holds that
x
2
−
y
2
2
x
2
+
1
+
y
2
−
z
2
2
y
2
+
1
+
z
2
−
x
2
2
z
2
+
1
≤
0
\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0
2
x
2
+
1
x
2
−
y
2
+
2
y
2
+
1
y
2
−
z
2
+
2
z
2
+
1
z
2
−
x
2
≤
0
Back to Problems
View on AoPS