MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey EGMO TST
2018 Turkey EGMO TST
5
Turkey Egmo tst 2018 p5
Turkey Egmo tst 2018 p5
Source:
February 9, 2018
inequalities
Problem Statement
Prove that
x
2
+
1
(
x
+
y
)
2
+
4
(
z
+
1
)
+
y
2
+
1
(
y
+
z
)
2
+
4
(
x
+
1
)
+
z
2
+
1
(
z
+
x
)
2
+
4
(
y
+
1
)
≥
1
2
\dfrac {x^2+1}{(x+y)^2+4 (z+1)}+\dfrac {y^2+1}{(y+z)^2+4 (x+1)}+\dfrac {z^2+1}{(z+x)^2+4 (y+1)} \ge \dfrac{1}{2}
(
x
+
y
)
2
+
4
(
z
+
1
)
x
2
+
1
+
(
y
+
z
)
2
+
4
(
x
+
1
)
y
2
+
1
+
(
z
+
x
)
2
+
4
(
y
+
1
)
z
2
+
1
≥
2
1
for all positive reals
x
,
y
,
z
x,y,z
x
,
y
,
z
Back to Problems
View on AoPS