MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2016 Macedonia JBMO TST
4
2016 JBMO TST- Macedonia, problem 4
2016 JBMO TST- Macedonia, problem 4
Source: 2016 JBMO TST- Macedonia
May 27, 2019
JMMO
Macedonia
Junior
2016
algebra
Inequality
Problem Statement
Let
x
x
x
,
y
y
y
, and
z
z
z
be positive real numbers. Prove that
x
y
x
2
+
y
2
+
2
z
2
+
y
z
y
2
+
z
2
+
2
x
2
+
z
x
z
2
+
x
2
+
2
y
2
≤
3
2
\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}
x
2
+
y
2
+
2
z
2
x
y
+
y
2
+
z
2
+
2
x
2
yz
+
z
2
+
x
2
+
2
y
2
z
x
≤
2
3
.When does equality hold?
Back to Problems
View on AoPS