MathDB
Problems
Contests
International Contests
JBMO ShortLists
2016 JBMO Shortlist
4
Inequality
Inequality
Source: JBMO 2016 shortlist
June 25, 2017
JBMO Shortlist
Inequality
Problem Statement
If the non-negative reals
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy
x
2
+
y
2
+
z
2
=
x
+
y
+
z
x^2+y^2+z^2=x+y+z
x
2
+
y
2
+
z
2
=
x
+
y
+
z
. Prove that
x
+
1
x
5
+
x
+
1
+
y
+
1
y
5
+
y
+
1
+
z
+
1
z
5
+
z
+
1
≥
3.
\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.
x
5
+
x
+
1
x
+
1
+
y
5
+
y
+
1
y
+
1
+
z
5
+
z
+
1
z
+
1
≥
3.
When does the equality occur?Proposed by Dorlir Ahmeti, Albania
Back to Problems
View on AoPS