MathDB
Problems
Contests
International Contests
Balkan MO Shortlist
2014 Balkan MO Shortlist
A7
BMO 2014 SL A7
BMO 2014 SL A7
Source: Balkan MO 2014 Shortlist
October 1, 2016
algebra
inequalities
Problem Statement
A
7
\boxed{A7}
A
7
Prove that for all
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
with
1
x
+
1
y
+
1
z
=
1
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1
x
1
+
y
1
+
z
1
=
1
and
0
≤
a
,
b
,
c
<
1
0\leq a,b,c<1
0
≤
a
,
b
,
c
<
1
the following inequality holds
x
2
+
y
2
1
−
a
z
+
y
2
+
z
2
1
−
b
x
+
z
2
+
x
2
1
−
c
y
≥
6
(
x
+
y
+
z
)
1
−
a
b
c
\frac{x^2+y^2}{1-a^z}+\frac{y^2+z^2}{1-b^x}+\frac{z^2+x^2}{1-c^y}\geq \frac{6(x+y+z)}{1-abc}
1
−
a
z
x
2
+
y
2
+
1
−
b
x
y
2
+
z
2
+
1
−
c
y
z
2
+
x
2
≥
1
−
ab
c
6
(
x
+
y
+
z
)
Back to Problems
View on AoPS