MathDB
Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2012 India Regional Mathematical Olympiad
8
8
Part of
2012 India Regional Mathematical Olympiad
Problems
(1)
1/(x-2)(y-2)(z-2)+8/(x+2)(y+2)(z+2)\le 1/ 32 if 2(xy + yz + zx) = xyz
Source: CRMO 2012 region 5 p8 Mumbai
9/30/2018
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive real numbers such that
2
(
x
y
+
y
z
+
z
x
)
=
x
y
z
2(xy + yz + zx) = xyz
2
(
x
y
+
yz
+
z
x
)
=
x
yz
. Prove that
1
(
x
−
2
)
(
y
−
2
)
(
z
−
2
)
+
8
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
≤
1
32
\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}
(
x
−
2
)
(
y
−
2
)
(
z
−
2
)
1
+
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
8
≤
32
1
inequalities
Inequality
3-variable inequality