MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Senior National Olympiad
2017 Azerbaijan Senior National Olympiad
A5
A5
Part of
2017 Azerbaijan Senior National Olympiad
Problems
(1)
Inequality where a^x=bc,b^y=ca, c^z=ab
Source: 2017 Azerbaijan Senior National Olympiad
4/29/2022
a
,
b
,
c
∈
(
0
,
1
)
a,b,c \in (0,1)
a
,
b
,
c
∈
(
0
,
1
)
and
x
,
y
,
z
∈
(
0
,
∞
)
x,y,z \in ( 0, \infty)
x
,
y
,
z
∈
(
0
,
∞
)
reals satisfies the condition
a
x
=
b
c
,
b
y
=
c
a
,
c
z
=
a
b
a^x=bc,b^y=ca,c^z=ab
a
x
=
b
c
,
b
y
=
c
a
,
c
z
=
ab
. Prove that
1
2
+
x
+
1
2
+
y
+
1
2
+
z
≤
3
4
\dfrac{1}{2+x}+\dfrac{1}{2+y}+\dfrac{1}{2+z} \leq \dfrac{3}{4}
2
+
x
1
+
2
+
y
1
+
2
+
z
1
≤
4
3
\\
Inequality
algebra
inequalities