MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Senior National Olympiad
2017 Azerbaijan Senior National Olympiad
2017 Azerbaijan Senior National Olympiad
Part of
Azerbaijan Senior National Olympiad
Subcontests
(2)
G4
1
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Minimum value of Hexagon's area
İn convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
's diagonals
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
intercepts each other at point
O
O
O
. If the area of triangles
A
O
B
,
C
O
D
,
E
O
F
AOB,COD,EOF
A
OB
,
CO
D
,
EOF
are
4
,
6
4,6
4
,
6
and
9
9
9
respectively, find the minimum possible value of area of hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
A5
1
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Inequality where a^x=bc,b^y=ca, c^z=ab
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
\\