MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2016 Korea National Olympiad
3
Weird Ineq
Weird Ineq
Source: 2016 KMO Senior #3
November 12, 2016
geometry
geometric inequality
Korea
Problem Statement
Acute triangle
△
A
B
C
\triangle ABC
△
A
BC
has area
S
S
S
and perimeter
L
L
L
. A point
P
P
P
inside
△
A
B
C
\triangle ABC
△
A
BC
has
d
i
s
t
(
P
,
B
C
)
=
1
,
d
i
s
t
(
P
,
C
A
)
=
1.5
,
d
i
s
t
(
P
,
A
B
)
=
2
dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2
d
i
s
t
(
P
,
BC
)
=
1
,
d
i
s
t
(
P
,
C
A
)
=
1.5
,
d
i
s
t
(
P
,
A
B
)
=
2
. Let
B
C
∩
A
P
=
D
BC \cap AP = D
BC
∩
A
P
=
D
,
C
A
∩
B
P
=
E
CA \cap BP = E
C
A
∩
BP
=
E
,
A
B
∩
C
P
=
F
AB \cap CP= F
A
B
∩
CP
=
F
. Let
T
T
T
be the area of
△
D
E
F
\triangle DEF
△
D
EF
. Prove the following inequality.
(
A
D
⋅
B
E
⋅
C
F
T
)
2
>
4
L
2
+
(
A
B
⋅
B
C
⋅
C
A
24
S
)
2
\left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2
(
T
A
D
⋅
BE
⋅
CF
)
2
>
4
L
2
+
(
24
S
A
B
⋅
BC
⋅
C
A
)
2
Back to Problems
View on AoPS