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International Contests
IMO Longlists
1990 IMO Longlists
21
21
Part of
1990 IMO Longlists
Problems
(1)
Area inequality - ILL 1990 GRE1
Source:
9/18/2010
Point
O
O
O
is interior to triangle
A
B
C
ABC
A
BC
. Through
O
O
O
, draw three lines
D
E
∥
B
C
,
F
G
∥
C
A
DE \parallel BC, FG \parallel CA
D
E
∥
BC
,
FG
∥
C
A
, and
H
I
∥
A
B
HI \parallel AB
H
I
∥
A
B
, where
D
,
G
D, G
D
,
G
are on
A
B
AB
A
B
,
I
,
F
I, F
I
,
F
are on
B
C
BC
BC
and
E
,
H
E, H
E
,
H
are on
C
A
CA
C
A
. Denote by
S
1
S_1
S
1
the area of hexagon
D
G
H
E
F
I
DGHEFI
D
G
H
EF
I
, and
S
2
S_2
S
2
the area of triangle
A
B
C
ABC
A
BC
. Prove that
S
1
≥
2
3
S
2
.
S_1 \geq \frac 23 S_2.
S
1
≥
3
2
S
2
.
geometry
inequalities