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Junior Balkan MO
2011 Junior Balkan MO
4
4
Part of
2011 Junior Balkan MO
Problems
(1)
Jbmo 2011 Problem 4
Source: Jbmo 2011
6/21/2011
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and points
E
E
E
and
F
F
F
on sides
A
B
,
C
D
AB,CD
A
B
,
C
D
such that
A
B
A
E
=
C
D
D
F
=
n
\tfrac{AB}{AE}=\tfrac{CD}{DF}=n
A
E
A
B
=
D
F
C
D
=
n
If
S
S
S
is the area of
A
E
F
D
AEFD
A
EF
D
show that
S
≤
A
B
⋅
C
D
+
n
(
n
−
1
)
A
D
2
+
n
2
D
A
⋅
B
C
2
n
2
{S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}
S
≤
2
n
2
A
B
⋅
C
D
+
n
(
n
−
1
)
A
D
2
+
n
2
D
A
⋅
BC
geometry
rectangle
inequalities
trigonometry
vector
triangle inequality
geometry unsolved