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Rioplatense Mathematical Olympiad, Level 3
2000 Rioplatense Mathematical Olympiad, Level 3
5
5
Part of
2000 Rioplatense Mathematical Olympiad, Level 3
Problems
(1)
Concurrency(little hard)
Source: Rioplatense Olympiad 2000
2/28/2018
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
, let
L
L
L
be midpoint of arc
B
C
BC
BC
(the point
A
A
A
is not in this arc) of the circumcircle
w
w
w
(
A
B
C
ABC
A
BC
). Let
E
E
E
be a point in
A
C
AC
A
C
where
A
E
=
A
B
+
A
C
2
AE = \frac{AB + AC}{2}
A
E
=
2
A
B
+
A
C
, the line
E
L
EL
E
L
intersects
w
w
w
in
P
P
P
. If
M
M
M
and
N
N
N
are the midpoints of
A
B
AB
A
B
and
B
C
BC
BC
, respectively, prove that
A
L
,
B
P
AL, BP
A
L
,
BP
and
M
N
MN
MN
are concurrents
geometry