MathDB
Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2018 Mediterranean Mathematics OIympiad
2
2
Part of
2018 Mediterranean Mathematics OIympiad
Problems
(1)
Geometrical equality
Source: Mediterranean math competition 2018
6/6/2018
Let
A
B
C
ABC
A
BC
be acute triangle. Let
E
E
E
and
F
F
F
be points on
B
C
BC
BC
, such that angles
B
A
E
BAE
B
A
E
and
F
A
C
FAC
F
A
C
are equal. Lines
A
E
AE
A
E
and
A
F
AF
A
F
intersect cirumcircle of
A
B
C
ABC
A
BC
at points
M
M
M
and
N
N
N
. On rays
A
B
AB
A
B
and
A
C
AC
A
C
we have points
P
P
P
and
R
R
R
, such that angle
P
E
A
PEA
PE
A
is equal to angle
B
B
B
and angle
A
E
R
AER
A
ER
is equal to angle
C
C
C
. Let
L
L
L
be intersection of
A
E
AE
A
E
and
P
R
PR
PR
and
D
D
D
be intersection of
B
C
BC
BC
and
L
N
LN
L
N
. Prove that
1
∣
M
N
∣
+
1
∣
E
F
∣
=
1
∣
E
D
∣
.
\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.
∣
MN
∣
1
+
∣
EF
∣
1
=
∣
E
D
∣
1
.
geometry
circumcircle
Ugly