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Bulgaria National Olympiad
2020 Bulgaria National Olympiad
P1
P1
Part of
2020 Bulgaria National Olympiad
Problems
(1)
Elementary Geometry
Source: Bulgaria National Olympiad 2020
6/30/2020
On the sides of
△
A
B
C
\triangle{ABC}
△
A
BC
points
P
,
Q
∈
A
B
P,Q \in{AB}
P
,
Q
∈
A
B
(
P
P
P
is between
A
A
A
and
Q
Q
Q
) and
R
∈
B
C
R\in{BC}
R
∈
BC
are chosen. The points
M
M
M
and
N
N
N
are defined as the intersection point of
A
R
AR
A
R
with the segments
C
P
CP
CP
and
C
Q
CQ
CQ
, respectively. If
B
C
=
B
Q
BC=BQ
BC
=
BQ
,
C
P
=
A
P
CP=AP
CP
=
A
P
,
C
R
=
C
N
CR=CN
CR
=
CN
and
∠
B
P
C
=
∠
C
R
A
\angle{BPC}=\angle{CRA}
∠
BPC
=
∠
CR
A
, prove that
M
P
+
N
Q
=
B
R
MP+NQ=BR
MP
+
NQ
=
BR
.
geometry