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National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2017 Moldova EGMO TST
4
4
Part of
2017 Moldova EGMO TST
Problems
(1)
Moldova egmo 2k17 tst
Source: MDA TST for EGMO 2017, problem 4
4/26/2017
The points
P
P
P
and
Q
Q
Q
are placed in the interior of the triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
such that
m
(
∠
P
A
B
)
=
m
(
∠
Q
A
C
)
<
1
2
m
(
∠
B
A
C
)
m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)
m
(
∠
P
A
B
)
=
m
(
∠
Q
A
C
)
<
2
1
m
(
∠
B
A
C
)
and similarly for the other
2
2
2
vertices(
P
P
P
and
Q
Q
Q
are isogonal conjugates). Let
P
A
P_{A}
P
A
and
Q
A
Q_{A}
Q
A
be the intersection points of
A
P
AP
A
P
and
A
Q
AQ
A
Q
with the circumcircle of
C
P
B
CPB
CPB
, respectively
C
Q
B
CQB
CQB
. Similarly the pairs of points
(
P
B
,
Q
B
)
(P_{B},Q_{B})
(
P
B
,
Q
B
)
and
(
P
C
,
Q
C
)
(P_{C},Q_{C})
(
P
C
,
Q
C
)
are defined. Let
P
Q
A
∩
Q
P
A
=
{
M
A
}
PQ_{A}\cap QP_{A}=\{M_{A}\}
P
Q
A
∩
Q
P
A
=
{
M
A
}
,
P
Q
B
∩
Q
P
B
=
{
M
B
}
PQ_{B}\cap QP_{B}=\{M_{B}\}
P
Q
B
∩
Q
P
B
=
{
M
B
}
,
P
Q
C
∩
Q
P
C
=
{
M
C
}
PQ_{C}\cap QP_{C}=\{M_{C}\}
P
Q
C
∩
Q
P
C
=
{
M
C
}
. Prove the following statements:
1.
1.
1.
Lines
A
M
A
AM_{A}
A
M
A
,
B
M
B
BM_{B}
B
M
B
,
C
M
C
CM_{C}
C
M
C
concur.
2.
2.
2.
M
A
∈
B
C
M_{A}\in BC
M
A
∈
BC
,
M
B
∈
C
A
M_{B}\in CA
M
B
∈
C
A
,
M
C
∈
A
B
M_{C}\in AB
M
C
∈
A
B