MathDB
Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2022 Iran MO (2nd round)
6
6
Part of
2022 Iran MO (2nd round)
Problems
(1)
Geometry that "looks" hard
Source: Iran 2nd round 2022 P6
5/9/2022
we have an isogonal triangle
A
B
C
ABC
A
BC
such that
B
C
=
A
B
BC=AB
BC
=
A
B
. take a random
P
P
P
on the altitude from
B
B
B
to
A
C
AC
A
C
. The circle
(
A
B
P
)
(ABP)
(
A
BP
)
intersects
A
C
AC
A
C
second time in
M
M
M
. Take
N
N
N
such that it's on the segment
A
C
AC
A
C
and
A
M
=
N
C
AM=NC
A
M
=
NC
and
M
≠
N
M \neq N
M
=
N
.The second intersection of
N
P
NP
NP
and circle
(
A
P
B
)
(APB)
(
A
PB
)
is
X
X
X
, (
X
≠
P
X \neq P
X
=
P
) and the second intersection of
A
B
AB
A
B
and circle
(
A
P
N
)
(APN)
(
A
PN
)
is
Y
Y
Y
,(
Y
≠
A
Y \neq A
Y
=
A
).The tangent from
A
A
A
to the circle
(
A
P
N
)
(APN)
(
A
PN
)
intersects the altitude from
B
B
B
at
Z
Z
Z
. Prove that
C
Z
CZ
CZ
is tangent to circle
(
P
X
Y
)
(PXY)
(
PX
Y
)
.
geometry