MathDB
Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2021 Philippine MO
8
8
Part of
2021 Philippine MO
Problems
(1)
pmo problem 8
Source: PMO 2021
3/20/2021
In right triangle
A
B
C
ABC
A
BC
,
∠
A
C
B
=
9
0
∘
\angle ACB = 90^{\circ}
∠
A
CB
=
9
0
∘
and
tan
A
>
2
\tan A > \sqrt{2}
tan
A
>
2
.
M
M
M
is the midpoint of
A
B
AB
A
B
,
P
P
P
is the foot of the altitude from
C
C
C
, and
N
N
N
is the midpoint of
C
P
CP
CP
. Line
A
B
AB
A
B
meets the circumcircle of
C
N
B
CNB
CNB
again at
Q
Q
Q
.
R
R
R
lies on line
B
C
BC
BC
such that
Q
R
QR
QR
and
C
P
CP
CP
are parallel,
S
S
S
lies on ray
C
A
CA
C
A
past
A
A
A
such that
B
R
=
R
S
BR = RS
BR
=
RS
, and
V
V
V
lies on segment
S
P
SP
SP
such that
A
V
=
V
P
AV = VP
A
V
=
V
P
. Line
S
P
SP
SP
meets the circumcircle of
C
P
B
CPB
CPB
again at
T
T
T
.
W
W
W
lies on ray
V
A
VA
V
A
past
A
A
A
such that
2
A
W
=
S
T
2AW = ST
2
A
W
=
ST
, and
O
O
O
is the circumcenter of
S
P
M
SPM
SPM
. Prove that lines
O
M
OM
OM
and
B
W
BW
B
W
are perpendicular.
geometry
PMO