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2007 Harvard-MIT Mathematics Tournament
36
36
Part of
2007 Harvard-MIT Mathematics Tournament
Problems
(1)
2007 Guts #36: The Marathon
Source:
6/22/2012
The Marathon. Let
ω
\omega
ω
denote the incircle of triangle
A
B
C
ABC
A
BC
. The segments
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
are tangent to
ω
\omega
ω
at
D
D
D
,
E
E
E
and
F
F
F
, respectively. Point
P
P
P
lies on
E
F
EF
EF
such that segment
P
D
PD
P
D
is perpendicular to
B
C
BC
BC
. The line
A
P
AP
A
P
intersects
B
C
BC
BC
at
Q
Q
Q
. The circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
pass through
B
B
B
and
C
C
C
, respectively, and are tangent to
A
Q
AQ
A
Q
at
Q
Q
Q
; the former meets
A
B
AB
A
B
again at
X
X
X
, and the latter meets
A
C
AC
A
C
again at
Y
Y
Y
. The line
X
Y
XY
X
Y
intersects
B
C
BC
BC
at
Z
Z
Z
. Given that
A
B
=
15
AB=15
A
B
=
15
,
B
C
=
14
BC=14
BC
=
14
, and
C
A
=
13
CA=13
C
A
=
13
, find
⌊
X
Z
⋅
Y
Z
⌋
\lfloor XZ\cdot YZ\rfloor
⌊
XZ
⋅
Y
Z
⌋
.
geometry
floor function