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2016 Harvard-MIT Mathematics Tournament
20
20
Part of
2016 Harvard-MIT Mathematics Tournament
Problems
(1)
2016 Guts #20
Source:
12/24/2016
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
13
AB=13
A
B
=
13
,
A
C
=
14
AC=14
A
C
=
14
, and
B
C
=
15
BC=15
BC
=
15
. Let
G
G
G
be the point on
A
C
AC
A
C
such that the reflection of
B
G
BG
BG
over the angle bisector of
∠
B
\angle B
∠
B
passes through the midpoint of
A
C
AC
A
C
. Let
Y
Y
Y
be the midpoint of
G
C
GC
GC
and
X
X
X
be a point on segment
A
G
AG
A
G
such that
A
X
X
G
=
3
\frac{AX}{XG}=3
XG
A
X
=
3
. Construct
F
F
F
and
H
H
H
on
A
B
AB
A
B
and
B
C
BC
BC
, respectively, such that
F
X
∥
B
G
∥
H
Y
FX \parallel BG \parallel HY
FX
∥
BG
∥
H
Y
. If
A
H
AH
A
H
and
C
F
CF
CF
concur at
Z
Z
Z
and
W
W
W
is on
A
C
AC
A
C
such that
W
Z
∥
B
G
WZ \parallel BG
W
Z
∥
BG
, find
W
Z
WZ
W
Z
.