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2013 Harvard-MIT Mathematics Tournament
24
24
Part of
2013 Harvard-MIT Mathematics Tournament
Problems
(1)
2013 HMMT Guts #24: Function in terms of Distance
Source:
3/26/2013
Given a point
p
p
p
and a line segment
l
l
l
, let
d
(
p
,
l
)
d(p,l)
d
(
p
,
l
)
be the distance between them. Let
A
A
A
,
B
B
B
, and
C
C
C
be points in the plane such that
A
B
=
6
AB=6
A
B
=
6
,
B
C
=
8
BC=8
BC
=
8
,
A
C
=
10
AC=10
A
C
=
10
. What is the area of the region in the
(
x
,
y
)
(x,y)
(
x
,
y
)
-plane formed by the ordered pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
such that there exists a point
P
P
P
inside triangle
A
B
C
ABC
A
BC
with
d
(
P
,
A
B
)
+
x
=
d
(
P
,
B
C
)
+
y
=
d
(
P
,
A
C
)
?
d(P,AB)+x=d(P,BC)+y=d(P,AC)?
d
(
P
,
A
B
)
+
x
=
d
(
P
,
BC
)
+
y
=
d
(
P
,
A
C
)?
HMMT
function
geometry