MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Princeton University Math Competition
2022 Princeton University Math Competition
A7
A7
Part of
2022 Princeton University Math Competition
Problems
(1)
2022 PUMaC Geometry A7
Source:
9/10/2023
Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle with
B
C
=
7
BC = 7
BC
=
7
,
C
A
=
6
CA = 6
C
A
=
6
, and,
A
B
=
5
AB = 5
A
B
=
5
. Let
I
I
I
be the incenter of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let the incircle of
△
A
B
C
\vartriangle ABC
△
A
BC
touch sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at points
D
,
E
D,E
D
,
E
, and
F
F
F
. Let the circumcircle of
△
A
E
F
\vartriangle AEF
△
A
EF
meet the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
for a second time at point
X
≠
A
X\ne A
X
=
A
. Let
P
P
P
denote the intersection of
X
I
XI
X
I
and
E
F
EF
EF
. If the product
X
P
⋅
I
P
XP \cdot IP
XP
⋅
I
P
can be written as
m
n
\frac{m}{n}
n
m
for relatively prime positive integers
m
,
n
m, n
m
,
n
, find
m
+
n
m + n
m
+
n
.
geometry