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2022 Princeton University Math Competition
A6 / B8
A6 / B8
Part of
2022 Princeton University Math Competition
Problems
(1)
2022 PUMaC Geometry A6 / B8
Source:
9/10/2023
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
has sidelengths
A
B
=
10
AB = 10
A
B
=
10
,
A
C
=
14
AC = 14
A
C
=
14
, and,
B
C
=
16
BC = 16
BC
=
16
. Circle
ω
1
\omega_1
ω
1
is tangent to rays
A
B
→
\overrightarrow{AB}
A
B
,
A
C
→
\overrightarrow{AC}
A
C
and passes through
B
B
B
. Circle
ω
2
\omega_2
ω
2
is tangent to rays
A
B
→
\overrightarrow{AB}
A
B
,
A
C
→
\overrightarrow{AC}
A
C
and passes through
C
C
C
. Let
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
intersect at points
X
,
Y
X, Y
X
,
Y
. The square of the perimeter of triangle
△
A
X
Y
\vartriangle AXY
△
A
X
Y
is equal to
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
, and,
d
d
d
are positive integers such that
a
a
a
and
d
d
d
are relatively prime, and
c
c
c
is not divisible by the square of any prime. Find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
.
geometry