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2018 PUMaC Live Round
7.3
7.3
Part of
2018 PUMaC Live Round
Problems
(1)
2018 PUMaC Live Round 7.3
Source:
1/13/2019
Kite
A
B
C
D
ABCD
A
BC
D
has right angles at
B
B
B
and
D
D
D
, and
A
B
<
B
C
AB<BC
A
B
<
BC
. Points
E
∈
A
B
E\in AB
E
∈
A
B
and
F
∈
A
D
F\in AD
F
∈
A
D
satisfy
A
E
=
4
AE=4
A
E
=
4
,
E
F
=
7
EF=7
EF
=
7
, and
F
A
=
5
FA=5
F
A
=
5
. If
A
B
=
8
AB=8
A
B
=
8
and points
X
X
X
lies outside
A
B
C
D
ABCD
A
BC
D
while satisfying
X
E
−
X
F
=
1
XE-XF=1
XE
−
XF
=
1
and
X
E
+
X
F
+
2
X
A
=
23
XE+XF+2XA=23
XE
+
XF
+
2
X
A
=
23
, then
X
A
XA
X
A
may be written as
a
−
b
c
d
\tfrac{a-b\sqrt{c}}{d}
d
a
−
b
c
for
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
positive integers with
gcd
(
a
2
,
b
2
,
c
,
d
2
)
=
1
\gcd(a^2,b^2,c,d^2)=1
g
cd
(
a
2
,
b
2
,
c
,
d
2
)
=
1
and
c
c
c
squarefree. Find
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
.
PuMAC
Live Round