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AIME Problems
2005 AIME Problems
7
7
Part of
2005 AIME Problems
Problems
(2)
Finding a Side Length, Quadrilaterals
Source:
11/10/2005
In quadrilateral
A
B
C
D
ABCD
A
BC
D
,
B
C
=
8
BC=8
BC
=
8
,
C
D
=
12
CD=12
C
D
=
12
,
A
D
=
10
AD=10
A
D
=
10
, and
m
∠
A
=
m
∠
B
=
6
0
∘
m\angle A= m\angle B = 60^\circ
m
∠
A
=
m
∠
B
=
6
0
∘
. Given that
A
B
=
p
+
q
AB=p + \sqrt{q}
A
B
=
p
+
q
, where
p
p
p
and
q
q
q
are positive integers, find
p
+
q
p+q
p
+
q
.
trigonometry
quadratics
algebra
quadratic formula
trig identities
Law of Cosines
A Giant Expansion
Source:
11/13/2005
Let
x
=
4
(
5
+
1
)
(
5
4
+
1
)
(
5
8
+
1
)
(
5
16
+
1
)
.
x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.
x
=
(
5
+
1
)
(
4
5
+
1
)
(
8
5
+
1
)
(
16
5
+
1
)
4
.
Find
(
x
+
1
)
48
(x+1)^{48}
(
x
+
1
)
48
.
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