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MAA AMC
AMC 12/AHSME
1991 AMC 12/AHSME
27
Algebra
Algebra
Source: AHSME 1991 problem 27
November 2, 2011
AMC
Problem Statement
If
x
+
x
2
−
1
+
1
x
−
x
2
−
1
=
20
x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20
x
+
x
2
−
1
+
x
−
x
2
−
1
1
=
20
then
x
2
+
x
4
−
1
+
1
x
2
+
x
4
−
1
=
x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} =
x
2
+
x
4
−
1
+
x
2
+
x
4
−
1
1
=
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
5.05
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
20
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
51.005
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
61.25
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
400
<span class='latex-bold'>(A)</span>\ 5.05\qquad<span class='latex-bold'>(B)</span>\ 20\qquad<span class='latex-bold'>(C)</span>\ 51.005\qquad<span class='latex-bold'>(D)</span>\ 61.25\qquad<span class='latex-bold'>(E)</span>\ 400
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
5.05
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
20
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
51.005
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
61.25
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
400
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