MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1956 AMC 12/AHSME
1956 AMC 12/AHSME
Part of
AMC 12/AHSME
Subcontests
(50)
50
1
Hide problems
Square on a Triangle
In triangle
A
B
C
ABC
A
BC
, \overline{CA} \equal{} \overline{CB}. On
C
B
CB
CB
square
B
C
D
E
BCDE
BC
D
E
is constructed away from the triangle. If
x
x
x
is the number of degrees in angle
D
A
B
DAB
D
A
B
, then
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
x
depends upon triangle
A
B
C
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
x
is independent of the triangle
<span class='latex-bold'>(A)</span>\ x\text{ depends upon triangle }ABC \qquad<span class='latex-bold'>(B)</span>\ x\text{ is independent of the triangle}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
x
depends upon triangle
A
BC
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
x
is independent of the triangle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
x
may equal angle
C
A
D
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
x
can never equal angle
C
A
B
<span class='latex-bold'>(C)</span>\ x\text{ may equal angle }CAD \qquad<span class='latex-bold'>(D)</span>\ x\text{ can never equal angle }CAB
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
x
may equal angle
C
A
D
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
x
can never equal angle
C
A
B
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
x
is greater than
4
5
∘
but less than
9
0
∘
<span class='latex-bold'>(E)</span>\ x\text{ is greater than }45^{\circ}\text{ but less than }90^{\circ}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
x
is greater than
4
5
∘
but less than
9
0
∘
49
1
Hide problems
Three Tangents to a Circle
Triangle
P
A
B
PAB
P
A
B
is formed by three tangents to circle
O
O
O
and < APB \equal{} 40^{\circ}; then angle
A
O
B
AOB
A
OB
equals:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
4
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
0
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
0
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
7
0
∘
<span class='latex-bold'>(A)</span>\ 45^{\circ} \qquad<span class='latex-bold'>(B)</span>\ 50^{\circ} \qquad<span class='latex-bold'>(C)</span>\ 55^{\circ} \qquad<span class='latex-bold'>(D)</span>\ 60^{\circ} \qquad<span class='latex-bold'>(E)</span>\ 70^{\circ}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
5
0
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
5
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6
0
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
7
0
∘
48
1
Hide problems
Fraction Reducing to Positive Integer
If
p
p
p
is a positive integer, then \frac {3p \plus{} 25}{2p \minus{} 5} can be a positive integer, if and only if
p
p
p
is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
at least
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
at least
3
and no more than
35
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
no more than
35
<span class='latex-bold'>(A)</span>\ \text{at least }3 \qquad<span class='latex-bold'>(B)</span>\ \text{at least }3\text{ and no more than }35 \qquad<span class='latex-bold'>(C)</span>\ \text{no more than }35
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
at least
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
at least
3
and no more than
35
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
no more than
35
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
equal to
35
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
equal to
3
or
35
<span class='latex-bold'>(D)</span>\ \text{equal to }35 \qquad<span class='latex-bold'>(E)</span>\ \text{equal to }3\text{ or }35
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
equal to
35
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
equal to
3
or
35
47
1
Hide problems
Highway Building Machines
An engineer said he could finish a highway section in
3
3
3
days with his present supply of a certain type of machine. However, with
3
3
3
more of these machines the job could be done in
2
2
2
days. If the machines all work at the same rate, how many days would it take to do the job with one machine?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
12
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
15
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
18
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
36
<span class='latex-bold'>(A)</span>\ 6 \qquad<span class='latex-bold'>(B)</span>\ 12 \qquad<span class='latex-bold'>(C)</span>\ 15 \qquad<span class='latex-bold'>(D)</span>\ 18 \qquad<span class='latex-bold'>(E)</span>\ 36
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
18
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
36
46
1
Hide problems
Keeping an Equation True
For the equation \frac {1 \plus{} x}{1 \minus{} x} \equal{} \frac {N \plus{} 1}{N} to be true where
N
N
N
is positive,
x
x
x
can have:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
any positive value less than
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
any value less than
1
<span class='latex-bold'>(A)</span>\ \text{any positive value less than }1 \qquad<span class='latex-bold'>(B)</span>\ \text{any value less than }1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
any positive value less than
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
any value less than
1
(C)
\ \text{the value zero only} \qquad
(D)
\ \text{any non \minus{} negative value} \qquad
(E)
\ \text{any value}
45
1
Hide problems
Rubber Tire
A wheel with a rubber tire has an outside diameter of
25
25
25
in. When the radius has been decreased a quarter of an inch, the number of revolutions in one mile will:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
be increased about
2
%
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
be increased about
1
%
<span class='latex-bold'>(A)</span>\ \text{be increased about }2\% \qquad<span class='latex-bold'>(B)</span>\ \text{be increased about }1\%
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
be increased about
2%
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
be increased about
1%
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
be increased about
20
%
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
be increased about
1
2
%
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
remain the same
<span class='latex-bold'>(C)</span>\ \text{be increased about }20\% \qquad<span class='latex-bold'>(D)</span>\ \text{be increased about }\frac {1}{2}\% \qquad<span class='latex-bold'>(E)</span>\ \text{remain the same}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
be increased about
20%
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
be increased about
2
1
%
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
remain the same
44
1
Hide problems
x<a<0
If
x
<
a
<
0
x < a < 0
x
<
a
<
0
means that
x
x
x
and
a
a
a
are numbers such that
x
x
x
is less than
a
a
a
and
a
a
a
is less than zero, then:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
x
2
<
a
x
<
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
x
2
>
a
x
>
a
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
x
2
<
a
2
<
0
<span class='latex-bold'>(A)</span>\ x^2 < ax < 0 \qquad<span class='latex-bold'>(B)</span>\ x^2 > ax > a^2 \qquad<span class='latex-bold'>(C)</span>\ x^2 < a^2 < 0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
x
2
<
a
x
<
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
x
2
>
a
x
>
a
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
x
2
<
a
2
<
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
x
2
>
a
x
but
a
x
<
0
<
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p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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a
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>
x
2
>
a
2
but
a
2
<
0
<span class='latex-bold'>(D)</span>\ x^2 > ax\text{ but }ax < 0 \qquad<span class='latex-bold'>(E)</span>\ x^2 > a^2\text{ but }a^2 < 0
<
s
p
an
c
l
a
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=
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a
t
e
x
−
b
o
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′
>
(
D
)
<
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>
x
2
>
a
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but
a
x
<
0
<
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ss
=
′
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a
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x
−
b
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>
(
E
)
<
/
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>
x
2
>
a
2
but
a
2
<
0
43
1
Hide problems
Integral Scalene Triangles
The number of scalene triangles having all sides of integral lengths, and perimeter less than
13
13
13
is:
<
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p
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n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
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n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
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>
3
<
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p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
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a
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>
4
<
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a
s
s
=
′
l
a
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x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
18
<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 18
<
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c
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=
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a
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x
−
b
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′
>
(
A
)
<
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>
1
<
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=
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a
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−
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>
(
B
)
<
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>
2
<
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a
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=
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a
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−
b
o
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d
′
>
(
C
)
<
/
s
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>
3
<
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p
an
c
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a
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=
′
l
a
t
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x
−
b
o
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′
>
(
D
)
<
/
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>
4
<
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=
′
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a
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e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
18
42
1
Hide problems
Radical Equation
The equation \sqrt {x \plus{} 4} \minus{} \sqrt {x \minus{} 3} \plus{} 1 \equal{} 0 has:
<
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(
A
)
<
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no root
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a
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x
−
b
o
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>
(
B
)
<
/
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p
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>
one real root
<span class='latex-bold'>(A)</span>\ \text{no root} \qquad<span class='latex-bold'>(B)</span>\ \text{one real root}
<
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=
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−
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>
(
A
)
<
/
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>
no root
<
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=
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a
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x
−
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′
>
(
B
)
<
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>
one real root
<
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s
=
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a
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x
−
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′
>
(
C
)
<
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>
one real root and one imaginary root
<span class='latex-bold'>(C)</span>\ \text{one real root and one imaginary root}
<
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=
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a
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−
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>
(
C
)
<
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>
one real root and one imaginary root
<
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=
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−
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>
(
D
)
<
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>
two imaginary roots
<
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=
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−
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o
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>
(
E
)
<
/
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>
two real roots
<span class='latex-bold'>(D)</span>\ \text{two imaginary roots} \qquad<span class='latex-bold'>(E)</span>\ \text{two real roots}
<
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c
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a
ss
=
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a
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−
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(
D
)
<
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>
two imaginary roots
<
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=
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>
(
E
)
<
/
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>
two real roots
41
1
Hide problems
Equation with y=2x
The equation 3y^2 \plus{} y \plus{} 4 \equal{} 2(6x^2 \plus{} y \plus{} 2) where y \equal{} 2x is satisfied by:
(A)
\ \text{no value of }x \qquad
(B)
\ \text{all values of }x \qquad
(C)
\ x \equal{} 0\text{ only}
<
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(
D
)
<
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>
all integral values of
x
only
<
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=
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−
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>
(
E
)
<
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>
all rational values of
x
only
<span class='latex-bold'>(D)</span>\ \text{all integral values of }x\text{ only} \qquad<span class='latex-bold'>(E)</span>\ \text{all rational values of }x\text{ only}
<
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a
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=
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(
D
)
<
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p
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>
all integral values of
x
only
<
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a
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=
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>
(
E
)
<
/
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>
all rational values of
x
only
40
1
Hide problems
Position, Velocity, Acceleration, and Time
If V \equal{} gt \plus{} V_0 and S \equal{} \frac {1}{2}gt^2 \plus{} V_0t, then
t
t
t
equals:
(A)
\ \frac {2S}{V \plus{} V_0} \qquad
(B)
\ \frac {2S}{V \minus{} V_0} \qquad
(C)
\ \frac {2S}{V_0 \minus{} V} \qquad
(D)
\ \frac {2S}{V} \qquad
(E)
\ 2S \minus{} V
39
1
Hide problems
Hypotenuse and Arm of Right Triangle
The hypotenuse
c
c
c
and one arm
a
a
a
of a right triangle are consecutive integers. The square of the second arm is:
(A)
\ ca \qquad
(B)
\ \frac {c}{a} \qquad
(C)
\ c \plus{} a \qquad
(D)
\ c \minus{} a \qquad
(E)
\ \text{none of these}
38
1
Hide problems
Altitude of Right Triangle
In a right triangle with sides
a
a
a
and
b
b
b
, and hypotenuse
c
c
c
, the altitude drawn on the hypotenuse is
x
x
x
. Then:
(A)
\ ab \equal{} x^2 \qquad
(B)
\ \frac {1}{a} \plus{} \frac {1}{b} \equal{} \frac {1}{x} \qquad
(C)
\ a^2 \plus{} b^2 \equal{} 2x^2
(D)
\ \frac {1}{x^2} \equal{} \frac {1}{a^2} \plus{} \frac {1}{b^2} \qquad
(E)
\ \frac {1}{x} \equal{} \frac {b}{a}
37
1
Hide problems
Estate on a Map
On a map whose scale is
400
400
400
miles to an inch and a half, a certain estate is represented by a rhombus having a
6
0
∘
60^{\circ}
6
0
∘
angle. The diagonal opposite
6
0
∘
60^{\circ}
6
0
∘
is
3
16
\frac {3}{16}
16
3
in. The area of the estate in square miles is:
<
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−
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>
(
A
)
<
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>
2500
3
<
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=
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−
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>
(
B
)
<
/
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>
1250
3
<
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a
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s
=
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x
−
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>
(
C
)
<
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>
1250
<
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=
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−
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>
(
D
)
<
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a
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>
5625
3
2
<
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(
E
)
<
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>
1250
3
<span class='latex-bold'>(A)</span>\ \frac {2500}{\sqrt {3}} \qquad<span class='latex-bold'>(B)</span>\ \frac {1250}{\sqrt {3}} \qquad<span class='latex-bold'>(C)</span>\ 1250 \qquad<span class='latex-bold'>(D)</span>\ \frac {5625\sqrt {3}}{2} \qquad<span class='latex-bold'>(E)</span>\ 1250\sqrt {3}
<
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(
A
)
<
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>
3
2500
<
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=
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−
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>
(
B
)
<
/
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>
3
1250
<
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c
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a
ss
=
′
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a
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−
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>
(
C
)
<
/
s
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>
1250
<
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c
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a
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=
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−
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>
(
D
)
<
/
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p
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>
2
5625
3
<
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=
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−
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>
(
E
)
<
/
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>
1250
3
36
1
Hide problems
Square Sum of Consecutive Integers
If the sum 1 \plus{} 2 \plus{} 3 \plus{} \cdots \plus{} K is a perfect square
N
2
N^2
N
2
and if
N
N
N
is less than
100
100
100
, then the possible values for
K
K
K
are:
<
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(
A
)
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only
1
<
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=
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−
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>
(
B
)
<
/
s
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a
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>
1
and
8
<
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a
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s
=
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x
−
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>
(
C
)
<
/
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>
only
8
<
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a
s
s
=
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t
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x
−
b
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′
>
(
D
)
<
/
s
p
a
n
>
8
and
49
<
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p
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l
a
s
s
=
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a
t
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x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
1
,
8
,
and
49
<span class='latex-bold'>(A)</span>\ \text{only }1 \qquad<span class='latex-bold'>(B)</span>\ 1\text{ and }8 \qquad<span class='latex-bold'>(C)</span>\ \text{only }8 \qquad<span class='latex-bold'>(D)</span>\ 8\text{ and }49 \qquad<span class='latex-bold'>(E)</span>\ 1,8,\text{ and }49
<
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=
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x
−
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(
A
)
<
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>
only
1
<
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p
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c
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a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
1
and
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
an
>
only
8
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
8
and
49
<
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c
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a
ss
=
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a
t
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x
−
b
o
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d
′
>
(
E
)
<
/
s
p
an
>
1
,
8
,
and
49
35
1
Hide problems
Rhombus of Radii and Chords
A rhombus is formed by two radii and two chords of a circle whose radius is
16
16
16
feet. The area of the rhombus in square feet is:
<
s
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l
a
s
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=
′
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a
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x
−
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o
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>
(
A
)
<
/
s
p
a
n
>
128
<
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c
l
a
s
s
=
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a
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x
−
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>
(
B
)
<
/
s
p
a
n
>
128
3
<
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
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a
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>
256
<
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c
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a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
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a
n
>
512
<
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c
l
a
s
s
=
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−
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>
(
E
)
<
/
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a
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>
512
3
<span class='latex-bold'>(A)</span>\ 128 \qquad<span class='latex-bold'>(B)</span>\ 128\sqrt {3} \qquad<span class='latex-bold'>(C)</span>\ 256 \qquad<span class='latex-bold'>(D)</span>\ 512 \qquad<span class='latex-bold'>(E)</span>\ 512\sqrt {3}
<
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p
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c
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a
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=
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−
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(
A
)
<
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>
128
<
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a
ss
=
′
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a
t
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x
−
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o
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d
′
>
(
B
)
<
/
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>
128
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
C
)
<
/
s
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an
>
256
<
s
p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
an
>
512
<
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a
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=
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a
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x
−
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′
>
(
E
)
<
/
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>
512
3
34
1
Hide problems
Divisibility of n^2(n^2-1)
If
n
n
n
is any whole number, n^2(n^2 \minus{} 1) is always divisible by
(A)
\ 12 \qquad
(B)
\ 24 \qquad
(C)
\ \text{any multiple of }12 \qquad
(D)
\ 12 \minus{} n \qquad
(E)
\ 12\text{ and }24
33
1
Hide problems
Decimal Representation of sqrt{2}
The number
2
\sqrt {2}
2
is equal to:
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s
=
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−
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(
A
)
<
/
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a
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>
a rational fraction
<
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a
s
s
=
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a
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x
−
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>
(
B
)
<
/
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a
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>
a finite decimal
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1.41421
<span class='latex-bold'>(A)</span>\ \text{a rational fraction} \qquad<span class='latex-bold'>(B)</span>\ \text{a finite decimal} \qquad<span class='latex-bold'>(C)</span>\ 1.41421
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a rational fraction
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a finite decimal
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1.41421
(D)
\ \text{an infinite repeating decimal} \qquad
(E)
\ \text{an infinite non \minus{} repeating decimal}
32
1
Hide problems
Racing in a Pool
George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time?
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9
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 4\frac {1}{2} \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 7\frac {1}{2} \qquad<span class='latex-bold'>(E)</span>\ 9
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9
31
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Counting in Base 4
In our number system the base is ten. If the base were changed to four you would count as follows:
1
,
2
,
3
,
10
,
11
,
12
,
13
,
20
,
21
,
22
,
23
,
30
,
…
1,2,3,10,11,12,13,20,21,22,23,30,\ldots
1
,
2
,
3
,
10
,
11
,
12
,
13
,
20
,
21
,
22
,
23
,
30
,
…
The twentieth number would be:
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38
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110
<span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 38 \qquad<span class='latex-bold'>(C)</span>\ 44 \qquad<span class='latex-bold'>(D)</span>\ 104 \qquad<span class='latex-bold'>(E)</span>\ 110
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30
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Hide problems
Altitude of Equilateral Triangle
If the altitude of an equilateral triangle is
6
\sqrt {6}
6
, then the area is:
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12
<span class='latex-bold'>(A)</span>\ 2\sqrt {2} \qquad<span class='latex-bold'>(B)</span>\ 2\sqrt {3} \qquad<span class='latex-bold'>(C)</span>\ 3\sqrt {3} \qquad<span class='latex-bold'>(D)</span>\ 6\sqrt {2} \qquad<span class='latex-bold'>(E)</span>\ 12
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29
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Hide problems
Connecting Points of Intersection
The points of intersection of xy \equal{} 12 and x^2 \plus{} y^2 \equal{} 25 are joined in succession. The resulting figure is:
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a straight line
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an equilateral triangle
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a parallelogram
<span class='latex-bold'>(A)</span>\ \text{a straight line} \qquad<span class='latex-bold'>(B)</span>\ \text{an equilateral triangle} \qquad<span class='latex-bold'>(C)</span>\ \text{a parallelogram}
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a straight line
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an equilateral triangle
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a parallelogram
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a square
<span class='latex-bold'>(D)</span>\ \text{a rectangle} \qquad<span class='latex-bold'>(E)</span>\ \text{a square}
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a square
28
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Hide problems
Mr. J's Estate
Mr. J left his entire estate to his wife, his daughter, his son, and the cook. His daughter and son got half the estate, sharing in the ratio of
4
4
4
to
3
3
3
. His wife got twice as much as the son. If the cook received a bequest of
$
500
\$500
$500
, then the entire estate was:
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$
3500
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$
5500
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$
6500
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$
7000
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$
7500
<span class='latex-bold'>(A)</span>\ \$3500 \qquad<span class='latex-bold'>(B)</span>\ \$5500 \qquad<span class='latex-bold'>(C)</span>\ \$6500 \qquad<span class='latex-bold'>(D)</span>\ \$7000 \qquad<span class='latex-bold'>(E)</span>\ \$7500
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$3500
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$5500
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$6500
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$7000
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$7500
27
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Doubling Triangle's Sides
If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by:
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more than
6
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 6 \qquad<span class='latex-bold'>(E)</span>\ \text{more than }6
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6
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more than
6
26
1
Hide problems
Uniquely Determining A Triangle
Which one of the following combinations of given parts does not determine the indicated triangle?
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base angle and vertex angle; isosceles triangle
<span class='latex-bold'>(A)</span>\ \text{base angle and vertex angle; isosceles triangle}
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base angle and vertex angle; isosceles triangle
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<
/
s
p
a
n
>
vertex angle and the base; isosceles triangle
<span class='latex-bold'>(B)</span>\ \text{vertex angle and the base; isosceles triangle}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
vertex angle and the base; isosceles triangle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
the radius of the circumscribed circle; equilateral triangle
<span class='latex-bold'>(C)</span>\ \text{the radius of the circumscribed circle; equilateral triangle}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
the radius of the circumscribed circle; equilateral triangle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
one arm and the radius of the inscribed circle; right triangle
<span class='latex-bold'>(D)</span>\ \text{one arm and the radius of the inscribed circle; right triangle}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
one arm and the radius of the inscribed circle; right triangle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
two angles and a side opposite one of them; scalene triangle
<span class='latex-bold'>(E)</span>\ \text{two angles and a side opposite one of them; scalene triangle}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
two angles and a side opposite one of them; scalene triangle
25
1
Hide problems
Sum of Odd Numbers
The sum of all numbers of the form 2k \plus{} 1, where
k
k
k
takes on integral values from
1
1
1
to
n
n
n
is:
(A)
\ n^2 \qquad
(B)
\ n(n \plus{} 1) \qquad
(C)
\ n(n \plus{} 2) \qquad
(D)
\ (n \plus{} 1)^2 \qquad
(E)
\ (n \plus{} 1)(n \plus{} 2)
24
1
Hide problems
Triangle With Angles
In the figure \overline{AB} \equal{} \overline{AC}, angle BAD \equal{} 30^{\circ}, and \overline{AE} \equal{} \overline{AD}. [asy]unitsize(20); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(3,3),B=(0,0),C=(6,0),D=(2,0),E=(5,1); draw(A--B--C--cycle); draw(A--D--E); label("
A
A
A
",A,N); label("
B
B
B
",B,W); label("
C
C
C
",C,E); label("
D
D
D
",D,S); label("
E
E
E
",E,NE);[/asy]Then angle
C
D
E
CDE
C
D
E
equals:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
7
1
2
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
0
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
12
1
2
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
1
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
2
0
∘
<span class='latex-bold'>(A)</span>\ 7\frac {1}{2}^{\circ} \qquad<span class='latex-bold'>(B)</span>\ 10^{\circ} \qquad<span class='latex-bold'>(C)</span>\ 12\frac {1}{2}^{\circ} \qquad<span class='latex-bold'>(D)</span>\ 15^{\circ} \qquad<span class='latex-bold'>(E)</span>\ 20^{\circ}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
7
2
1
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
0
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
12
2
1
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
1
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2
0
∘
23
1
Hide problems
Zero Discriminant
About the equation ax^2 \minus{} 2x\sqrt {2} \plus{} c \equal{} 0, with
a
a
a
and
c
c
c
real constants, we are told that the discriminant is zero. The roots are necessarily:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
equal and integral
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
equal and rational
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
equal and real
<span class='latex-bold'>(A)</span>\ \text{equal and integral} \qquad<span class='latex-bold'>(B)</span>\ \text{equal and rational} \qquad<span class='latex-bold'>(C)</span>\ \text{equal and real}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
equal and integral
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
equal and rational
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
equal and real
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
equal and irrational
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
equal and imaginary
<span class='latex-bold'>(D)</span>\ \text{equal and irrational} \qquad<span class='latex-bold'>(E)</span>\ \text{equal and imaginary}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
equal and irrational
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
equal and imaginary
22
1
Hide problems
Jones' Trip
Jones covered a distance of
50
50
50
miles on his first trip. On a later trip he traveled
300
300
300
miles while going three times as fast. His new time compared with the old time was:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
three times as much
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
twice as much
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
the same
<span class='latex-bold'>(A)</span>\ \text{three times as much} \qquad<span class='latex-bold'>(B)</span>\ \text{twice as much} \qquad<span class='latex-bold'>(C)</span>\ \text{the same}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
three times as much
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
twice as much
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
the same
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
half as much
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
a third as much
<span class='latex-bold'>(D)</span>\ \text{half as much} \qquad<span class='latex-bold'>(E)</span>\ \text{a third as much}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
half as much
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
a third as much
21
1
Hide problems
Two Lines and a Hyperbola
If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
or
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
or
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
or
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
2
,
3
,
or
4
<span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 2\text{ or }3 \qquad<span class='latex-bold'>(C)</span>\ 2\text{ or }4 \qquad<span class='latex-bold'>(D)</span>\ 3\text{ or }4 \qquad<span class='latex-bold'>(E)</span>\ 2,3,\text{ or }4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
or
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
or
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
or
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2
,
3
,
or
4
20
1
Hide problems
Estimating x
If (0.2)^x \equal{} 2 and \log 2 \equal{} 0.3010, then the value of
x
x
x
to the nearest tenth is:
(A)
\ \minus{} 10.0 \qquad
(B)
\ \minus{} 0.5 \qquad
(C)
\ \minus{} 0.4 \qquad
(D)
\ \minus{} 0.2 \qquad
(E)
\ 10.0
19
1
Hide problems
Two Burning Candles
Two candles of the same height are lighted at the same time. The first is consumed in
4
4
4
hours and the second in
3
3
3
hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
1
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2
2
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
2
1
2
<span class='latex-bold'>(A)</span>\ \frac {3}{4} \qquad<span class='latex-bold'>(B)</span>\ 1\frac {1}{2} \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 2\frac {2}{5} \qquad<span class='latex-bold'>(E)</span>\ 2\frac {1}{2}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
2
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
5
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2
2
1
18
1
Hide problems
Power of 10
If 10^{2y} \equal{} 25, then 10^{ \minus{} y} equals:
(A)
\ \minus{} \frac {1}{5} \qquad
(B)
\ \frac {1}{625} \qquad
(C)
\ \frac {1}{50} \qquad
(D)
\ \frac {1}{25} \qquad
(E)
\ \frac {1}{5}
17
1
Hide problems
Partial Fractions
The fraction \frac {5x \minus{} 11}{2x^2 \plus{} x \minus{} 6} was obtained by adding the two fractions \frac {A}{x \plus{} 2} and \frac {B}{2x \minus{} 3}. The values of
A
A
A
and
B
B
B
must be, respectively:
(A)
\ 5x, \minus{} 11 \qquad
(B)
\ \minus{} 11,5x \qquad
(C)
\ \minus{} 1,3 \qquad
(D)
\ 3, \minus{} 1 \qquad
(E)
\ 5, \minus{} 11
16
1
Hide problems
Three Numbers With Sum 98
The sum of three numbers is
98
98
98
. The ratio of the first to the second is
2
3
\frac {2}{3}
3
2
, and the ratio of the second to the third is
5
8
\frac {5}{8}
8
5
. The second number is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
15
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
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>
20
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(
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30
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32
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33
<span class='latex-bold'>(A)</span>\ 15 \qquad<span class='latex-bold'>(B)</span>\ 20 \qquad<span class='latex-bold'>(C)</span>\ 30 \qquad<span class='latex-bold'>(D)</span>\ 32 \qquad<span class='latex-bold'>(E)</span>\ 33
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(
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)
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15
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20
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30
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32
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33
15
1
Hide problems
Root(s) of a Fractional Equation
The root(s) of \frac {15}{x^2 \minus{} 4} \minus{} \frac {2}{x \minus{} 2} \equal{} 1 is (are):
(A)
\ \minus{} 5\text{ and }3 \qquad
(B)
\ \pm 2 \qquad
(C)
\ 2\text{ only} \qquad
(D)
\ \minus{} 3\text{ and }5 \qquad
(E)
\ 3\text{ only}
14
1
Hide problems
Points A,B,C on Circle O
The points
A
,
B
,
C
A,B,C
A
,
B
,
C
are on a circle
O
O
O
. The tangent line at
A
A
A
and the secant
B
C
BC
BC
intersect at
P
P
P
,
B
B
B
lying between
C
C
C
and
P
P
P
. If \overline{BC} \equal{} 20 and \overline{PA} \equal{} 10\sqrt {3}, then
P
B
‾
\overline{PB}
PB
equals:
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5
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10
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10
3
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20
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30
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 10 \qquad<span class='latex-bold'>(C)</span>\ 10\sqrt {3} \qquad<span class='latex-bold'>(D)</span>\ 20 \qquad<span class='latex-bold'>(E)</span>\ 30
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5
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(
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)
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10
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10
3
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20
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30
13
1
Hide problems
Percent Less
Given two positive integers
x
x
x
and
y
y
y
with
x
<
y
x < y
x
<
y
. The percent that
x
x
x
is less than
y
y
y
is:
(A)
\ \frac {100(y \minus{} x)}{x} \qquad
(B)
\ \frac {100(x \minus{} y)}{x} \qquad
(C)
\ \frac {100(y \minus{} x)}{y} \qquad
(D)
\ 100(y \minus{} x)
(E)
\ 100(x \minus{} y)
12
1
Hide problems
Dividing x^{-1}-1 by x-1
If x^{ \minus{} 1} \minus{} 1 is divided by x \minus{} 1 the quotient is:
(A)
\ 1 \qquad
(B)
\ \frac {1}{x \minus{} 1} \qquad
(C)
\ \frac { \minus{} 1}{x \minus{} 1} \qquad
(D)
\ \frac {1}{x} \qquad
(E)
\ \minus{} \frac {1}{x}
11
1
Hide problems
Simplifying Radical Fractions
The expression 1 \minus{} \frac {1}{1 \plus{} \sqrt {3}} \plus{} \frac {1}{1 \minus{} \sqrt {3}} equals:
(A)
\ 1 \minus{} \sqrt {3} \qquad
(B)
\ 1 \qquad
(C)
\ \minus{} \sqrt {3} \qquad
(D)
\ \sqrt {3} \qquad
(E)
\ 1 \plus{} \sqrt {3}
10
1
Hide problems
Circle with Center C
A circle of radius
10
10
10
inches has its center at the vertex
C
C
C
of an equilateral triangle
A
B
C
ABC
A
BC
and passes through the other two vertices. The side
A
C
AC
A
C
extended through
C
C
C
intersects the circle at
D
D
D
. The number of degrees of angle
A
D
B
ADB
A
D
B
is:
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120
<span class='latex-bold'>(A)</span>\ 15 \qquad<span class='latex-bold'>(B)</span>\ 30 \qquad<span class='latex-bold'>(C)</span>\ 60 \qquad<span class='latex-bold'>(D)</span>\ 90 \qquad<span class='latex-bold'>(E)</span>\ 120
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)
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15
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(
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30
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(
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)
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60
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(
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)
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90
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>
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)
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120
9
1
Hide problems
Simplifying Power of Radical of Radical of Power
Simplify
[
a
9
6
3
]
4
[
a
9
3
6
]
4
\left[ \sqrt [3]{\sqrt [6]{a^9}} \right]^4\left[ \sqrt [6]{\sqrt [3]{a^9}} \right]^4
[
3
6
a
9
]
4
[
6
3
a
9
]
4
; the result is:
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16
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12
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8
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4
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2
<span class='latex-bold'>(A)</span>\ a^{16} \qquad<span class='latex-bold'>(B)</span>\ a^{12} \qquad<span class='latex-bold'>(C)</span>\ a^8 \qquad<span class='latex-bold'>(D)</span>\ a^4 \qquad<span class='latex-bold'>(E)</span>\ a^2
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12
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8
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a
2
8
1
Hide problems
Exponential x,y
If 8\cdot2^x \equal{} 5^{y \plus{} 8}, then when y \equal{} \minus{} 8,x \equal{}
(A)
\ \minus{} 4 \qquad
(B)
\ \minus{} 3 \qquad
(C)
\ 0 \qquad
(D)
\ 4 \qquad
(E)
\ 8
7
1
Hide problems
Reciprocal Roots of a Quadratic
The roots of the equation ax^2 \plus{} bx \plus{} c \equal{} 0 will be reciprocal if:
(A)
\ a \equal{} b \qquad
(B)
\ a \equal{} bc \qquad
(C)
\ c \equal{} a \qquad
(D)
\ c \equal{} b \qquad
(E)
\ c \equal{} ab
6
1
Hide problems
Cows and Chickens
In a group of cows and chickens, the number of legs was
14
14
14
more than twice the number of heads. The number of cows was:
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14
<span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 14
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10
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5
1
Hide problems
Nickles on a Table
A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:
<
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)
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>
4
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>
(
B
)
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>
5
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>
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C
)
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>
6
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)
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>
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<span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 8 \qquad<span class='latex-bold'>(E)</span>\ 12
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)
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(
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)
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(
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)
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6
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4
1
Hide problems
Investing 10,000 Dollars
A man has
$
10
,
000
\$10,000
$10
,
000
to invest. He invests
$
4000
\$4000
$4000
at
5
%
5\%
5%
and
$
3500
\$3500
$3500
at
4
%
4\%
4%
. In order to have a yearly income of
$
500
\$500
$500
, he must invest the remainder at:
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o
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(
A
)
<
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6
%
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e
x
−
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(
B
)
<
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6.1
%
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c
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a
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a
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e
x
−
b
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(
C
)
<
/
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>
6.2
%
<
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c
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a
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=
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a
t
e
x
−
b
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d
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(
D
)
<
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>
6.3
%
<
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(
E
)
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6.4
%
<span class='latex-bold'>(A)</span>\ 6\% \qquad<span class='latex-bold'>(B)</span>\ 6.1\% \qquad<span class='latex-bold'>(C)</span>\ 6.2\% \qquad<span class='latex-bold'>(D)</span>\ 6.3\% \qquad<span class='latex-bold'>(E)</span>\ 6.4\%
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a
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(
A
)
<
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6%
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a
ss
=
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a
t
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x
−
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′
>
(
B
)
<
/
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p
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>
6.1%
<
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p
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l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6.2%
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
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>
6.3%
<
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a
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′
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a
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x
−
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(
E
)
<
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6.4%
3
1
Hide problems
Light's Distance over 100 Years
The distance light travels in one year is approximately
5
,
870
,
000
,
000
,
000
5,870,000,000,000
5
,
870
,
000
,
000
,
000
miles. The distance light travels in
100
100
100
years is:
(A)
\ 587 * 10^8 \text{ miles} \qquad
(B)
\ 587 * 10^{10} \text{ miles} \qquad
(C)
\ 587*10^{ \minus{} 10} \text{ miles}
(D)
\ 587 * 10^{12} \text{ miles} \qquad
(E)
\ 587* 10^{ \minus{} 12} \text{ miles}
2
1
Hide problems
Mr. Jones' Pipes
Mr. Jones sold two pipes at
$
1.20
\$ 1.20
$1.20
each. Based on the cost, his profit one was
20
%
20 \%
20%
and his loss on the other was
20
%
20 \%
20%
. On the sale of the pipes, he:
<
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A
)
<
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broke even
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−
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(
B
)
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lost
4
cents
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=
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x
−
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C
)
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gained
4
cents
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(
D
)
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/
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lost
10
cents
<
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(
E
)
<
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>
gained
10
cents
<span class='latex-bold'>(A)</span>\ \text{broke even} \qquad<span class='latex-bold'>(B)</span>\ \text{lost } 4\text{ cents} \qquad<span class='latex-bold'>(C)</span>\ \text{gained } 4\text{ cents} \qquad<span class='latex-bold'>(D)</span>\ \text{lost } 10 \text{ cents} \qquad<span class='latex-bold'>(E)</span>\ \text{gained } 10 \text{ cents}
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A
)
<
/
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>
broke even
<
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=
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a
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x
−
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(
B
)
<
/
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p
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>
lost
4
cents
<
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p
an
c
l
a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
(
C
)
<
/
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p
an
>
gained
4
cents
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
an
>
lost
10
cents
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
gained
10
cents
1
1
Hide problems
Expression With x's
The value of x \plus{} x(x^x) when x \equal{} 2 is:
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A
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10
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16
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C
)
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18
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(
D
)
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36
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64
<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 16 \qquad<span class='latex-bold'>(C)</span>\ 18 \qquad<span class='latex-bold'>(D)</span>\ 36 \qquad<span class='latex-bold'>(E)</span>\ 64
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10
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(
B
)
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16
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(
C
)
<
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18
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D
)
<
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36
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E
)
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64