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Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1962 AMC 12/AHSME
1962 AMC 12/AHSME
Part of
AMC 12/AHSME
Subcontests
(40)
36
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Diophantine Equation
If both
x
x
x
and
y
y
y
are both integers, how many pairs of solutions are there of the equation (x\minus{}8)(x\minus{}10) \equal{} 2^y?
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more than 3
<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ \text{more than 3}
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more than 3
40
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Limiting Sum of an Infinite Series
The limiting sum of the infinite series, \frac{1}{10} \plus{} \frac{2}{10^2} \plus{} \frac{3}{10^3} \plus{} \dots whose
n
n
n
th term is
n
1
0
n
\frac{n}{10^n}
1
0
n
n
is:
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9
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8
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larger than any finite quantity
<span class='latex-bold'>(A)</span>\ \frac19 \qquad <span class='latex-bold'>(B)</span>\ \frac{10}{81} \qquad <span class='latex-bold'>(C)</span>\ \frac18 \qquad <span class='latex-bold'>(D)</span>\ \frac{17}{72} \qquad <span class='latex-bold'>(E)</span>\ \text{larger than any finite quantity}
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larger than any finite quantity
39
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Length of a Median
Two medians of a triangle with unequal sides are
3
3
3
inches and
6
6
6
inches. Its area is
3
15
3 \sqrt{15}
3
15
square inches. The length of the third median in inches, is:
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<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 3 \sqrt{3} \qquad <span class='latex-bold'>(C)</span>\ 3 \sqrt{6} \qquad <span class='latex-bold'>(D)</span>\ 6 \sqrt{3} \qquad <span class='latex-bold'>(E)</span>\ 6 \sqrt{6}
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38
1
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Perfect Square Population
The population of Nosuch Junction at one time was a perfect square. Later, with an increase of 100, the population was one more than a perfect square. Now, with an additional increase of 100, the population is again a perfect square. The original population is a multiple of:
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<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 7 \qquad <span class='latex-bold'>(C)</span>\ 9 \qquad <span class='latex-bold'>(D)</span>\ 11 \qquad <span class='latex-bold'>(E)</span>\ 17
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Maximization of the Area of a Quadrilateral
A
B
C
D
ABCD
A
BC
D
is a square with side of unit length. Points
E
E
E
and
F
F
F
are taken respectively on sides
A
B
AB
A
B
and
A
D
AD
A
D
so that AE \equal{} AF and the quadrilateral
C
D
F
E
CDFE
C
D
FE
has maximum area. In square units this maximum area is:
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3
<span class='latex-bold'>(A)</span>\ \frac12 \qquad <span class='latex-bold'>(B)</span>\ \frac {9}{16} \qquad <span class='latex-bold'>(C)</span>\ \frac{19}{32} \qquad <span class='latex-bold'>(D)</span>\ \frac {5}{8} \qquad <span class='latex-bold'>(E)</span>\ \frac23
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2
35
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Angle Measure on a Clock
A man on his way to dinner short after
6
:
00
6: 00
6
:
00
p.m. observes that the hands of his watch form an angle of
11
0
∘
.
110^{\circ}.
11
0
∘
.
Returning before
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00
7: 00
7
:
00
p.m. he notices that again the hands of his watch form an angle of
11
0
∘
.
110^{\circ}.
11
0
∘
.
The number of minutes that he has been away is:
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3
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<span class='latex-bold'>(A)</span>\ 36 \frac23 \qquad <span class='latex-bold'>(B)</span>\ 40 \qquad <span class='latex-bold'>(C)</span>\ 42 \qquad <span class='latex-bold'>(D)</span>\ 42.4 \qquad <span class='latex-bold'>(E)</span>\ 45
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Real Roots of a Quadratic
For what real values of
K
K
K
does x \equal{} K^2 (x\minus{}1)(x\minus{}2) have real roots?
(A)
\ \text{none} \qquad
(B)
\ \minus{}2
(C)
\ \minus{}2 \sqrt{2} < K < 2 \sqrt{2} \qquad
(D)
\ K>1 \text{ or } K<\minus{}2 \qquad
(E)
\ \text{all}
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Inequality
The set of
x
x
x
-values satisfying the inequality 2 \leq |x\minus{}1| \leq 5 is:
(A)
\ \minus{}4 \leq x \leq \minus{}1 \text{ or } 3 \leq x \leq 6 \qquad
(B)
\ 3 \leq x \leq 6 \text{ or } \minus{}6 \leq x \leq \minus{}3 \qquad
(C)
\ x \leq \minus{}1 \text{ or } x \geq 3 \qquad
(D)
\ \minus{}1 \leq x \leq 3 \qquad
(E)
\ \minus{}4 \leq x \leq 6
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Recursion
If x_{k\plus{}1} \equal{} x_k \plus{} \frac12 for k\equal{}1, 2, \dots, n\minus{}1 and x_1\equal{}1, find x_1 \plus{} x_2 \plus{} \dots \plus{} x_n.
(A)
\ \frac{n\plus{}1}{2} \qquad
(B)
\ \frac{n\plus{}3}{2} \qquad
(C)
\ \frac{n^2\minus{}1}{2} \qquad
(D)
\ \frac{n^2\plus{}n}{4} \qquad
(E)
\ \frac{n^2\plus{}3n}{4}
31
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Ratio of Interior Angles
The ratio of the interior angles of two regular polygons with sides of unit length is
3
:
2
3: 2
3
:
2
. How many such pairs are there?
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e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
s
p
a
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
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
infinitely many
<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>\ \text{infinitely many}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
infinitely many
30
1
Hide problems
Truth Value of Statements
Consider the statements:
<
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o
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>
(
1
)
<
/
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a
n
>
p and q are both true
<
s
p
a
n
c
l
a
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s
=
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l
a
t
e
x
−
b
o
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′
>
(
2
)
<
/
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a
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>
p is true and q is false
<
s
p
a
n
c
l
a
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s
=
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t
e
x
−
b
o
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d
′
>
(
3
)
<
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a
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>
p is false and q is true
<
s
p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
4
)
<
/
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a
n
>
p is false and q is false.
<span class='latex-bold'>(1)</span>\ \text{p and q are both true} \qquad <span class='latex-bold'>(2)</span>\ \text{p is true and q is false} \qquad <span class='latex-bold'>(3)</span>\ \text{p is false and q is true} \qquad <span class='latex-bold'>(4)</span>\ \text{p is false and q is false.}
<
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p
an
c
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a
ss
=
′
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a
t
e
x
−
b
o
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d
′
>
(
1
)
<
/
s
p
an
>
p and q are both true
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
2
)
<
/
s
p
an
>
p is true and q is false
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
3
)
<
/
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an
>
p is false and q is true
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
4
)
<
/
s
p
an
>
p is false and q is false.
How many of these imply the negative of the statement "p and q are both true?"
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
<
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
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
4
<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
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
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
29
1
Hide problems
Quadratic Inequality
Which of the following sets of
x
x
x
-values satisfy the inequality 2x^2 \plus{} x < 6?
(A)
\ \minus{} 2 < x < \frac{3}{2} \qquad
(B)
\ x > \frac32 \text{ or }x < \minus{} 2 \qquad
(C)
\ x < \frac32 \qquad
(D)
\ \frac32 < x < 2 \qquad
(E)
\ x < \minus{} 2
28
1
Hide problems
Algebraic Equation
The set of
x
x
x
-values satisfying the equation x^{\log_{10} x} \equal{} \frac{x^3}{100} consists of:
<
s
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a
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l
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−
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>
(
A
)
<
/
s
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a
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>
1
10
<
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n
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l
a
s
s
=
′
l
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t
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x
−
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o
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′
>
(
B
)
<
/
s
p
a
n
>
10, only
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
100, only
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
10 or 100, only
<
s
p
a
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
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a
n
>
more than two real numbers.
<span class='latex-bold'>(A)</span>\ \frac{1}{10} \qquad <span class='latex-bold'>(B)</span>\ \text{10, only} \qquad <span class='latex-bold'>(C)</span>\ \text{100, only} \qquad <span class='latex-bold'>(D)</span>\ \text{10 or 100, only} \qquad <span class='latex-bold'>(E)</span>\ \text{more than two real numbers.}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
10
1
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
10, only
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
100, only
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
10 or 100, only
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
more than two real numbers.
27
1
Hide problems
Binary Operator
Let
a
@
b
a @ b
a
@
b
represent the operation on two numbers,
a
a
a
and
b
b
b
, which selects the larger of the two numbers, with a@a \equal{} a. Let
a
!
b
a ! b
a
!
b
represent the operator which selects the smaller of the two numbers, with a ! a \equal{} a. Which of the following three rules is (are) correct?
(1)
\ a@b \equal{} b@a \qquad
(2)
\ a@(b@c) \equal{} (a@b)@c \qquad
(3)
\ a ! (b@c) \equal{} (a ! b) @ (a ! c)
<
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>
(
A
)
<
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>
(
1
)
only
<
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l
a
s
s
=
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x
−
b
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>
(
B
)
<
/
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p
a
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>
(
2
)
only
<
s
p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
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>
(
C
)
<
/
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>
(1) and (2) only
<
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p
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c
l
a
s
s
=
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a
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e
x
−
b
o
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′
>
(
D
)
<
/
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a
n
>
(1) and (3) only
<
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c
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a
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s
=
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−
b
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>
(
E
)
<
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>
all three
<span class='latex-bold'>(A)</span>\ (1)\text{ only} \qquad <span class='latex-bold'>(B)</span>\ (2) \text{ only} \qquad <span class='latex-bold'>(C)</span>\ \text{(1) and (2) only} \qquad <span class='latex-bold'>(D)</span>\ \text{(1) and (3) only} \qquad <span class='latex-bold'>(E)</span>\ \text{all three}
<
s
p
an
c
l
a
ss
=
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x
−
b
o
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>
(
A
)
<
/
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p
an
>
(
1
)
only
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
(
2
)
only
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
(1) and (2) only
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
(1) and (3) only
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
all three
26
1
Hide problems
Maximization of an Expression
For any real value of
x
x
x
the maximum value of 8x \minus{} 3x^2 is:
<
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a
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c
l
a
s
s
=
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−
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(
A
)
<
/
s
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a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
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>
(
B
)
<
/
s
p
a
n
>
8
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
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a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
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a
n
>
5
<
s
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a
s
s
=
′
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x
−
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>
(
E
)
<
/
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a
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>
16
3
<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ \frac83 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ \frac{16}{3}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
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>
(
A
)
<
/
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an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
3
8
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
<
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
an
>
5
<
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p
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c
l
a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
(
E
)
<
/
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p
an
>
3
16
25
1
Hide problems
Radius of a Circle
Given square
A
B
C
D
ABCD
A
BC
D
with side
8
8
8
feet. A circle is drawn through vertices
A
A
A
and
D
D
D
and tangent to side
B
C
.
BC.
BC
.
The radius of the circle, in feet, is:
<
s
p
a
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c
l
a
s
s
=
′
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(
A
)
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>
4
<
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c
l
a
s
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=
′
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x
−
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o
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′
>
(
B
)
<
/
s
p
a
n
>
4
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
5
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
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>
6
<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 4 \sqrt{2} \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 5 \sqrt{2} \qquad <span class='latex-bold'>(E)</span>\ 6
<
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c
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=
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−
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>
(
A
)
<
/
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>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
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an
>
5
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
6
24
1
Hide problems
Three Machines
Three machines
P, Q, and R,
\text{P, Q, and R,}
P, Q, and R,
working together, can do a job in
x
x
x
hours. When working alone,
P
\text{P}
P
needs an additional
6
6
6
hours to do the job;
Q
\text{Q}
Q
, one additional hour; and
R
R
R
,
x
x
x
additional hours. The value of
x
x
x
is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
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>
(
A
)
<
/
s
p
a
n
>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
11
12
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
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
<
s
p
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n
c
l
a
s
s
=
′
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a
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x
−
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d
′
>
(
E
)
<
/
s
p
a
n
>
3
<span class='latex-bold'>(A)</span>\ \frac23 \qquad <span class='latex-bold'>(B)</span>\ \frac{11}{12} \qquad <span class='latex-bold'>(C)</span>\ \frac32 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ 3
<
s
p
an
c
l
a
ss
=
′
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a
t
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x
−
b
o
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d
′
>
(
A
)
<
/
s
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an
>
3
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
12
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
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a
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−
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o
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d
′
>
(
E
)
<
/
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>
3
23
1
Hide problems
Length of a Segment
In triangle
A
B
C
ABC
A
BC
,
C
D
CD
C
D
is the altitude to
A
B
AB
A
B
and
A
E
AE
A
E
is the altitude to
B
C
.
BC.
BC
.
If the lengths of
A
B
,
C
D
,
AB, CD,
A
B
,
C
D
,
and
A
E
AE
A
E
are known, the length of
D
B
DB
D
B
is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
not determined by the information given
<span class='latex-bold'>(A)</span>\ \text{not determined by the information given} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
not determined by the information given
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
determined only if A is an acute angle
<span class='latex-bold'>(B)</span>\ \text{determined only if A is an acute angle} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
determined only if A is an acute angle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
determined only if B is an acute angle
<span class='latex-bold'>(C)</span>\ \text{determined only if B is an acute angle} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
determined only if B is an acute angle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
determined only in ABC is an acute triangle
<span class='latex-bold'>(D)</span>\ \text{determined only in ABC is an acute triangle} \qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
determined only in ABC is an acute triangle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
none of these is correct
<span class='latex-bold'>(E)</span>\ \text{none of these is correct}
<
s
p
an
c
l
a
ss
=
′
l
a
t
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−
b
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d
′
>
(
E
)
<
/
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an
>
none of these is correct
22
1
Hide problems
Integral Base b
The number
12
1
b
121_b
12
1
b
, written in the integral base
b
b
b
, is the square of an integer, for
(A)
\ b \equal{} 10,\text{ only} \qquad
(B)
\ b \equal{} 10 \text{ and } b \equal{} 5, \text{ only} \qquad
(C)
\ 2 \leq b \leq 10 \qquad
(D)
\ b > 2 \qquad
(E)
\ \text{no value of }b
21
1
Hide problems
Root of a Quadratic
It is given that one root of 2x^2 \plus{} rx \plus{} s \equal{} 0, with
r
r
r
and
s
s
s
real numbers, is 3\plus{}2i (i \equal{} \sqrt{\minus{}1}). The value of
s
s
s
is:
(A)
\ \text{undetermined} \qquad
(B)
\ 5 \qquad
(C)
\ 6 \qquad
(D)
\ \minus{}13 \qquad
(E)
\ 26
20
1
Hide problems
Angles of a Pentagon in an Arithmetic Progression
The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:
<
s
p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
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′
>
(
A
)
<
/
s
p
a
n
>
108
<
s
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c
l
a
s
s
=
′
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a
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e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
a
n
>
90
<
s
p
a
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c
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a
s
s
=
′
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a
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e
x
−
b
o
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d
′
>
(
C
)
<
/
s
p
a
n
>
72
<
s
p
a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
a
n
>
54
<
s
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a
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l
a
s
s
=
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>
(
E
)
<
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>
36
<span class='latex-bold'>(A)</span>\ 108 \qquad <span class='latex-bold'>(B)</span>\ 90 \qquad <span class='latex-bold'>(C)</span>\ 72 \qquad <span class='latex-bold'>(D)</span>\ 54 \qquad <span class='latex-bold'>(E)</span>\ 36
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
A
)
<
/
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p
an
>
108
<
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a
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=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
90
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
72
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
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o
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d
′
>
(
D
)
<
/
s
p
an
>
54
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
36
19
1
Hide problems
Parabola
If the parabola y \equal{} ax^2 \plus{} bx \plus{} c passes through the points ( \minus{} 1, 12), (0, 5), and (2, \minus{} 3), the value of a \plus{} b \plus{} c is:
(A)
\ \minus{} 4 \qquad
(B)
\ \minus{} 2 \qquad
(C)
\ 0 \qquad
(D)
\ 1 \qquad
(E)
\ 2
18
1
Hide problems
Regular Dodecagon
A regular dodecagon (
12
12
12
sides) is inscribed in a circle with radius
r
r
r
inches. The area of the dodecagon, in square inches, is:
<
s
p
a
n
c
l
a
s
s
=
′
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a
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e
x
−
b
o
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′
>
(
A
)
<
/
s
p
a
n
>
3
r
2
<
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c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
a
n
>
2
r
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
r
2
3
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
a
n
>
r
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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′
>
(
E
)
<
/
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p
a
n
>
3
r
2
3
<span class='latex-bold'>(A)</span>\ 3r^2 \qquad <span class='latex-bold'>(B)</span>\ 2r^2 \qquad <span class='latex-bold'>(C)</span>\ \frac{3r^2 \sqrt{3}}{4} \qquad <span class='latex-bold'>(D)</span>\ r^2 \sqrt{3} \qquad <span class='latex-bold'>(E)</span>\ 3r^2 \sqrt{3}
<
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a
ss
=
′
l
a
t
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x
−
b
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>
(
A
)
<
/
s
p
an
>
3
r
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
r
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
3
r
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
r
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
p
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>
3
r
2
3
17
1
Hide problems
Logarithms
If a \equal{} \log_8 225 and b \equal{} \log_2 15, then
a
a
a
, in terms of
b
,
b,
b
,
is:
<
s
p
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l
a
s
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−
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>
(
A
)
<
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>
b
2
<
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l
a
s
s
=
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−
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o
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d
′
>
(
B
)
<
/
s
p
a
n
>
2
b
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
D
)
<
/
s
p
a
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>
3
b
2
<
s
p
a
n
c
l
a
s
s
=
′
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a
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e
x
−
b
o
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d
′
>
(
E
)
<
/
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p
a
n
>
2
b
<span class='latex-bold'>(A)</span>\ \frac{b}{2} \qquad <span class='latex-bold'>(B)</span>\ \frac{2b}{3}\qquad <span class='latex-bold'>(C)</span>\ b \qquad <span class='latex-bold'>(D)</span>\ \frac{3b}{2} \qquad <span class='latex-bold'>(E)</span>\ 2b
<
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a
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=
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−
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>
(
A
)
<
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>
2
b
<
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=
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x
−
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>
(
B
)
<
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>
3
2
b
<
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c
l
a
ss
=
′
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a
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−
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>
(
C
)
<
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>
b
<
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p
an
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l
a
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=
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a
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x
−
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′
>
(
D
)
<
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>
2
3
b
<
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a
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=
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−
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′
>
(
E
)
<
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>
2
b
16
1
Hide problems
Area of a Rectangle
Given rectangle
R
1
R_1
R
1
with one side
2
2
2
inches and area
12
12
12
square inches. Rectangle
R
2
R_2
R
2
with diagonal
15
15
15
inches is similar to
R
1
.
R_1.
R
1
.
Expressed in square inches the area of
R
2
R_2
R
2
is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
9
2
<
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a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
36
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
135
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
9
10
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
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a
n
>
27
10
4
<span class='latex-bold'>(A)</span>\ \frac92 \qquad <span class='latex-bold'>(B)</span>\ 36 \qquad <span class='latex-bold'>(C)</span>\ \frac{135}{2} \qquad <span class='latex-bold'>(D)</span>\ 9 \sqrt{10} \qquad <span class='latex-bold'>(E)</span>\ \frac{27 \sqrt{10}}{4}
<
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p
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c
l
a
ss
=
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−
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>
(
A
)
<
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>
2
9
<
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p
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c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
36
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
135
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
9
10
<
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27
10
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Moving Vertex on a Triangle
Given triangle
A
B
C
ABC
A
BC
with base
A
B
AB
A
B
fixed in length and position. As the vertex
C
C
C
moves on a straight line, the intersection point of the three medians moves on:
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a circle
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a parabola
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an ellipse
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a straight line
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a curve here not listed
<span class='latex-bold'>(A)</span>\ \text{a circle} \qquad <span class='latex-bold'>(B)</span>\ \text{a parabola} \qquad <span class='latex-bold'>(C)</span>\ \text{an ellipse} \qquad <span class='latex-bold'>(D)</span>\ \text{a straight line} \qquad <span class='latex-bold'>(E)</span>\ \text{a curve here not listed}
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a circle
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a parabola
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an ellipse
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a straight line
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a curve here not listed
14
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Infinite Geometric Series
Let
s
s
s
be the limiting sum of the geometric series 4\minus{} \frac83 \plus{} \frac{16}{9} \minus{} \dots, as the number of terms increases without bound. Then
s
s
s
equals:
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a number between 0 and 1
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2.4
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2.5
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3.6
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12
<span class='latex-bold'>(A)</span>\ \text{a number between 0 and 1} \qquad <span class='latex-bold'>(B)</span>\ 2.4 \qquad <span class='latex-bold'>(C)</span>\ 2.5 \qquad <span class='latex-bold'>(D)</span>\ 3.6 \qquad <span class='latex-bold'>(E)</span>\ 12
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a number between 0 and 1
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2.4
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2.5
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3.6
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12
13
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Direct and Inverse Variance
R
R
R
varies directly as
S
S
S
and inverse as
T
T
T
. When R \equal{} \frac43 and T \equal{} \frac {9}{14}, S \equal{} \frac37. Find
S
S
S
when R \equal{} \sqrt {48} and T \equal{} \sqrt {75}.
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28
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40
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42
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60
<span class='latex-bold'>(A)</span>\ 28 \qquad <span class='latex-bold'>(B)</span>\ 30 \qquad <span class='latex-bold'>(C)</span>\ 40 \qquad <span class='latex-bold'>(D)</span>\ 42 \qquad <span class='latex-bold'>(E)</span>\ 60
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28
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30
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40
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42
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60
12
1
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Expansion of a Binomial
When \left ( 1 \minus{} \frac{1}{a} \right ) ^6 is expanded the sum of the last three coefficients is:
(A)
\ 22 \qquad
(B)
\ 11 \qquad
(C)
\ 10 \qquad
(D)
\ \minus{}10 \qquad
(E)
\ \minus{}11
11
1
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Differences of zeroes
The difference between the larger root and the smaller root of x^2 \minus{} px \plus{} (p^2 \minus{} 1)/4 \equal{} 0 is:
(A)
\ 0 \qquad
(B)
\ 1 \qquad
(C)
\ 2 \qquad
(D)
\ p \qquad
(E)
\ p\plus{}1
10
1
Hide problems
Average Rate
A man drives
150
150
150
miles to the seashore in
3
3
3
hours and
20
20
20
minutes. He returns from the shore to the starting point in
4
4
4
hours and
10
10
10
minutes. Let
r
r
r
be the average rate for the entire trip. Then the average rate for the trip going exceeds
r
r
r
in miles per hour, by:
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5
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1
2
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1
<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 4 \frac{1}{2} \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ 1
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2
1
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9
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Factorization of a binomial
When x^9\minus{}x is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:
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more than 5
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<span class='latex-bold'>(A)</span>\ \text{more than 5} \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 2
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more than 5
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2
8
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Arithmetic Mean of a Set of n Numbers
Given the set of
n
n
n
numbers;
n
>
1
n > 1
n
>
1
, of which one is 1 \minus{} \frac {1}{n} and all the others are
1.
1.
1.
The arithmetic mean of the
n
n
n
numbers is:
(A)
\ 1 \qquad
(B)
\ n \minus{} \frac {1}{n} \qquad
(C)
\ n \minus{} \frac {1}{n^2} \qquad
(D)
\ 1 \minus{} \frac {1}{n^2} \qquad
(E)
\ 1 \minus{} \frac {1}{n} \minus{} \frac {1}{n^2}
7
1
Hide problems
Angle Bisectors
Let the bisectors of the exterior angles at
B
B
B
and
C
C
C
of triangle
A
B
C
ABC
A
BC
meet at
D
.
D.
D
.
Then, if all measurements are in degrees, angle
B
D
C
BDC
B
D
C
equals:
(A)
\ \frac {1}{2} (90 \minus{} A) \qquad
(B)
\ 90 \minus{} A \qquad
(C)
\ \frac {1}{2} (180 \minus{} A) \qquad
(D)
\ 180 \minus{} A \qquad
(E)
\ 180 \minus{} 2A
6
1
Hide problems
Equal Perimeters of a Square and Triangle
A square and an equilateral triangle have equal perimeters. The area of the triangle is
9
3
9 \sqrt{3}
9
3
square inches. Expressed in inches the diagonal of the square is:
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2
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5
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4
2
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2
2
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none of these
<span class='latex-bold'>(A)</span>\ \frac{9}{2} \qquad <span class='latex-bold'>(B)</span>\ 2 \sqrt{5} \qquad <span class='latex-bold'>(C)</span>\ 4 \sqrt{2} \qquad <span class='latex-bold'>(D)</span>\ \frac{9 \sqrt{2}}{2} \qquad <span class='latex-bold'>(E)</span>\ \text{none of these}
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9
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5
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2
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2
9
2
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none of these
5
1
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Radii of circles
If the radius of a circle is increased by
1
1
1
unit, the ratio of the new circumference to the new diameter is:
(A)
\ \pi \plus{} 2 \qquad
(B)
\ \frac{2 \pi \plus{} 1}{2} \qquad
(C)
\ \pi \qquad
(D)
\ \frac{2 \pi \minus{} 1}{2} \qquad
(E)
\ \pi \minus{} 2
4
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Algebraic Equation
If 8^x \equal{} 32, then
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equals:
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<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ \frac{5}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{3}{2} \qquad <span class='latex-bold'>(D)</span>\ \frac{3}{5} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{4}
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Arithmetic Sequence
The first three terms of an arithmetic progression are x \minus{} 1, x \plus{} 1, 2x \plus{} 3, in the order shown. The value of
x
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is:
(A)
\ \minus{} 2 \qquad
(B)
\ 0 \qquad
(C)
\ 2 \qquad
(D)
\ 4 \qquad
(E)
\ \text{undetermined}
2
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Algebraic Expression
The expression
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4
\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}
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−
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is equal to:
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<span class='latex-bold'>(A)</span>\ \frac{\sqrt{3}}{6} \qquad <span class='latex-bold'>(B)</span>\ \frac{-\sqrt{3}}{6} \qquad <span class='latex-bold'>(C)</span>\ \frac{\sqrt{-3}}{6} \qquad <span class='latex-bold'>(D)</span>\ \frac{5 \sqrt{3}}{6} \qquad <span class='latex-bold'>(E)</span>\ 1
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Algebraic Expression
The expression \frac{1^{4y\minus{}1}}{5^{\minus{}1}\plus{}3^{\minus{}1}} is equal to:
(A)
\ \frac{4y\minus{}1}{8} \qquad
(B)
\ 8 \qquad
(C)
\ \frac{15}{2} \qquad
(D)
\ \frac{15}{8} \qquad
(E)
\ \frac{1}{8}