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Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1958 AMC 12/AHSME
1958 AMC 12/AHSME
Part of
AMC 12/AHSME
Subcontests
(50)
50
1
Hide problems
Associating AB to A'B'
In this diagram a scheme is indicated for associating all the points of segment
A
B
‾
\overline{AB}
A
B
with those of segment
A
′
B
′
‾
\overline{A'B'}
A
′
B
′
, and reciprocally. To described this association scheme analytically, let
x
x
x
be the distance from a point
P
P
P
on
A
B
‾
\overline{AB}
A
B
to
D
D
D
and let
y
y
y
be the distance from the associated point
P
′
P'
P
′
of
A
′
B
′
‾
\overline{A'B'}
A
′
B
′
to
D
′
D'
D
′
. Then for any pair of associated points, if x \equal{} a,\, x \plus{} y equals: [asy]defaultpen(linewidth(.8pt)); unitsize(.8cm);pair D= (0,9); pair E = origin; pair A = (3,9); pair P = (3.6,9); pair B = (4,9); pair F = (1,0); pair G = (2.6,0); pair H = (5,0);dot((0,0));dot((1,0));dot((2,0));dot((3,0));dot((4,0));dot((5,0)); dot((0,9));dot((1,9));dot((2,9));dot((3,9));dot((4,9));dot((5,9)); draw((D+(0,0.5))--(0,-0.5)); draw(A--H); draw(P--G); draw(B--F); draw(F--H); draw(A--B);label("
D
D
D
",D,NW); label("
D
′
D'
D
′
",E,NW); label("0",(0,0),SE); label("1",(1,0),SE); label("2",(2,0),SE); label("3",(3,0),SE); label("4",(4,0),SE); label("5",(5,0),SE); label("0",(0,9),SE); label("1",(1,9),SE); label("2",(2,9),SE); label("3",(3,9),SW); label("4",(4,9),SE); label("5",(5,9),SE); label("
B
′
B'
B
′
",F,NW); label("
P
′
P'
P
′
",G,S); label("
A
′
A'
A
′
",H,NE); label("
A
A
A
",A,NW); label("
P
P
P
",P,N); label("
B
B
B
",B,NE);[/asy]
(A)
\ 13a\qquad
(B)
\ 17a \minus{} 51\qquad
(C)
\ 17 \minus{} 3a\qquad
(D)
\ \frac {17 \minus{} 3a}{4}\qquad
(E)
\ 12a \minus{} 34
49
1
Hide problems
Binomial Expansion
In the expansion of (a \plus{} b)^n there are n \plus{} 1 dissimilar terms. The number of dissimilar terms in the expansion of (a \plus{} b \plus{} c)^{10} is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
11
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
33
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
55
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
66
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
132
<span class='latex-bold'>(A)</span>\ 11\qquad <span class='latex-bold'>(B)</span>\ 33\qquad <span class='latex-bold'>(C)</span>\ 55\qquad <span class='latex-bold'>(D)</span>\ 66\qquad <span class='latex-bold'>(E)</span>\ 132
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
33
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
55
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
66
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
132
48
1
Hide problems
Broken Line Path
Diameter
A
B
‾
\overline{AB}
A
B
of a circle with center
O
O
O
is
10
10
10
units.
C
C
C
is a point
4
4
4
units from
A
A
A
, and on
A
B
‾
\overline{AB}
A
B
.
D
D
D
is a point
4
4
4
units from
B
B
B
, and on
A
B
‾
\overline{AB}
A
B
.
P
P
P
is any point on the circle. Then the broken-line path from
C
C
C
to
P
P
P
to
D
D
D
:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
has the same length for all positions of
P
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
exceeds
10
units for all positions of
P
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
cannot exceed
10
units
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
is shortest when
△
C
P
D
is a 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
>
is longest when
P
is equidistant from
C
and
D
.
<span class='latex-bold'>(A)</span>\ \text{has the same length for all positions of }{P}\qquad\\ <span class='latex-bold'>(B)</span>\ \text{exceeds }{10}\text{ units for all positions of }{P}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{cannot exceed }{10}\text{ units}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{is shortest when }{\triangle CPD}\text{ is a right triangle}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{is longest when }{P}\text{ is equidistant from }{C}\text{ and }{D}.
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
has the same length for all positions of
P
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
exceeds
10
units for all positions of
P
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
cannot exceed
10
units
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
is shortest when
△
CP
D
is a right triangle
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
is longest when
P
is equidistant from
C
and
D
.
47
1
Hide problems
Rectangle with a Lot of Perpendiculars
A
B
C
D
ABCD
A
BC
D
is a rectangle (see the accompanying diagram) with
P
P
P
any point on
A
B
‾
\overline{AB}
A
B
.
P
S
‾
⊥
B
D
‾
\overline{PS} \perp \overline{BD}
PS
⊥
B
D
and
P
R
‾
⊥
A
C
‾
\overline{PR} \perp \overline{AC}
PR
⊥
A
C
.
A
F
‾
⊥
B
D
‾
\overline{AF} \perp \overline{BD}
A
F
⊥
B
D
and
P
Q
‾
⊥
A
F
‾
\overline{PQ} \perp \overline{AF}
PQ
⊥
A
F
. Then PR \plus{} PS is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(3cm);pair D = origin; pair C = (2,0); pair B = (2,1); pair A = (0,1); pair P = waypoint(B--A,0.2); pair S = foot(P,D,B); pair R = foot(P,A,C); pair F = foot(A,D,B); pair Q = foot(P,A,F); pair T = intersectionpoint(P--Q,A--C); pair X = intersectionpoint(A--C,B--D);draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(P--S); draw(A--F); draw(P--R); draw(P--Q);label("
A
A
A
",A,NW); label("
B
B
B
",B,NE); label("
C
C
C
",C,SE); label("
D
D
D
",D,SW); label("
P
P
P
",P,N); label("
S
S
S
",S,SE); label("
T
T
T
",T,N); label("
E
E
E
",X,SW+SE); label("
R
R
R
",R,SW); label("
F
F
F
",F,SE); label("
Q
Q
Q
",Q,SW);[/asy]
(A)
\ PQ\qquad
(B)
\ AE\qquad
(C)
\ PT \plus{} AT\qquad
(D)
\ AF\qquad
(E)
\ EF
46
1
Hide problems
Optimization of a Rational Function
For values of
x
x
x
less than
1
1
1
but greater than \minus{}4, the expression \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2} has:
(A)
\ \text{no maximum or minimum value}\qquad \\
(B)
\ \text{a minimum value of }{\plus{}1}\qquad \\
(C)
\ \text{a maximum value of }{\plus{}1}\qquad \\
(D)
\ \text{a minimum value of }{\minus{}1}\qquad \\
(E)
\ \text{a maximum value of }{\minus{}1}
45
1
Hide problems
Error in a Check
A check is written for
x
x
x
dollars and
y
y
y
cents,
x
x
x
and
y
y
y
both two-digit numbers. In error it is cashed for
y
y
y
dollars and
x
x
x
cents, the incorrect amount exceeding the correct amount by
$
17.82
\$17.82
$17.82
. Then:
<
s
p
a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
x
cannot exceed
70
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
y
can equal
2
x
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
the amount of the check cannot be a multiple of
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
the incorrect amount can equal twice the correct amount
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
the sum of the digits of the correct amount is divisible by
9
<span class='latex-bold'>(A)</span>\ {x}\text{ cannot exceed }{70}\qquad \\ <span class='latex-bold'>(B)</span>\ {y}\text{ can equal }{2x}\qquad\\ <span class='latex-bold'>(C)</span>\ \text{the amount of the check cannot be a multiple of }{5}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{the incorrect amount can equal twice the correct amount}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{the sum of the digits of the correct amount is divisible by }{9}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
x
cannot exceed
70
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
y
can equal
2
x
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
the amount of the check cannot be a multiple of
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
the incorrect amount can equal twice the correct amount
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
the sum of the digits of the correct amount is divisible by
9
44
1
Hide problems
Conclusion From Statements
Given the true statements: (1) If
a
a
a
is greater than
b
b
b
, then
c
c
c
is greater than
d
d
d
(2) If
c
c
c
is less than
d
d
d
, then
e
e
e
is greater than
f
f
f
. A valid conclusion is:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
If
a
is less than
b
, then
e
is greater than
f
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
If
e
is greater than
f
, then
a
is less than
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
If
e
is less than
f
, then
a
is greater than
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
If
a
is greater than
b
, then
e
is less than
f
<
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
<span class='latex-bold'>(A)</span>\ \text{If }{a}\text{ is less than }{b}\text{, then }{e}\text{ is greater than }{f}\qquad \\ <span class='latex-bold'>(B)</span>\ \text{If }{e}\text{ is greater than }{f}\text{, then }{a}\text{ is less than }{b}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{If }{e}\text{ is less than }{f}\text{, then }{a}\text{ is greater than }{b}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{If }{a}\text{ is greater than }{b}\text{, then }{e}\text{ is less than }{f}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{none of these}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
If
a
is less than
b
, then
e
is greater than
f
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
If
e
is greater than
f
, then
a
is less than
b
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
If
e
is less than
f
, then
a
is greater than
b
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
If
a
is greater than
b
, then
e
is less than
f
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
none of these
43
1
Hide problems
Medians in a Right Triangle
A
B
‾
\overline{AB}
A
B
is the hypotenuse of a right triangle
A
B
C
ABC
A
BC
. Median
A
D
‾
\overline{AD}
A
D
has length
7
7
7
and median
B
E
‾
\overline{BE}
BE
has length
4
4
4
. The length of
A
B
‾
\overline{AB}
A
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
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
2
<
s
p
a
n
c
l
a
s
s
=
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13
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<span class='latex-bold'>(A)</span>\ 10\qquad <span class='latex-bold'>(B)</span>\ 5\sqrt{3}\qquad <span class='latex-bold'>(C)</span>\ 5\sqrt{2}\qquad <span class='latex-bold'>(D)</span>\ 2\sqrt{13}\qquad <span class='latex-bold'>(E)</span>\ 2\sqrt{15}
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3
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2
15
42
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Chords in a Circle
In a circle with center
O
O
O
, chord
A
B
‾
\overline{AB}
A
B
equals chord
A
C
‾
\overline{AC}
A
C
. Chord
A
D
‾
\overline{AD}
A
D
cuts
B
C
‾
\overline{BC}
BC
in
E
E
E
. If AC \equal{} 12 and AE \equal{} 8, then
A
D
AD
A
D
equals:
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27
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24
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21
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18
<span class='latex-bold'>(A)</span>\ 27\qquad <span class='latex-bold'>(B)</span>\ 24\qquad <span class='latex-bold'>(C)</span>\ 21\qquad <span class='latex-bold'>(D)</span>\ 20\qquad <span class='latex-bold'>(E)</span>\ 18
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27
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24
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21
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20
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18
41
1
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Transforming Polynomial
The roots of Ax^2 \plus{} Bx \plus{} C \equal{} 0 are
r
r
r
and
s
s
s
. For the roots of x^2 \plus{} px \plus{} q \equal{} 0 to be
r
2
r^2
r
2
and
s
2
s^2
s
2
,
p
p
p
must equal:
(A)
\ \frac{B^2 \minus{} 4AC}{A^2}\qquad
(B)
\ \frac{B^2 \minus{} 2AC}{A^2}\qquad
(C)
\ \frac{2AC \minus{} B^2}{A^2}\qquad \\
(D)
\ B^2 \minus{} 2C\qquad
(E)
\ 2C \minus{} B^2
40
1
Hide problems
Recursive Relation
Given a_0 \equal{} 1, a_1 \equal{} 3, and the general relation a_n^2 \minus{} a_{n \minus{} 1}a_{n \plus{} 1} \equal{} (\minus{}1)^n for
n
≥
1
n \ge 1
n
≥
1
. Then
a
3
a_3
a
3
equals:
(A)
\ \frac{13}{27}\qquad
(B)
\ 33\qquad
(C)
\ 21\qquad
(D)
\ 10\qquad
(E)
\ \minus{}17
39
1
Hide problems
Absolute Value Quadratic Equation
We may say concerning the solution of |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 that:
(A)
\ \text{there is only one root}\qquad
(B)
\ \text{the sum of the roots is }{\plus{}1}\qquad
(C)
\ \text{the sum of the roots is }{0}\qquad \\
(D)
\ \text{the product of the roots is }{\plus{}4}\qquad
(E)
\ \text{the product of the roots is }{\minus{}6}
38
1
Hide problems
Implicit Trigonometry
Let
r
r
r
be the distance from the origion to a point
P
P
P
with coordinates
x
x
x
and
y
y
y
. Designate the ratio
y
r
\frac{y}{r}
r
y
by
s
s
s
and the ratio
x
r
\frac{x}{r}
r
x
by
c
c
c
. Then the values of s^2 \minus{} c^2 are limited to the numbers:
(A)
\ \text{less than }{\minus{}1}\text{ are greater than }{\plus{}1}\text{, both excluded}\qquad\\
(B)
\ \text{less than }{\minus{}1}\text{ are greater than }{\plus{}1}\text{, both included}\qquad \\
(C)
\ \text{between }{\minus{}1}\text{ and }{\plus{}1}\text{, both excluded}\qquad \\
(D)
\ \text{between }{\minus{}1}\text{ and }{\plus{}1}\text{, both included}\qquad \\
(E)
\ {\minus{}1}\text{ and }{\plus{}1}\text{ only}
37
1
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Arithmetic Series
The first term of an arithmetic series of consecutive integers is k^2 \plus{} 1. The sum of 2k \plus{} 1 terms of this series may be expressed as:
(A)
\ k^3 \plus{} (k \plus{} 1)^3\qquad
(B)
\ (k \minus{} 1)^3 \plus{} k^3\qquad
(C)
\ (k \plus{} 1)^3\qquad \\
(D)
\ (k \plus{} 1)^2\qquad
(E)
\ (2k \plus{} 1)(k \plus{} 1)^2
36
1
Hide problems
Larger of a Segment Cut By a Point
The sides of a triangle are
30
30
30
,
70
70
70
, and
80
80
80
units. If an altitude is dropped upon the side of length
80
80
80
, the larger segment cut off on this side is:
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64
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66
<span class='latex-bold'>(A)</span>\ 62\qquad <span class='latex-bold'>(B)</span>\ 63\qquad <span class='latex-bold'>(C)</span>\ 64\qquad <span class='latex-bold'>(D)</span>\ 65\qquad <span class='latex-bold'>(E)</span>\ 66
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62
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63
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64
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)
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66
35
1
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A Triangle in the Cartesian Plane
A triangle is formed by joining three points whose coordinates are integers. If the
x
x
x
-coordinate and the
y
y
y
-coordinate each have a value of
1
1
1
, then the area of the triangle, in square units:
<
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(
A
)
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must be an integer
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−
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(
B
)
<
/
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a
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>
may be irrational
<
s
p
a
n
c
l
a
s
s
=
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t
e
x
−
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o
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>
(
C
)
<
/
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>
must be irrational
<
s
p
a
n
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a
s
s
=
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a
t
e
x
−
b
o
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>
(
D
)
<
/
s
p
a
n
>
must be rational
<
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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>
(
E
)
<
/
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p
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>
will be an integer only if the triangle is equilateral.
<span class='latex-bold'>(A)</span>\ \text{must be an integer}\qquad <span class='latex-bold'>(B)</span>\ \text{may be irrational}\qquad <span class='latex-bold'>(C)</span>\ \text{must be irrational}\qquad <span class='latex-bold'>(D)</span>\ \text{must be rational}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{will be an integer only if the triangle is equilateral.}
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a
t
e
x
−
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d
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>
(
A
)
<
/
s
p
an
>
must be an integer
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
may be irrational
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
must be irrational
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
must be rational
<
s
p
an
c
l
a
ss
=
′
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a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
will be an integer only if the triangle is equilateral.
34
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A Fraction Inequality
The numerator of a fraction is 6x \plus{} 1, then denominator is 7 \minus{} 4x, and
x
x
x
can have any value between \minus{}2 and
2
2
2
, both included. The values of
x
x
x
for which the numerator is greater than the denominator are:
(A)
\ \frac{3}{5} < x \le 2\qquad
(B)
\ \frac{3}{5} \le x \le 2\qquad
(C)
\ 0 < x \le 2\qquad \\
(D)
\ 0 \le x \le 2\qquad
(E)
\ \minus{}2 \le x \le 2
33
1
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Condition for a Double Root of a Quadratic
For one root of ax^2 \plus{} bx \plus{} c \equal{} 0 to be double the other, the coefficients
a
,
b
,
c
a,\,b,\,c
a
,
b
,
c
must be related as follows:
(A)
\ 4b^2 \equal{} 9c\qquad
(B)
\ 2b^2 \equal{} 9ac\qquad
(C)
\ 2b^2 \equal{} 9a\qquad \\
(D)
\ b^2 \minus{} 8ac \equal{} 0\qquad
(E)
\ 9b^2 \equal{} 2ac
32
1
Hide problems
Steers and Cows
With
$
1000
\$1000
$1000
a rancher is to buy steers at
$
25
\$25
$25
each and cows at
$
26
\$26
$26
each. If the number of steers
s
s
s
and the number of cows
c
c
c
are both positive integers, then:
<
s
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l
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s
s
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(
A
)
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>
this problem has no solution
<
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a
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=
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e
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−
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o
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>
(
B
)
<
/
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a
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>
there are two solutions with
s
exceeding
c
<
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−
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(
C
)
<
/
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>
there are two solutions with
c
exceeding
s
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
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>
(
D
)
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/
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>
there is one solution with
s
exceeding
c
<
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(
E
)
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/
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a
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>
there is one solution with
c
exceeding
s
<span class='latex-bold'>(A)</span>\ \text{this problem has no solution}\qquad\\ <span class='latex-bold'>(B)</span>\ \text{there are two solutions with }{s}\text{ exceeding }{c}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{there are two solutions with }{c}\text{ exceeding }{s}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{there is one solution with }{s}\text{ exceeding }{c}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{there is one solution with }{c}\text{ exceeding }{s}
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c
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a
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>
(
A
)
<
/
s
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>
this problem has no solution
<
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=
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a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
there are two solutions with
s
exceeding
c
<
s
p
an
c
l
a
ss
=
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t
e
x
−
b
o
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d
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(
C
)
<
/
s
p
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>
there are two solutions with
c
exceeding
s
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
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an
>
there is one solution with
s
exceeding
c
<
s
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an
c
l
a
ss
=
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a
t
e
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>
(
E
)
<
/
s
p
an
>
there is one solution with
c
exceeding
s
31
1
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Area of a Triangle
The altitude drawn to the base of an isosceles triangle is
8
8
8
, and the perimeter
32
32
32
. The area of the triangle is:
<
s
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(
A
)
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>
56
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(
B
)
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>
48
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o
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(
C
)
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>
40
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o
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>
(
D
)
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/
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>
32
<
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)
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24
<span class='latex-bold'>(A)</span>\ 56\qquad <span class='latex-bold'>(B)</span>\ 48\qquad <span class='latex-bold'>(C)</span>\ 40\qquad <span class='latex-bold'>(D)</span>\ 32\qquad <span class='latex-bold'>(E)</span>\ 24
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(
A
)
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>
56
<
s
p
an
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Algebraic Manipulations
If xy \equal{} b and \frac{1}{x^2} \plus{} \frac{1}{y^2} \equal{} a, then (x \plus{} y)^2 equals:
(A)
\ (a \plus{} 2b)^2\qquad
(B)
\ a^2 \plus{} b^2\qquad
(C)
\ b(ab \plus{} 2)\qquad
(D)
\ ab(b \plus{} 2)\qquad
(E)
\ \frac{1}{a} \plus{} 2b
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Relations of Angles in a Triangle
In a general triangle
A
D
E
ADE
A
D
E
(as shown) lines
E
B
‾
\overline{EB}
EB
and
E
C
‾
\overline{EC}
EC
are drawn. Which of the following angle relations is true? [asy]defaultpen(linewidth(.8pt)); unitsize(2cm);pair A = origin; pair B = (1,0); pair C = (2,0); pair D = (3,0); pair E = (1.25,1.75);draw(A--D--E--cycle); draw(E--B); draw(E--C);label("
A
A
A
",A,SW); label("
B
B
B
",B,S); label("
C
C
C
",C,S); label("
D
D
D
",D,SE); label("
E
E
E
",E,N); label("
y
y
y
",E,3SW + 3S); label("
w
w
w
",E,7S + E); label("
b
b
b
",E,3SE + 4S + E); label("
x
x
x
",A,NE); label("
z
z
z
",B,NW); label("
m
m
m
",B,NE); label("
n
n
n
",C,NW); label("
c
c
c
",C,NE); label("
a
a
a
",D,NW+W);[/asy]
(A)
\ x \plus{} z \equal{} a \plus{} b\qquad
(B)
\ y \plus{} z \equal{} a \plus{} b\qquad
(C)
\ m \plus{} x \equal{} w \plus{} n\qquad \\
(D)
\ x \plus{} z \plus{} n \equal{} w \plus{} c \plus{} m\qquad
(E)
\ x \plus{} y \plus{} n \equal{} a \plus{} b \plus{} m
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A 16-Quart Radiator
A
16
16
16
-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is:
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256
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<span class='latex-bold'>(A)</span>\ \frac{1}{4}\qquad <span class='latex-bold'>(B)</span>\ \frac{81}{256}\qquad <span class='latex-bold'>(C)</span>\ \frac{27}{64}\qquad <span class='latex-bold'>(D)</span>\ \frac{37}{64}\qquad <span class='latex-bold'>(E)</span>\ \frac{175}{256}
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Three Collinear Points
The points (2,\minus{}3),
(
4
,
3
)
(4,3)
(
4
,
3
)
, and
(
5
,
k
/
2
)
(5, k/2)
(
5
,
k
/2
)
are on the same straight line. The value(s) of
k
k
k
is (are):
(A)
\ 12\qquad
(B)
\ \minus{}12\qquad
(C)
\ \pm 12\qquad
(D)
\ {12}\text{ or }{6}\qquad
(E)
\ {6}\text{ or }{6\frac{2}{3}}
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Change in the Sum of the Numbers in a Set
A set of
n
n
n
numbers has the sum
s
s
s
. Each number of the set is increased by
20
20
20
, then multiplied by
5
5
5
, and then decreased by
20
20
20
. The sum of the numbers in the new set thus obtained is:
(A)
\ s \plus{} 20n\qquad
(B)
\ 5s \plus{} 80n\qquad
(C)
\ s\qquad
(D)
\ 5s\qquad
(E)
\ 5s \plus{} 4n
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Logarithmic Equation
If \log_{k}{x}\cdot \log_{5}{k} \equal{} 3, then
x
x
x
equals:
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<span class='latex-bold'>(A)</span>\ k^6\qquad <span class='latex-bold'>(B)</span>\ 5k^3\qquad <span class='latex-bold'>(C)</span>\ k^3\qquad <span class='latex-bold'>(D)</span>\ 243\qquad <span class='latex-bold'>(E)</span>\ 125
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Average Rate
A man travels
m
m
m
feet due north at
2
2
2
minutes per mile. He returns due south to his starting point at
2
2
2
miles per minute. The average rate in miles per hour for the entire trip is:
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impossible to determine without knowing the value of
m
<span class='latex-bold'>(A)</span>\ 75\qquad <span class='latex-bold'>(B)</span>\ 48\qquad <span class='latex-bold'>(C)</span>\ 45\qquad <span class='latex-bold'>(D)</span>\ 24\qquad\\ <span class='latex-bold'>(E)</span>\ \text{impossible to determine without knowing the value of }{m}
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impossible to determine without knowing the value of
m
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Change in the Value of an Expression
If, in the expression x^2 \minus{} 3,
x
x
x
increases or decreases by a positive amount of
a
a
a
, the expression changes by an amount:
(A)
\ {\pm 2ax \plus{} a^2}\qquad
(B)
\ {2ax \pm a^2}\qquad
(C)
\ {\pm a^2 \minus{} 3} \qquad
(D)
\ {(x \plus{} a)^2 \minus{} 3}\qquad\\
(E)
\ {(x \minus{} a)^2 \minus{} 3}
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A Particle in the Cartesian Plane
A particle is placed on the parabola y \equal{} x^2 \minus{} x \minus{} 6 at a point
P
P
P
whose
y
y
y
-coordinate is
6
6
6
. It is allowed to roll along the parabola until it reaches the nearest point
Q
Q
Q
whose
y
y
y
-coordinate is \minus{}6. The horizontal distance traveled by the particle (the numerical value of the difference in the
x
x
x
-coordinates of
P
P
P
and
Q
Q
Q
) is:
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<span class='latex-bold'>(A)</span>\ 5\qquad <span class='latex-bold'>(B)</span>\ 4\qquad <span class='latex-bold'>(C)</span>\ 3\qquad <span class='latex-bold'>(D)</span>\ 2\qquad <span class='latex-bold'>(E)</span>\ 1
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Ratio of Triangles in a Circle
In the accompanying figure
C
E
‾
\overline{CE}
CE
and
D
E
‾
\overline{DE}
D
E
are equal chords of a circle with center
O
O
O
. Arc
A
B
AB
A
B
is a quarter-circle. Then the ratio of the area of triangle
C
E
D
CED
CE
D
to the area of triangle
A
O
B
AOB
A
OB
is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm);pair O = origin; pair C = (-1,0); pair D = (1,0); pair E = (0,1); pair A = dir(-135); pair B = dir(-60);draw(Circle(O,1)); draw(C--E--D--cycle); draw(A--O--B--cycle);label("
A
A
A
",A,SW); label("
C
C
C
",C,W); label("
E
E
E
",E,N); label("
D
D
D
",D,NE); label("
B
B
B
",B,SE); label("
O
O
O
",O,N);[/asy]
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<span class='latex-bold'>(A)</span>\ \sqrt {2} : 1\qquad <span class='latex-bold'>(B)</span>\ \sqrt {3} : 1\qquad <span class='latex-bold'>(C)</span>\ 4 : 1\qquad <span class='latex-bold'>(D)</span>\ 3 : 1\qquad <span class='latex-bold'>(E)</span>\ 2 : 1
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:
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Evalute an Expression Given an Exponential Equation
If 4^x \minus{} 4^{x \minus{} 1} \equal{} 24, then
(
2
x
)
x
(2x)^x
(
2
x
)
x
equals:
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5
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25
<span class='latex-bold'>(A)</span>\ 5\sqrt{5}\qquad <span class='latex-bold'>(B)</span>\ \sqrt{5}\qquad <span class='latex-bold'>(C)</span>\ 25\sqrt{5}\qquad <span class='latex-bold'>(D)</span>\ 125\qquad <span class='latex-bold'>(E)</span>\ 25
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19
1
Hide problems
Ratio of Segments in a Triangle
The sides of a right triangle are
a
a
a
and
b
b
b
and the hypotenuse is
c
c
c
. A perpendicular from the vertex divides
c
c
c
into segments
r
r
r
and
s
s
s
, adjacent respectively to
a
a
a
and
b
b
b
. If a : b \equal{} 1 : 3, then the ratio of
r
r
r
to
s
s
s
is:
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:
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3
:
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:
10
<span class='latex-bold'>(A)</span>\ 1 : 3\qquad <span class='latex-bold'>(B)</span>\ 1 : 9\qquad <span class='latex-bold'>(C)</span>\ 1 : 10\qquad <span class='latex-bold'>(D)</span>\ 3 : 10\qquad <span class='latex-bold'>(E)</span>\ 1 : \sqrt{10}
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:
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Change in the Area of a Circle
The area of a circle is doubled when its radius
r
r
r
is increased by
n
n
n
. Then
r
r
r
equals:
(A)
\ n(\sqrt{2} \plus{} 1)\qquad
(B)
\ n(\sqrt{2} \minus{} 1)\qquad
(C)
\ n\qquad
(D)
\ n(2 \minus{} \sqrt{2})\qquad
(E)
\ \frac{n\pi}{\sqrt{2} \plus{} 1}
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Logarithm Single-Variable Inequality
If
x
x
x
is positive and \log{x} \ge \log{2} \plus{} \frac{1}{2}\log{x}, then:
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<span class='latex-bold'>(A)</span>\ {x}\text{ has no minimum or maximum value}\qquad \\ <span class='latex-bold'>(B)</span>\ \text{the maximum value of }{x}\text{ is }{1}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{the minimum value of }{x}\text{ is }{1}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{the maximum value of }{x}\text{ is }{4}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{the minimum value of }{x}\text{ is }{4}
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the minimum value of
x
is
4
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Area of a Hexagon Circumscribed About a Circle
The area of a circle inscribed in a regular hexagon is
100
π
100\pi
100
π
. The area of hexagon is:
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5
<span class='latex-bold'>(A)</span>\ 600\qquad <span class='latex-bold'>(B)</span>\ 300\qquad <span class='latex-bold'>(C)</span>\ 200\sqrt{2}\qquad <span class='latex-bold'>(D)</span>\ 200\sqrt{3}\qquad <span class='latex-bold'>(E)</span>\ 120\sqrt{5}
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5
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Sum of the Measures of Inscribed Angles
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:
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1080
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360
<span class='latex-bold'>(A)</span>\ 1080\qquad <span class='latex-bold'>(B)</span>\ 900\qquad <span class='latex-bold'>(C)</span>\ 720\qquad <span class='latex-bold'>(D)</span>\ 540\qquad <span class='latex-bold'>(E)</span>\ 360
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1080
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900
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1
Hide problems
A Dance Party
At a dance party a group of boys and girls exchange dances as follows: one boy dances with
5
5
5
girls, a second boy dances with
6
6
6
girls, and so on, the last boy dancing with all the girls. If
b
b
b
represents the number of boys and
g
g
g
the number of girls, then:
(A)
\ b \equal{} g\qquad
(B)
\ b \equal{} \frac{g}{5}\qquad
(C)
\ b \equal{} g \minus{} 4\qquad
(D)
\ b \equal{} g \minus{} 5\qquad \\
(E)
\ \text{It is impossible to determine a relation between }{b}\text{ and }{g}\text{ without knowing }{b \plus{} g.}
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A Sum, Product, and Reciprocal Sum
The sum of two numbers is
10
10
10
; their product is
20
20
20
. The sum of their reciprocals is:
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10
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<span class='latex-bold'>(A)</span>\ \frac{1}{10}\qquad <span class='latex-bold'>(B)</span>\ \frac{1}{2}\qquad <span class='latex-bold'>(C)</span>\ 1\qquad <span class='latex-bold'>(D)</span>\ 2\qquad <span class='latex-bold'>(E)</span>\ 4
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Solving an Expression with an Exponent for n
If P \equal{} \frac{s}{(1 \plus{} k)^n} then
n
n
n
equals:
(A)
\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 \plus{} k)}}\qquad
(B)
\ \log{\left(\frac{s}{P(1 \plus{} k)}\right)}\qquad
(C)
\ \log{\left(\frac{s \minus{} P}{1 \plus{} k}\right)}\qquad \\
(D)
\ \log{\left(\frac{s}{P}\right)} \plus{} \log{(1 \plus{} k)}\qquad
(E)
\ \frac{\log{(s)}}{\log{(P(1 \plus{} k))}}
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Number of Roots of an Equation
The number of roots satisfying the equation \sqrt{5 \minus{} x} \equal{} x\sqrt{5 \minus{} x} is:
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unlimited
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Condition for Real Solutions
For what real values of
k
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, other than k \equal{} 0, does the equation x^2 \plus{} kx \plus{} k^2 \equal{} 0 have real roots?
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<span class='latex-bold'>(A)</span>\ {k < 0}\qquad <span class='latex-bold'>(B)</span>\ {k > 0} \qquad <span class='latex-bold'>(C)</span>\ {k \ge 1} \qquad <span class='latex-bold'>(D)</span>\ \text{all values of }{k}\qquad <span class='latex-bold'>(E)</span>\ \text{no values of }{k}
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Solution to an Equation in Terms of Constants
A value of
x
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x
satisfying the equation x^2 \plus{} b^2 \equal{} (a \minus{} x)^2 is:
(A)
\ \frac{b^2 \plus{} a^2}{2a}\qquad
(B)
\ \frac{b^2 \minus{} a^2}{2a}\qquad
(C)
\ \frac{a^2 \minus{} b^2}{2a}\qquad
(D)
\ \frac{a \minus{} b}{2}\qquad
(E)
\ \frac{a^2 \minus{} b^2}{2}
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Finding Rational Numbers
Which of these four numbers \sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{\minus{}1}\cdot \sqrt{(.09)^{\minus{}1}}, is (are) rational:
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X-Intercept of a Line
A straight line joins the points (\minus{}1,1) and
(
3
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9
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(3,9)
(
3
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9
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. Its
x
x
x
-intercept is:
(A)
\ \minus{}\frac{3}{2}\qquad
(B)
\ \minus{}\frac{2}{3}\qquad
(C)
\ \frac{2}{5}\qquad
(D)
\ 2\qquad
(E)
\ 3
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Arithmetic Mean
The arithmetic mean between \frac {x \plus{} a}{x} and \frac {x \minus{} a}{x}, when x \not \equal{} 0, is:
(A)
\ {2}\text{, if }{a \not \equal{} 0}\qquad
(B)
\ 1\qquad
(C)
\ {1}\text{, only if }{a \equal{} 0}\qquad
(D)
\ \frac {a}{x}\qquad
(E)
\ x
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Sum of Radical Expressions
The expression 2 \plus{} \sqrt{2} \plus{} \frac{1}{2 \plus{} \sqrt{2}} \plus{} \frac{1}{\sqrt{2} \minus{} 2} equals:
(A)
\ 2\qquad
(B)
\ 2 \minus{} \sqrt{2}\qquad
(C)
\ 2 \plus{} \sqrt{2}\qquad
(D)
\ 2\sqrt{2}\qquad
(E)
\ \frac{\sqrt{2}}{2}
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Complex Fraction Substitution
In the expression \frac{x \plus{} 1}{x \minus{} 1} each
x
x
x
is replaced by \frac{x \plus{} 1}{x \minus{} 1}. The resulting expression, evaluated for x \equal{} \frac{1}{2}, equals:
(A)
\ 3\qquad
(B)
\ \minus{}3\qquad
(C)
\ 1\qquad
(D)
\ \minus{}1\qquad
(E)
\ \text{none of these}
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Simplifying an Expression
Of the following expressions the one equal to \frac{a^{\minus{}1}b^{\minus{}1}}{a^{\minus{}3} \minus{} b^{\minus{}3}} is:
(A)
\ \frac{a^2b^2}{b^2 \minus{} a^2}\qquad
(B)
\ \frac{a^2b^2}{b^3 \minus{} a^3}\qquad
(C)
\ \frac{ab}{b^3 \minus{} a^3}\qquad
(D)
\ \frac{a^3 \minus{} b^3}{ab}\qquad
(E)
\ \frac{a^2b^2}{a \minus{} b}
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Solving for a Variable
If \frac {1}{x} \minus{} \frac {1}{y} \equal{} \frac {1}{z}, then
z
z
z
equals:
(A)
\ y \minus{} x\qquad
(B)
\ x \minus{} y\qquad
(C)
\ \frac {y \minus{} x}{xy}\qquad
(D)
\ \frac {xy}{y \minus{} x}\qquad
(E)
\ \frac {xy}{x \minus{} y}
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Expression with Negative Exponents
The value of [2 \minus{} 3(2 \minus{} 3)^{\minus{}1}]^{\minus{}1} is:
(A)
\ 5\qquad
(B)
\ \minus{}5\qquad
(C)
\ \frac{1}{5}\qquad
(D)
\ \minus{}\frac{1}{5}\qquad
(E)
\ \frac{5}{3}