MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1957 AMC 12/AHSME
1957 AMC 12/AHSME
Part of
AMC 12/AHSME
Subcontests
(49)
50
1
Hide problems
The Path of Point O'
In circle
O
O
O
,
G
G
G
is a moving point on diameter
A
B
‾
\overline{AB}
A
B
.
A
A
′
‾
\overline{AA'}
A
A
′
is drawn perpendicular to
A
B
‾
\overline{AB}
A
B
and equal to
A
G
‾
\overline{AG}
A
G
.
B
B
′
‾
\overline{BB'}
B
B
′
is drawn perpendicular to
A
B
‾
\overline{AB}
A
B
, on the same side of diameter
A
B
‾
\overline{AB}
A
B
as
A
A
′
‾
\overline{AA'}
A
A
′
, and equal to
B
G
BG
BG
. Let
O
′
O'
O
′
be the midpoint of
A
′
B
′
‾
\overline{A'B'}
A
′
B
′
. Then, as
G
G
G
moves from
A
A
A
to
B
B
B
, point
O
′
O'
O
′
:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
moves on a straight line parallel to
A
B
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
remains stationary
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
moves on a straight line perpendicular to
A
B
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
moves in a small circle intersecting the given circle
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
follows a path which is neither a circle nor a straight line
<span class='latex-bold'>(A)</span>\ \text{moves on a straight line parallel to }{AB}\qquad \\ <span class='latex-bold'>(B)</span>\ \text{remains stationary}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{moves on a straight line perpendicular to }{AB}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{moves in a small circle intersecting the given circle}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{follows a path which is neither a circle nor a straight line}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
moves on a straight line parallel to
A
B
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
remains stationary
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
moves on a straight line perpendicular to
A
B
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
moves in a small circle intersecting the given circle
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
follows a path which is neither a circle nor a straight line
49
1
Hide problems
Two Trapezoids
The parallel sides of a trapezoid are
3
3
3
and
9
9
9
. The non-parallel sides are
4
4
4
and
6
6
6
. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm);pair A = origin; pair B = (2.25,0); pair C = (2,1); pair D = (1,1); pair E = waypoint(A--D,0.25); pair F = waypoint(B--C,0.25);draw(A--B--C--D--cycle); draw(E--F);label("6",midpoint(A--D),NW); label("3",midpoint(C--D),N); label("4",midpoint(C--B),NE); label("9",midpoint(A--B),S);[/asy]
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
4
:
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
:
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
4
:
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
:
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
6
:
1
<span class='latex-bold'>(A)</span>\ 4: 3\qquad <span class='latex-bold'>(B)</span>\ 3: 2\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>\ 6: 1
<
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
>
3
:
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
:
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
:
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
6
:
1
48
1
Hide problems
Equilateral Triangle in a Circle
Let
A
B
C
ABC
A
BC
be an equilateral triangle inscribed in circle
O
O
O
.
M
M
M
is a point on arc
B
C
BC
BC
. Lines
A
M
‾
\overline{AM}
A
M
,
B
M
‾
\overline{BM}
BM
, and
C
M
‾
\overline{CM}
CM
are drawn. Then
A
M
AM
A
M
is: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm);pair O = origin; pair B = (1,0); pair C = dir(120); pair A = dir(240); pair M = dir(90 - 18);draw(Circle(O,1)); draw(A--C--M--B--cycle); draw(B--C); draw(A--M); dot(O);label("
A
A
A
",A,SW); label("
B
B
B
",B,E); label("
M
M
M
",M,NE); label("
C
C
C
",C,NW); label("
O
O
O
",O,SE);[/asy]
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
equal to
B
M
+
C
M
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
less than
B
M
+
C
M
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
greater than
B
M
+
C
M
<span class='latex-bold'>(A)</span>\ \text{equal to }{BM + CM}\qquad <span class='latex-bold'>(B)</span>\ \text{less than }{BM + CM}\qquad <span class='latex-bold'>(C)</span>\ \text{greater than }{BM + CM}\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
equal to
BM
+
CM
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
less than
BM
+
CM
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
greater than
BM
+
CM
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
equal, less than, or greater than
B
M
+
C
M
, depending upon the position of
M
<span class='latex-bold'>(D)</span>\ \text{equal, less than, or greater than }{BM + CM}\text{, depending upon the position of }{ {M}\qquad}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
equal, less than, or greater than
BM
+
CM
, depending upon the position of
M
<
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'>(E)</span>\ \text{none of these}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
none of these
41
1
Hide problems
System of Equations
Given the system of equations ax \plus{} (a \minus{} 1)y \equal{} 1 \\ (a \plus{} 1)x \minus{} ay \equal{} 1. For which one of the following values of
a
a
a
is there no solution
x
x
x
and
y
y
y
?
(A)
\ 1\qquad
(B)
\ 0\qquad
(C)
\ \minus{} 1\qquad
(D)
\ \frac {\pm \sqrt {2}}{2}\qquad
(E)
\ \pm\sqrt {2}
47
1
Hide problems
Bunch of Lines and Arcs in a Circle
In circle
O
O
O
, the midpoint of radius
O
X
OX
OX
is
Q
Q
Q
; at
Q
Q
Q
,
A
B
‾
⊥
X
Y
‾
\overline{AB} \perp \overline{XY}
A
B
⊥
X
Y
. The semi-circle with
A
B
‾
\overline{AB}
A
B
as diameter intersects
X
Y
‾
\overline{XY}
X
Y
in
M
M
M
. Line
A
M
‾
\overline{AM}
A
M
intersects circle
O
O
O
in
C
C
C
, and line
B
M
‾
\overline{BM}
BM
intersects circle
O
O
O
in
D
D
D
. Line
A
D
‾
\overline{AD}
A
D
is drawn. Then, if the radius of circle
O
O
O
is
r
r
r
,
A
D
AD
A
D
is: [asy]defaultpen(linewidth(.8pt)); unitsize(2.5cm);real m = 0; real b = 0;pair O = origin; pair X = (-1,0); pair Y = (1,0); pair Q = midpoint(O--X); pair A = (Q.x, -1*sqrt(3)/2); pair B = (Q.x, -1*A.y); pair M = (Q.x + sqrt(3)/2,0);m = (B.y - M.y)/(B.x - M.x); b = (B.y - m*B.x);pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));m = (A.y - M.y)/(A.x - M.x); b = (A.y - m*A.x);pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));draw(Circle(O,1)); draw(Arc(Q,sqrt(3)/2,-90,90)); draw(A--B); draw(X--Y); draw(B--D); draw(A--C); draw(A--D); dot(O);dot(M);label("
B
B
B
",B,NW); label("
C
C
C
",C,NE); label("
Y
Y
Y
",Y,E); label("
D
D
D
",D,SE); label("
A
A
A
",A,SW); label("
X
X
X
",X,W); label("
Q
Q
Q
",Q,SW); label("
O
O
O
",O,SW); label("
M
M
M
",M,NE+2N);[/asy]
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
r
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
r
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
not a side of an inscribed regular polygon
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
r
3
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
r
3
<span class='latex-bold'>(A)</span>\ r\sqrt {2} \qquad <span class='latex-bold'>(B)</span>\ r\qquad <span class='latex-bold'>(C)</span>\ \text{not a side of an inscribed regular polygon}\qquad <span class='latex-bold'>(D)</span>\ \frac {r\sqrt {3}}{2}\qquad <span class='latex-bold'>(E)</span>\ r\sqrt {3}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
r
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
r
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
not a side of an inscribed regular polygon
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
r
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
r
3
45
1
Hide problems
Conclusion From a Relation
If two real numbers
x
x
x
and
y
y
y
satisfy the equation \frac{x}{y} \equal{} x \minus{} y, 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
≥
4
and
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
>
y
can equal
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
both
x
and
y
must be irrational
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
x
and
y
cannot both be integers
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
both
x
and
y
must be rational
<span class='latex-bold'>(A)</span>\ {x \ge 4}\text{ and }{x \le 0}\qquad \\ <span class='latex-bold'>(B)</span>\ {y}\text{ can equal }{1}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{both }{x}\text{ and }{y}\text{ must be irrational}\qquad \\ <span class='latex-bold'>(D)</span>\ {x}\text{ and }{y}\text{ cannot both be integers}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{both }{x}\text{ and }{y}\text{ must be rational}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
x
≥
4
and
x
≤
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
y
can equal
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
both
x
and
y
must be irrational
<
s
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l
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=
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x
and
y
cannot both be integers
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<
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both
x
and
y
must be rational
44
1
Hide problems
Angle Chasing in a Triangle
In triangle
A
B
C
ABC
A
BC
, AC \equal{} CD and \angle CAB \minus{} \angle ABC \equal{} 30^\circ. Then
∠
B
A
D
\angle BAD
∠
B
A
D
is: [asy]defaultpen(linewidth(.8pt)); unitsize(2.5cm); pair A = origin; pair B = (2,0); pair C = (0.5,0.75); pair D = midpoint(C--B);draw(A--B--C--cycle); draw(A--D);label("
A
A
A
",A,SW); label("
B
B
B
",B,SE); label("
C
C
C
",C,N); label("
D
D
D
",D,NE);[/asy]
<
s
p
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n
c
l
a
s
s
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−
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(
A
)
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>
3
0
∘
<
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l
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s
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(
B
)
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a
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>
2
0
∘
<
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a
n
c
l
a
s
s
=
′
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a
t
e
x
−
b
o
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′
>
(
C
)
<
/
s
p
a
n
>
22
1
2
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
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′
>
(
D
)
<
/
s
p
a
n
>
1
0
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
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o
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′
>
(
E
)
<
/
s
p
a
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>
1
5
∘
<span class='latex-bold'>(A)</span>\ 30^\circ\qquad <span class='latex-bold'>(B)</span>\ 20^\circ\qquad <span class='latex-bold'>(C)</span>\ 22\frac {1}{2}^\circ\qquad <span class='latex-bold'>(D)</span>\ 10^\circ\qquad <span class='latex-bold'>(E)</span>\ 15^\circ
<
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(
A
)
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s
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>
3
0
∘
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s
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an
c
l
a
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=
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x
−
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>
(
B
)
<
/
s
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an
>
2
0
∘
<
s
p
an
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l
a
ss
=
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a
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x
−
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′
>
(
C
)
<
/
s
p
an
>
22
2
1
∘
<
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
1
0
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
1
5
∘
43
1
Hide problems
Lattice Points Bounded By a Parabola and a Line
We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the
x
x
x
-axis, the line x \equal{} 4, and the parabola y \equal{} x^2 is:
<
s
p
a
n
c
l
a
s
s
=
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t
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−
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o
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′
>
(
A
)
<
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s
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a
n
>
24
<
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a
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c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
a
n
>
35
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
34
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
30
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
not finite
<span class='latex-bold'>(A)</span>\ 24 \qquad <span class='latex-bold'>(B)</span>\ 35\qquad <span class='latex-bold'>(C)</span>\ 34\qquad <span class='latex-bold'>(D)</span>\ 30\qquad <span class='latex-bold'>(E)</span>\ \text{not finite}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
A
)
<
/
s
p
an
>
24
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
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d
′
>
(
B
)
<
/
s
p
an
>
35
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
34
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
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p
an
>
not finite
42
1
Hide problems
Possible Values of Powers of i
If S \equal{} i^n \plus{} i^{\minus{}n}, where i \equal{} \sqrt{\minus{}1} and
n
n
n
is an integer, then the total number of possible distinct values for
S
S
S
is:
<
s
p
a
n
c
l
a
s
s
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(
A
)
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>
1
<
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c
l
a
s
s
=
′
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e
x
−
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o
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′
>
(
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
>
more than 4
<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{more than 4}
<
s
p
an
c
l
a
ss
=
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a
t
e
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−
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(
A
)
<
/
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>
1
<
s
p
an
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l
a
ss
=
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l
a
t
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−
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>
(
B
)
<
/
s
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>
2
<
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c
l
a
ss
=
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l
a
t
e
x
−
b
o
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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
)
<
/
s
p
an
>
more than 4
40
1
Hide problems
The Set That a Variable Belongs To
If the parabola y \equal{} \minus{}x^2 \plus{} bx \minus{} 8 has its vertex on the
x
x
x
-axis, then
b
b
b
must be:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
a positive integer
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
a positive or a negative rational number
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
a positive rational number
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
a positive or a negative irrational number
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
a negative irrational number
<span class='latex-bold'>(A)</span>\ \text{a positive integer}\qquad \\ <span class='latex-bold'>(B)</span>\ \text{a positive or a negative rational number}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{a positive rational number}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{a positive or a negative irrational number}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{a negative irrational number}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
a positive integer
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
a positive or a negative rational number
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
a positive rational number
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
a positive or a negative irrational number
<
s
p
an
c
l
a
ss
=
′
l
a
t
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x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
a negative irrational number
39
1
Hide problems
Men Walking in A.P.
Two men set out at the same time to walk towards each other from
M
M
M
and
N
N
N
,
72
72
72
miles apart. The first man walks at the rate of
4
4
4
mph. The second man walks
2
2
2
miles the first hour,
2
1
2
2\frac {1}{2}
2
2
1
miles the second hour,
3
3
3
miles the third hour, and so on in arithmetic progression. Then the men will meet:
<
s
p
a
n
c
l
a
s
s
=
′
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t
e
x
−
b
o
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d
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>
(
A
)
<
/
s
p
a
n
>
in 7 hours
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
in
8
1
4
hours
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
nearer
M
than
N
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
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>
(
D
)
<
/
s
p
a
n
>
nearer
N
than
M
<
s
p
a
n
c
l
a
s
s
=
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l
a
t
e
x
−
b
o
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d
′
>
(
E
)
<
/
s
p
a
n
>
midway between
M
and
N
<span class='latex-bold'>(A)</span>\ \text{in 7 hours} \qquad <span class='latex-bold'>(B)</span>\ \text{in }{8\frac {1}{4}}\text{ hours}\qquad <span class='latex-bold'>(C)</span>\ \text{nearer }{M}\text{ than }{N}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{nearer }{N}\text{ than }{M}\qquad <span class='latex-bold'>(E)</span>\ \text{midway between }{M}\text{ and }{N}
<
s
p
an
c
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a
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=
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l
a
t
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−
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o
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d
′
>
(
A
)
<
/
s
p
an
>
in 7 hours
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
in
8
4
1
hours
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
nearer
M
than
N
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
nearer
N
than
M
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
midway between
M
and
N
38
1
Hide problems
Subtracting a Number with Reversed Digits
From a two-digit number
N
N
N
we subtract the number with the digits reversed and find that the result is a positive perfect cube. Then:
<
s
p
a
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c
l
a
s
s
=
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l
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−
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o
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>
(
A
)
<
/
s
p
a
n
>
N
cannot end in 5
<
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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>
(
B
)
<
/
s
p
a
n
>
N
can end in any digit other than 5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
N
does not exist
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
there are exactly 7 values for
N
<
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
n
>
there are exactly 10 values for
N
<span class='latex-bold'>(A)</span>\ {N}\text{ cannot end in 5}\qquad\\ <span class='latex-bold'>(B)</span>\ {N}\text{ can end in any digit other than 5}\qquad \\ <span class='latex-bold'>(C)</span>\ {N}\text{ does not exist}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{there are exactly 7 values for }{N}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{there are exactly 10 values for }{N}
<
s
p
an
c
l
a
ss
=
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a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
N
cannot end in 5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
N
can end in any digit other than 5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
N
does not exist
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
there are exactly 7 values for
N
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
there are exactly 10 values for
N
37
1
Hide problems
Rectangle in a Triangle
In right triangle
A
B
C
ABC
A
BC
, BC \equal{} 5, AC \equal{} 12, and AM \equal{} x;
M
N
‾
⊥
A
C
‾
\overline{MN} \perp \overline{AC}
MN
⊥
A
C
,
N
P
‾
⊥
B
C
‾
\overline{NP} \perp \overline{BC}
NP
⊥
BC
;
N
N
N
is on
A
B
AB
A
B
. If y \equal{} MN \plus{} NP, one-half the perimeter of rectangle
M
C
P
N
MCPN
MCPN
, then: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair M = (1,0); pair C = (2,0); pair P = (2,0.5); pair B = (2,1); pair Q = (1,0.5);draw(A--B--C--cycle); draw(M--Q--P);label("
A
A
A
",A,SW); label("
M
M
M
",M,S); label("
C
C
C
",C,SE); label("
P
P
P
",P,E); label("
B
B
B
",B,NE); label("
N
N
N
",Q,NW);[/asy]
(A)
\ y \equal{} \frac {1}{2}(5 \plus{} 12) \qquad
(B)
\ y \equal{} \frac {5x}{12} \plus{} \frac {12}{5}\qquad
(C)
\ y \equal{} \frac {144 \minus{} 7x}{12}\qquad
(D)
\ y \equal{} 12\qquad \qquad \,\,
(E)
\ y \equal{} \frac {5x}{12} \plus{} 6
36
1
Hide problems
Optimization Problem
If x \plus{} y \equal{} 1, then the largest value of
x
y
xy
x
y
is:
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1
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>
0.5
<
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=
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t
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x
−
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>
(
C
)
<
/
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>
an irrational number about
0.4
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(
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0.25
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E
)
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0
<span class='latex-bold'>(A)</span>\ 1\qquad <span class='latex-bold'>(B)</span>\ 0.5\qquad <span class='latex-bold'>(C)</span>\ \text{an irrational number about }{0.4}\qquad <span class='latex-bold'>(D)</span>\ 0.25\qquad <span class='latex-bold'>(E)</span>\ 0
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1
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)
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>
0.5
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>
(
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)
<
/
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>
an irrational number about
0.4
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a
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0.25
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(
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0
35
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Hide problems
Triangle Divided Into 8 Parts
Side
A
C
AC
A
C
of right triangle
A
B
C
ABC
A
BC
is divide into
8
8
8
equal parts. Seven line segments parallel to
B
C
BC
BC
are drawn to
A
B
AB
A
B
from the points of division. If BC \equal{} 10, then the sum of the lengths of the seven line segments:
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(
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)
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/
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>
cannot be found from the given information
<
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a
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s
=
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x
−
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>
(
B
)
<
/
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>
is
33
<
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a
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=
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>
(
C
)
<
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>
is
34
<
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a
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s
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(
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)
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>
is
35
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is
45
<span class='latex-bold'>(A)</span>\ \text{cannot be found from the given information} \qquad <span class='latex-bold'>(B)</span>\ \text{is }{33}\qquad <span class='latex-bold'>(C)</span>\ \text{is }{34}\qquad <span class='latex-bold'>(D)</span>\ \text{is }{35}\qquad <span class='latex-bold'>(E)</span>\ \text{is }{45}
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>
cannot be found from the given information
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=
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(
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)
<
/
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>
is
33
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a
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=
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)
<
/
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is
34
<
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(
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)
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is
35
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>
(
E
)
<
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>
is
45
34
1
Hide problems
Intersection of an Equation and a Inequality
The points that satisfy the system x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25, constitute the following set:
(A)
\ \text{only two points} \qquad \\
(B)
\ \text{an arc of a circle}\qquad \\
(C)
\ \text{a straight line segment not including the end\minus{}points}\qquad \\
(D)
\ \text{a straight line segment including the end\minus{}points}\qquad \\
(E)
\ \text{a single point}
33
1
Hide problems
Exponential Equation
If 9^{x \plus{} 2} \equal{} 240 \plus{} 9^x, then the value of
x
x
x
is:
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)
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>
0.1
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0.2
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0.3
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0.4
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0.5
<span class='latex-bold'>(A)</span>\ 0.1 \qquad <span class='latex-bold'>(B)</span>\ 0.2\qquad <span class='latex-bold'>(C)</span>\ 0.3\qquad <span class='latex-bold'>(D)</span>\ 0.4\qquad <span class='latex-bold'>(E)</span>\ 0.5
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(
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0.5
32
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GCD of a Sequence
The largest of the following integers which divides each of the numbers of the sequence 1^5 \minus{} 1,\, 2^5 \minus{} 2,\, 3^5 \minus{} 3,\, \cdots, n^5 \minus{} n, \cdots is:
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)
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=
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(
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)
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>
60
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>
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>
15
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30
<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 60 \qquad <span class='latex-bold'>(C)</span>\ 15 \qquad <span class='latex-bold'>(D)</span>\ 120\qquad <span class='latex-bold'>(E)</span>\ 30
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60
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30
31
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Right Triangles and an Octagon
A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square. If the square has sides of one unit, the leg of each of the triangles has length:
(A)
\ \frac{2 \plus{} \sqrt{2}}{3} \qquad
(B)
\ \frac{2 \minus{} \sqrt{2}}{2}\qquad
(C)
\ \frac{1 \plus{} \sqrt{2}}{2}\qquad
(D)
\ \frac{1 \plus{} \sqrt{2}}{3}\qquad
(E)
\ \frac{2 \minus{} \sqrt{2}}{3}
30
1
Hide problems
Expression for the Sum of Squares
The sum of the squares of the first
n
n
n
positive integers is given by the expression \frac{n(n \plus{} c)(2n \plus{} k)}{6}, if
c
c
c
and
k
k
k
are, respectively:
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)
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1
and
2
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)
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3
and
5
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2
and
2
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1
and
1
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(
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2
and
1
<span class='latex-bold'>(A)</span>\ {1}\text{ and }{2} \qquad <span class='latex-bold'>(B)</span>\ {3}\text{ and }{5}\qquad <span class='latex-bold'>(C)</span>\ {2}\text{ and }{2}\qquad <span class='latex-bold'>(D)</span>\ {1}\text{ and }{1}\qquad <span class='latex-bold'>(E)</span>\ {2}\text{ and }{1}
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and
2
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(
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)
<
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3
and
5
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2
and
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and
1
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2
and
1
29
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Quartic Inequality
The relation x^2(x^2 \minus{} 1)\ge 0 is true only for:
(A)
\ x \ge 1\qquad
(B)
\ \minus{} 1 \le x \le 1\qquad
(C)
\ x \equal{} 0,\, x \equal{} 1,\, x \equal{} \minus{} 1\qquad \\
(D)
\ x \equal{} 0,\, x \le \minus{} 1,\, x \ge 1\qquad
(E)
\ x \ge 0
28
1
Hide problems
Dependency of a Logarithm
If
a
a
a
and
b
b
b
are positive and a\not\equal{} 1,\,b\not\equal{} 1, then the value of
b
log
b
a
b^{\log_b{a}}
b
l
o
g
b
a
is:
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(
A
)
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dependent upon
b
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>
(
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)
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>
dependent upon
a
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=
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−
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(
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)
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dependent upon
a
and
b
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zero
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one
<span class='latex-bold'>(A)</span>\ \text{dependent upon }{b} \qquad <span class='latex-bold'>(B)</span>\ \text{dependent upon }{a}\qquad <span class='latex-bold'>(C)</span>\ \text{dependent upon }{a}\text{ and }{b}\qquad <span class='latex-bold'>(D)</span>\ \text{zero}\qquad <span class='latex-bold'>(E)</span>\ \text{one}
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(
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dependent upon
b
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(
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)
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dependent upon
a
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(
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)
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dependent upon
a
and
b
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=
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−
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(
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)
<
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zero
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one
27
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Sum of Reciprocal Roots
The sum of the reciprocals of the roots of the equation x^2 \plus{} px \plus{} q \equal{} 0 is:
(A)
\ \minus{}\frac{p}{q} \qquad
(B)
\ \frac{q}{p}\qquad
(C)
\ \frac{p}{q}\qquad
(D)
\ \minus{}\frac{q}{p}\qquad
(E)
\ pq
26
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Equal Areas in a Triangle
From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be:
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(
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the center of the inscribed circle
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the center of the circumscribed circle
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(
C
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such that the three angles fromed at the point each be
12
0
∘
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(
D
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the intersection of the altitudes of the triangle
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the intersection of the medians of the triangle
<span class='latex-bold'>(A)</span>\ \text{the center of the inscribed circle} \qquad \\ <span class='latex-bold'>(B)</span>\ \text{the center of the circumscribed circle}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{such that the three angles fromed at the point each be }{120^\circ}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{the intersection of the altitudes of the triangle}\qquad \\ <span class='latex-bold'>(E)</span>\ \text{the intersection of the medians of the triangle}
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the center of the inscribed circle
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a
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=
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>
(
B
)
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>
the center of the circumscribed circle
<
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a
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=
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x
−
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o
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>
(
C
)
<
/
s
p
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>
such that the three angles fromed at the point each be
12
0
∘
<
s
p
an
c
l
a
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=
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a
t
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x
−
b
o
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>
(
D
)
<
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>
the intersection of the altitudes of the triangle
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a
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=
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(
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)
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the intersection of the medians of the triangle
25
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Hide problems
Triangle in the Cartesian Plane
The vertices of triangle
P
Q
R
PQR
PQR
have coordinates as follows:
P
(
0
,
a
)
,
Q
(
b
,
0
)
,
R
(
c
,
d
)
,
P(0,a),\,Q(b,0),\,R(c,d),
P
(
0
,
a
)
,
Q
(
b
,
0
)
,
R
(
c
,
d
)
,
where
a
,
b
,
c
a,\,b,\,c
a
,
b
,
c
and
d
d
d
are positive. The origin and point
R
R
R
lie on opposite sides of
P
Q
PQ
PQ
. The area of triangle
P
Q
R
PQR
PQR
may be found from the expression:
(A)
\ \frac{ab \plus{} ac \plus{} bc \plus{} cd}{2} \qquad
(B)
\ \frac{ac \plus{} bd \minus{} ab}{2}\qquad
(C)
\ \frac{ab \minus{} ac \minus{} bd}{2}\qquad
(D)
\ \frac{ac \plus{} bd \plus{} ab}{2}\qquad
(E)
\ \frac{ac \plus{} bd \minus{} ab \minus{} cd}{2}
24
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Hide problems
Square of a Two-Digit Number
If the square of a number of two digits is decreased by the square of the number formed by reversing the digits, then the result is not always divisible by:
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9
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(
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the product of the digits
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(
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the sum of the digits
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(
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the difference of the digits
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11
<span class='latex-bold'>(A)</span>\ 9 \qquad <span class='latex-bold'>(B)</span>\ \text{the product of the digits}\qquad <span class='latex-bold'>(C)</span>\ \text{the sum of the digits}\qquad <span class='latex-bold'>(D)</span>\ \text{the difference of the digits}\qquad <span class='latex-bold'>(E)</span>\ 11
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9
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(
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)
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the product of the digits
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=
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(
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)
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>
the sum of the digits
<
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a
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=
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>
(
D
)
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>
the difference of the digits
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11
23
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Intersection of Graphs
The graph of x^2 \plus{} y \equal{} 10 and the graph of x \plus{} y \equal{} 10 meet in two points. The distance between these two points is:
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less than 1
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1
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2
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2
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(
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more than 2
<span class='latex-bold'>(A)</span>\ \text{less than 1} \qquad <span class='latex-bold'>(B)</span>\ 1\qquad <span class='latex-bold'>(C)</span>\ \sqrt{2}\qquad <span class='latex-bold'>(D)</span>\ 2\qquad <span class='latex-bold'>(E)</span>\ \text{more than 2}
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less than 1
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1
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2
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(
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2
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(
E
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more than 2
22
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Radical Equation
If \sqrt{x \minus{} 1} \minus{} \sqrt{x \plus{} 1} \plus{} 1 \equal{} 0, then
4
x
4x
4
x
equals:
(A)
\ 5 \qquad
(B)
\ 4\sqrt{\minus{}1}\qquad
(C)
\ 0\qquad
(D)
\ 1\frac{1}{4}\qquad
(E)
\ \text{no real value}
21
1
Hide problems
Equivalency of a Theorem
Start with the theorem "If two angles of a triangle are equal, the triangle is isosceles," and the following four statements: 1. If two angles of a triangle are not equal, the triangle is not isosceles. 2. The base angles of an isosceles triangle are equal. 3. If a triangle is not isosceles, then two of its angles are not equal. 4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles. Which combination of statements contains only those which are logically equivalent to the given theorem?
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1
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3
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2
,
3
,
4
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1
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2
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3
,
4
<span class='latex-bold'>(A)</span>\ 1,\,2,\,3,\,4 \qquad <span class='latex-bold'>(B)</span>\ 1,\,2,\,3\qquad <span class='latex-bold'>(C)</span>\ 2,\,3,\,4\qquad <span class='latex-bold'>(D)</span>\ 1,\,2\qquad <span class='latex-bold'>(E)</span>\ 3,\,4
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(
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1
,
2
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3
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2
,
3
,
4
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,
2
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(
E
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3
,
4
20
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Trip by Automobile
A man makes a trip by automobile at an average speed of
50
50
50
mph. He returns over the same route at an average speed of
45
45
45
mph. His average speed for the entire trip is:
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47
7
19
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47
1
4
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47
1
2
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47
11
19
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none of these
<span class='latex-bold'>(A)</span>\ 47\frac{7}{19}\qquad <span class='latex-bold'>(B)</span>\ 47\frac{1}{4}\qquad <span class='latex-bold'>(C)</span>\ 47\frac{1}{2}\qquad <span class='latex-bold'>(D)</span>\ 47\frac{11}{19}\qquad <span class='latex-bold'>(E)</span>\ \text{none of these}
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19
7
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47
4
1
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47
2
1
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47
19
11
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none of these
19
1
Hide problems
The Binary System
The base of the decimal number system is ten, meaning, for example, that 123 \equal{} 1\cdot 10^2 \plus{} 2\cdot 10 \plus{} 3. In the binary system, which has base two, the first five positive integers are
1
,
10
,
11
,
100
,
101
1,\,10,\,11,\,100,\,101
1
,
10
,
11
,
100
,
101
. The numeral
10011
10011
10011
in the binary system would then be written in the decimal system as:
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19
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40
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10011
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11
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7
<span class='latex-bold'>(A)</span>\ 19 \qquad <span class='latex-bold'>(B)</span>\ 40\qquad <span class='latex-bold'>(C)</span>\ 10011\qquad <span class='latex-bold'>(D)</span>\ 11\qquad <span class='latex-bold'>(E)</span>\ 7
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19
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(
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(
C
)
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10011
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(
D
)
<
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18
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Hide problems
A Circle and a Product
Circle
O
O
O
has diameters
A
B
AB
A
B
and
C
D
CD
C
D
perpendicular to each other.
A
M
AM
A
M
is any chord intersecting
C
D
CD
C
D
at
P
P
P
. Then
A
P
⋅
A
M
AP\cdot AM
A
P
⋅
A
M
is equal to: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair O = origin; pair A = (-1,0); pair B = (1,0); pair C = (0,1); pair D = (0,-1); pair M = dir(45); pair P = intersectionpoint(O--C,A--M);draw(Circle(O,1)); draw(A--B); draw(C--D); draw(A--M);label("
A
A
A
",A,W); label("
B
B
B
",B,E); label("
C
C
C
",C,N); label("
D
D
D
",D,S); label("
M
M
M
",M,NE); label("
O
O
O
",O,NE); label("
P
P
P
",P,NW);[/asy]
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<span class='latex-bold'>(A)</span>\ AO\cdot OB \qquad <span class='latex-bold'>(B)</span>\ AO\cdot AB\qquad <span class='latex-bold'>(C)</span>\ CP\cdot CD \qquad <span class='latex-bold'>(D)</span>\ CP\cdot PD\qquad
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<span class='latex-bold'>(E)</span>\ CO\cdot OP
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17
1
Hide problems
A Cube Made from Wire
A cube is made by soldering twelve
3
3
3
-inch lengths of wire properly at the vertices of the cube. If a fly alights at one of the vertices and then walks along the edges, the greatest distance it could travel before coming to any vertex a second time, without retracing any distance, is:
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24
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36
in.
<span class='latex-bold'>(A)</span>\ 24\text{ in.}\qquad <span class='latex-bold'>(B)</span>\ 12\text{ in.}\qquad <span class='latex-bold'>(C)</span>\ 30\text{ in.}\qquad <span class='latex-bold'>(D)</span>\ 18\text{ in.}\qquad <span class='latex-bold'>(E)</span>\ 36\text{ in.}
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36
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16
1
Hide problems
Cost of Goldfish
Goldfish are sold at
15
15
15
cents each. The rectangular coordinate graph showing the cost of
1
1
1
to
12
12
12
goldfish is:
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a straight line segment
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a finite set of distinct points
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<span class='latex-bold'>(A)</span>\ \text{a straight line segment} \qquad \\ <span class='latex-bold'>(B)</span>\ \text{a set of horizontal parallel line segments}\qquad \\ <span class='latex-bold'>(C)</span>\ \text{a set of vertical parallel line segments}\qquad \\ <span class='latex-bold'>(D)</span>\ \text{a finite set of distinct points}\qquad <span class='latex-bold'>(E)</span>\ \text{a straight line}
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a straight line segment
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a straight line
15
1
Hide problems
Table of Values of a Ball
The table below shows the distance
s
s
s
in feet a ball rolls down an inclined plane in
t
t
t
seconds. \begin{tabular}{|c|c|c|c|c|c|c|} \hline t & 0 & 1 & 2 & 3 & 4 & 5\\ \hline s & 0 & 10 & 40 & 90 & 160 & 250\\ \hline \end{tabular} The distance
s
s
s
for t \equal{} 2.5 is:
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45
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62.5
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82.5
<span class='latex-bold'>(A)</span>\ 45\qquad <span class='latex-bold'>(B)</span>\ 62.5\qquad <span class='latex-bold'>(C)</span>\ 70\qquad <span class='latex-bold'>(D)</span>\ 75\qquad <span class='latex-bold'>(E)</span>\ 82.5
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82.5
14
1
Hide problems
Simplification of Radicals
If y \equal{} \sqrt{x^2 \minus{} 2x \plus{} 1} \plus{} \sqrt{x^2 \plus{} 2x \plus{} 1}, then
y
y
y
is:
(A)
\ 2x\qquad
(B)
\ 2(x \plus{} 1)\qquad
(C)
\ 0\qquad
(D)
\ |x \minus{} 1| \plus{} |x \plus{} 1|\qquad
(E)
\ \text{none of these}
13
1
Hide problems
Rational Number Between Two Irrational Numbers
A rational number between
2
\sqrt{2}
2
and
3
\sqrt{3}
3
is:
(A)
\ \frac{\sqrt{2} \plus{} \sqrt{3}}{2} \qquad
(B)
\ \frac{\sqrt{2} \cdot \sqrt{3}}{2}\qquad
(C)
\ 1.5\qquad
(D)
\ 1.8\qquad
(E)
\ 1.4
12
1
Hide problems
Comparing Numbers in Scientific Notation
Comparing the numbers 10^{\minus{}49} and 2\cdot 10^{\minus{}50} we may say:
(A)
\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}1}}\qquad\\
(B)
\ \text{the first exceeds the second by }{2\cdot 10^{\minus{}1}}\qquad \\
(C)
\ \text{the first exceeds the second by }{8\cdot 10^{\minus{}50}}\qquad \\
(D)
\ \text{the second is five times the first}\qquad \\
(E)
\ \text{the first exceeds the second by }{5}
11
1
Hide problems
Angles in a Clock
The angle formed by the hands of a clock at
2
:
15
2: 15
2
:
15
is:
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27
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157
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172
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none of these
<span class='latex-bold'>(A)</span>\ 30^\circ \qquad <span class='latex-bold'>(B)</span>\ 27\frac{1}{2}^\circ\qquad <span class='latex-bold'>(C)</span>\ 157\frac{1}{2}^\circ\qquad <span class='latex-bold'>(D)</span>\ 172\frac{1}{2}^\circ\qquad <span class='latex-bold'>(E)</span>\ \text{none of these}
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none of these
10
1
Hide problems
Minimum/Maximum of a Parabola
The graph of y \equal{} 2x^2 \plus{} 4x \plus{} 3 has its:
(A)
\ \text{lowest point at } {(\minus{}1,9)}\qquad
(B)
\ \text{lowest point at } {(1,1)}\qquad \\
(C)
\ \text{lowest point at } {(\minus{}1,1)}\qquad
(D)
\ \text{highest point at } {(\minus{}1,9)}\qquad \\
(E)
\ \text{highest point at } {(\minus{}1,1)}
9
1
Hide problems
Evaluating an Expression
The value of x \minus{} y^{x \minus{} y} when x \equal{} 2 and y \equal{} \minus{}2 is:
(A)
\ \minus{}18 \qquad
(B)
\ \minus{}14\qquad
(C)
\ 14\qquad
(D)
\ 18\qquad
(E)
\ 256
8
1
Hide problems
Proportionality Between Variables
The numbers
x
,
y
,
z
x,\,y,\,z
x
,
y
,
z
are proportional to
2
,
3
,
5
2,\,3,\,5
2
,
3
,
5
. The sum of
x
x
x
,
y
y
y
, and
z
z
z
is
100
100
100
. The number
y
y
y
is given by the equation y \equal{} ax \minus{} 10. Then
a
a
a
is:
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<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ \frac{3}{2}\qquad <span class='latex-bold'>(C)</span>\ 3\qquad <span class='latex-bold'>(D)</span>\ \frac{5}{2}\qquad <span class='latex-bold'>(E)</span>\ 4
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Perimeter of a Triangle
The area of a circle inscribed in an equilateral triangle is
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48\pi
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. The perimeter of this triangle is:
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<span class='latex-bold'>(A)</span>\ 72\sqrt{3} \qquad <span class='latex-bold'>(B)</span>\ 48\sqrt{3}\qquad <span class='latex-bold'>(C)</span>\ 36\qquad <span class='latex-bold'>(D)</span>\ 24\qquad <span class='latex-bold'>(E)</span>\ 72
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A Box from a Sheet of Metal
An open box is constructed by starting with a rectangular sheet of metal
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10
10
in. by
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in. and cutting a square of side
x
x
x
inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:
(A)
\ 140x \minus{} 48x^2 \plus{} 4x^3 \qquad
(B)
\ 140x \plus{} 48x^2 \plus{} 4x^3\qquad \\
(C)
\ 140x \plus{} 24x^2 \plus{} x^3\qquad
(D)
\ 140x \minus{} 24x^2 \plus{} x^3\qquad
(E)
\ \text{none of these}
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Logarithm Simplification
Through the use of theorems on logarithms \log{\frac{a}{b}} \plus{} \log{\frac{b}{c}} \plus{} \log{\frac{c}{d}} \minus{} \log{\frac{ay}{dx}} can be reduced to:
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Distributive Property
The first step in finding the product (3x \plus{} 2)(x \minus{} 5) by use of the distributive property in the form a(b \plus{} c) \equal{} ab \plus{} ac is:
(A)
\ 3x^2 \minus{} 13x \minus{} 10 \qquad
(B)
\ 3x(x \minus{} 5) \plus{} 2(x \minus{} 5)\qquad \\
(C)
\ (3x \plus{} 2)x \plus{} (3x \plus{} 2)( \minus{} 5)\qquad
(D)
\ 3x^2 \minus{} 17x \minus{} 10\qquad
(E)
\ 3x^2 \plus{} 2x \minus{} 15x \minus{} 10
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Complex Fraction
The simplest form of 1 \minus{} \frac{1}{1 \plus{} \frac{a}{1 \minus{} a}} is:
(A)
\ {a}\text{ if }{a\not\equal{} 0} \qquad
(B)
\ 1\qquad
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\ {a}\text{ if }{a\not\equal{} \minus{}1}\qquad
(D)
\ {1 \minus{} a}\text{ with not restriction on }{a}\qquad
(E)
\ {a}\text{ if }{a\not\equal{} 1}
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Quadratic Equation with Unknown Coefficients
In the equation 2x^2 \minus{} hx \plus{} 2k \equal{} 0, the sum of the roots is
4
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4
and the product of the roots is \minus{}3. Then
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and
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have the values, respectively:
(A)
\ 8\text{ and }{\minus{}6} \qquad
(B)
\ 4\text{ and }{\minus{}3}\qquad
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\ {\minus{}3}\text{ and }4\qquad
(D)
\ {\minus{}3}\text{ and }8\qquad
(E)
\ 8\text{ and }{\minus{}3}
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Special Lines in a Triangle
The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:
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<span class='latex-bold'>(A)</span>\ 9\qquad <span class='latex-bold'>(B)</span>\ 7\qquad <span class='latex-bold'>(C)</span>\ 6\qquad <span class='latex-bold'>(D)</span>\ 5\qquad <span class='latex-bold'>(E)</span>\ 3
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