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Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 10
2008 AMC 10
2008 AMC 10
Part of
AMC 10
Subcontests
(25)
23
2
Hide problems
Intersection + Union
Two subsets of the set S\equal{}\{a,b,c,d,e\} are to be chosen so that their union is
S
S
S
and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?
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<span class='latex-bold'>(A)</span>\ 20 \qquad <span class='latex-bold'>(B)</span>\ 40 \qquad <span class='latex-bold'>(C)</span>\ 60 \qquad <span class='latex-bold'>(D)</span>\ 160 \qquad <span class='latex-bold'>(E)</span>\ 320
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320
Rectangular Floor
A rectangular floor measures
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feet, where
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and
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are positive integers with
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. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width
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foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair
(
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(a,b)
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<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>\ 5
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22
2
Hide problems
Jacob's Complicated Notation
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be
6
6
6
. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts
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1
1
. If it comes up tails, he takes half of the previous term and subtracts
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1
1
. What is the probability that the fourth term in Jacob's sequence is an integer?
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<span class='latex-bold'>(A)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(D)</span>\ \frac{5}{8} \qquad <span class='latex-bold'>(E)</span>\ \frac{3}{4}
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3
Flag-Colored Beads
Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?
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<span class='latex-bold'>(A)</span>\ \frac{1}{12} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{10} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{3} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{2}
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21
2
Hide problems
Cube Slicing
A cube with side length
1
1
1
is sliced by a plane that passes through two diagonally opposite vertices
A
A
A
and
C
C
C
and the midpoints
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B
B
and
D
D
D
of two opposite edges not containing
A
A
A
and
C
C
C
, ac shown. What is the area of quadrilateral
A
B
C
D
ABCD
A
BC
D
? [asy]import three; size(200); defaultpen(fontsize(8)+linewidth(0.7)); currentprojection=obliqueX; dotfactor=4;draw((0.5,0,0)--(0,0,0)--(0,0,1)--(0,0,0)--(0,1,0),linetype("4 4")); draw((0.5,0,1)--(0,0,1)--(0,1,1)--(0.5,1,1)--(0.5,0,1)--(0.5,0,0)--(0.5,1,0)--(0.5,1,1)); draw((0.5,1,0)--(0,1,0)--(0,1,1)); dot((0.5,0,0)); label("
A
A
A
",(0.5,0,0),WSW); dot((0,1,1)); label("
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C
C
",(0,1,1),NE); dot((0.5,1,0.5)); label("
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",(0.5,1,0.5),ESE); dot((0,0,0.5)); label("
B
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B
",(0,0,0.5),NW);[/asy]
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<span class='latex-bold'>(A)</span>\ \frac {\sqrt6}{2} \qquad <span class='latex-bold'>(B)</span>\ \frac {5}{4} \qquad <span class='latex-bold'>(C)</span>\ \sqrt2 \qquad <span class='latex-bold'>(D)</span>\ \frac {3}{2} \qquad <span class='latex-bold'>(E)</span>\ \sqrt3
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Couples at a Table
Ten chairs are evenly spaced around a round table and numbered clockwise from
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1
through
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10
. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?
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<span class='latex-bold'>(A)</span>\ 240\qquad <span class='latex-bold'>(B)</span>\ 360\qquad <span class='latex-bold'>(C)</span>\ 480\qquad <span class='latex-bold'>(D)</span>\ 540\qquad <span class='latex-bold'>(E)</span>\ 720
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20
2
Hide problems
Trapezoidal Diagonals
Trapezoid
A
B
C
D
ABCD
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BC
D
has bases
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B
‾
\overline{AB}
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B
and
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‾
\overline{CD}
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D
and diagonals intersecting at
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. Suppose that AB\equal{}9, DC\equal{}12, and the area of
△
A
K
D
\triangle AKD
△
A
KD
is
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24
24
. What is the area of trapezoid
A
B
C
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ABCD
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BC
D
?
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94
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98
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100
<span class='latex-bold'>(A)</span>\ 92 \qquad <span class='latex-bold'>(B)</span>\ 94 \qquad <span class='latex-bold'>(C)</span>\ 96 \qquad <span class='latex-bold'>(D)</span>\ 98 \qquad <span class='latex-bold'>(E)</span>\ 100
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92
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94
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96
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98
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100
Cubical Dice
The faces of a cubical die are marked with the numbers
1
1
1
,
2
2
2
,
2
2
2
,
3
3
3
,
3
3
3
, and
4
4
4
. The faces of a second cubical die are marked with the numbers
1
1
1
,
3
3
3
,
4
4
4
,
5
5
5
,
6
6
6
, and
8
8
8
. Both dice are thrown. What is the probability that the sum of the two top numbers will be
5
5
5
,
7
7
7
, or
9
9
9
?
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18
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18
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18
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4
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9
<span class='latex-bold'>(A)</span>\ \frac {5}{18} \qquad <span class='latex-bold'>(B)</span>\ \frac {7}{18} \qquad <span class='latex-bold'>(C)</span>\ \frac {11}{18} \qquad <span class='latex-bold'>(D)</span>\ \frac {3}{4} \qquad <span class='latex-bold'>(E)</span>\ \frac {8}{9}
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5
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7
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11
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8
19
2
Hide problems
Rotating Rectangle
Rectangle
P
Q
R
S
PQRS
PQRS
lies in a plane with
P
Q
=
R
S
=
2
PQ = RS = 2
PQ
=
RS
=
2
and
Q
R
=
S
P
=
6
QR = SP = 6
QR
=
SP
=
6
. The rectangle is rotated
9
0
∘
90^\circ
9
0
∘
clockwise about
R
R
R
, then rotated
9
0
∘
90^\circ
9
0
∘
clockwise about the point that
S
S
S
moved to after the first rotation. What is the length of the path traveled by point
P
P
P
?
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(
2
3
+
5
)
π
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6
π
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(
3
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10
)
π
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(
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π
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2
10
π
{ <span class='latex-bold'>(A)</span>\ (2\sqrt3 + \sqrt5})\pi \qquad <span class='latex-bold'>(B)</span>\ 6\pi \qquad <span class='latex-bold'>(C)</span>\ (3 + \sqrt {10})\pi \qquad <span class='latex-bold'>(D)</span>\ (\sqrt3 + 2\sqrt5)\pi \\ <span class='latex-bold'>(E)</span>\ 2\sqrt {10}\pi
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(
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)
π
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6
π
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(
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π
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10
π
Cylindrical Tank
A cylindrical tank with radius
4
4
4
feet and height
9
9
9
feet is lying on its side. The tank is filled with water to a depth of
2
2
2
feet. What is the volume of the water, in cubic feet?
(A)
\ 24\pi \minus{} 36 \sqrt {2} \qquad
(B)
\ 24\pi \minus{} 24 \sqrt {3} \qquad
(C)
\ 36\pi \minus{} 36 \sqrt {3} \qquad
(D)
\ 36\pi \minus{} 24 \sqrt {2} \\
(E)
\ 48\pi \minus{} 36 \sqrt {3}
18
2
Hide problems
Perimeter and Area of Right Triangle
A right triangle has perimeter
32
32
32
and area
20
20
20
. What is the length of its hypotenuse?
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4
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59
4
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61
4
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63
4
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65
4
<span class='latex-bold'>(A)</span>\ \frac{57}{4} \qquad <span class='latex-bold'>(B)</span>\ \frac{59}{4} \qquad <span class='latex-bold'>(C)</span>\ \frac{61}{4} \qquad <span class='latex-bold'>(D)</span>\ \frac{63}{4} \qquad <span class='latex-bold'>(E)</span>\ \frac{65}{4}
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4
57
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4
59
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61
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4
63
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4
65
Bricklayers
Bricklayer Brenda would take
9
9
9
hours to build a chimney alone, and bricklayer Brandon would take
10
10
10
hours to build it alone. When they work together they talk a lot, and their combined output is decreased by
10
10
10
bricks per hour. Working together, they build the chimney in
5
5
5
hours. How many bricks are in the chimney?
<
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(
A
)
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>
500
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x
−
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o
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>
(
B
)
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/
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>
900
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>
950
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)
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1000
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1900
<span class='latex-bold'>(A)</span>\ 500 \qquad <span class='latex-bold'>(B)</span>\ 900 \qquad <span class='latex-bold'>(C)</span>\ 950 \qquad <span class='latex-bold'>(D)</span>\ 1000 \qquad <span class='latex-bold'>(E)</span>\ 1900
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)
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900
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950
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a
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−
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)
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>
1000
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1900
17
2
Hide problems
Region Outside Triangle
An equilateral triangle has side length
6
6
6
. What is the area of the region containing all points that are outside the triangle and not more than
3
3
3
units from a point of the triangle?
(A)
\ 36\plus{}24\sqrt{3} \qquad
(B)
\ 54\plus{}9\pi \qquad
(C)
\ 54\plus{}18\sqrt{3}\plus{}6\pi \qquad
(D)
\ \left(2\sqrt{3}\plus{}3\right)^2\pi \\
(E)
\ 9\left(\sqrt{3}\plus{}1\right)^2\pi
Approval Ratings
A poll shows that
70
%
70\%
70%
of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?
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A
)
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/
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n
>
0.063
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x
−
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)
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/
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>
0.189
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0.233
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0.333
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0.441
<span class='latex-bold'>(A)</span>\ 0.063 \qquad <span class='latex-bold'>(B)</span>\ 0.189 \qquad <span class='latex-bold'>(C)</span>\ 0.233 \qquad <span class='latex-bold'>(D)</span>\ 0.333 \qquad <span class='latex-bold'>(E)</span>\ 0.441
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0.063
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0.189
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0.233
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0.333
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0.441
15
2
Hide problems
Han, Ian, and Jan
Yesterday Han drove
1
1
1
hour longer than Ian at an average speed
5
5
5
miles per hour faster than Ian. Jan drove
2
2
2
hours longer than Ian at an average speed
10
10
10
miles per hour faster than Ian. Han drove
70
70
70
miles more than Ian. How many more miles did Jan drive than Ian?
<
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)
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120
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(
B
)
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130
<
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<span class='latex-bold'>(A)</span>\ 120 \qquad <span class='latex-bold'>(B)</span>\ 130 \qquad <span class='latex-bold'>(C)</span>\ 140 \qquad <span class='latex-bold'>(D)</span>\ 150 \qquad <span class='latex-bold'>(E)</span>\ 160
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160
Integer Right Triangles
How many right triangles have integer leg lengths
a
a
a
and
b
b
b
and a hypotenuse of length b\plus{}1, where
b
<
100
b<100
b
<
100
?
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<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 7 \qquad <span class='latex-bold'>(C)</span>\ 8 \qquad <span class='latex-bold'>(D)</span>\ 9 \qquad <span class='latex-bold'>(E)</span>\ 10
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10
12
2
Hide problems
Collection of Marbles
In a collection of red, blue, and green marbles, there are
25
%
25\%
25%
more red marbles than blue marbles, and there are
60
%
60\%
60%
more green marbles than red marbles. Suppose that there are
r
r
r
red marbles. What is the total number of marbles in that collection?
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<span class='latex-bold'>(A)</span>\ 2.85r \qquad <span class='latex-bold'>(B)</span>\ 3r \qquad <span class='latex-bold'>(C)</span>\ 3.4r \qquad <span class='latex-bold'>(D)</span>\ 3.85r \qquad <span class='latex-bold'>(E)</span>\ 4.25r
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Postman Pete
Postman Pete has a pedometer to count his steps. The pedometer records up to
99999
99999
99999
steps, then flips over to
00000
00000
00000
on the next step. Pete plans to determine his mileage for a year. On January
1
1
1
Pete sets the pedometer to
00000
00000
00000
. During the year, the pedometer flips from
99999
99999
99999
to
00000
00000
00000
forty-four times. On December
31
31
31
the pedometer reads
50000
50000
50000
. Pete takes
1800
1800
1800
steps per mile. Which of the following is closest to the number of miles Pete walked during the year?
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<span class='latex-bold'>(A)</span>\ 2500 \qquad <span class='latex-bold'>(B)</span>\ 3000 \qquad <span class='latex-bold'>(C)</span>\ 3500 \qquad <span class='latex-bold'>(D)</span>\ 4000 \qquad <span class='latex-bold'>(E)</span>\ 4500
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10
2
Hide problems
Bisecting Square Sides
Each of the sides of a square
S
1
S_1
S
1
with area
16
16
16
is bisected, and a smaller square
S
2
S_2
S
2
is constructed using the bisection points as vertices. The same process is carried out on
S
2
S_2
S
2
to construct an even smaller square
S
3
S_3
S
3
. What is the area of
S
3
S_3
S
3
?
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<span class='latex-bold'>(A)</span>\ \frac {1}{2} \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4
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4
Arc Segment
Points
A
A
A
and
B
B
B
are on a circle of radius
5
5
5
and AB\equal{}6. Point
C
C
C
is the midpoint of the minor arc
A
B
AB
A
B
. What is the length of the line segment
A
C
AC
A
C
?
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<span class='latex-bold'>(A)</span>\ \sqrt{10} \qquad <span class='latex-bold'>(B)</span>\ \frac{7}{2} \qquad <span class='latex-bold'>(C)</span>\ \sqrt{14} \qquad <span class='latex-bold'>(D)</span>\ \sqrt{15} \qquad <span class='latex-bold'>(E)</span>\ 4
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7
2
Hide problems
Simplification of Fraction
The fraction
(
3
2008
)
2
−
(
3
2006
)
2
(
3
2007
)
2
−
(
3
2005
)
2
\frac {(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2}
(
3
2007
)
2
−
(
3
2005
)
2
(
3
2008
)
2
−
(
3
2006
)
2
simplifies to which of the following?
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4
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2
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<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ \frac {9}{4} \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ \frac {9}{2} \qquad <span class='latex-bold'>(E)</span>\ 9
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9
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9
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9
Equilateral Triangle Filling
An equilateral triangle of side length
10
10
10
is completely filled in by non-overlapping equilateral triangles of side length
1
1
1
. How many small triangles are required?
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1000
<span class='latex-bold'>(A)</span>\ 10 \qquad <span class='latex-bold'>(B)</span>\ 25 \qquad <span class='latex-bold'>(C)</span>\ 100 \qquad <span class='latex-bold'>(D)</span>\ 250 \qquad <span class='latex-bold'>(E)</span>\ 1000
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25
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100
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250
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1000
6
2
Hide problems
Triathlon
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of
3
3
3
kilometers per hour, bikes at a rate of
20
20
20
kilometers per hour, and runs at a rate of
10
10
10
kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
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7
<span class='latex-bold'>(A)</span>\ 3 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 7
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4
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7
Segment Ratios
Points
B
B
B
and
C
C
C
lie on
A
D
‾
\overline{AD}
A
D
. The length of
A
B
‾
\overline{AB}
A
B
is
4
4
4
times the length of
B
D
‾
\overline{BD}
B
D
, and the length of
A
C
‾
\overline{AC}
A
C
is
9
9
9
times the length of
C
D
‾
\overline{CD}
C
D
. The length of
B
C
‾
\overline{BC}
BC
is what fraction of the length of
A
D
‾
\overline{AD}
A
D
?
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36
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10
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36
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5
<span class='latex-bold'>(A)</span>\ \frac{1}{36} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{13} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{10} \qquad <span class='latex-bold'>(D)</span>\ \frac{5}{36} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{5}
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1
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1
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5
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5
1
3
2
Hide problems
Sum of Divisors
For the positive integer
n
n
n
, let
<
n
>
\left< n \right>
⟨
n
⟩
denote the sum of all the positive divisors of
n
n
n
with the exception of
n
n
n
itself. For example,
<
4
>
=
1
+
2
=
3
\left<4\right> = 1+2=3
⟨
4
⟩
=
1
+
2
=
3
and
<
12
>
=
1
+
2
+
3
+
4
+
6
=
16
\left<12\right>=1+2+3+4+6=16
⟨
12
⟩
=
1
+
2
+
3
+
4
+
6
=
16
What is
<
<
<
6
>
>
>
\left< \left< \left< 6 \right>\right>\right>
⟨
⟨
⟨
6
⟩
⟩
⟩
?
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6
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24
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<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 12 \qquad <span class='latex-bold'>(C)</span>\ 24 \qquad <span class='latex-bold'>(D)</span>\ 32 \qquad <span class='latex-bold'>(E)</span>\ 36
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36
Cube Root
Assume that
x
x
x
is a positive real number. Which is equivalent to
x
x
3
\sqrt[3]{x\sqrt{x}}
3
x
x
?
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<span class='latex-bold'>(A)</span>\ x^{1/6} \qquad <span class='latex-bold'>(B)</span>\ x^{1/4} \qquad <span class='latex-bold'>(C)</span>\ x^{3/8} \qquad <span class='latex-bold'>(D)</span>\ x^{1/2} \qquad <span class='latex-bold'>(E)</span>\ x
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2
2
Hide problems
Square in a Rectangle
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is
2
:
1
2: 1
2
:
1
. The ratio of the rectangle's length to its width is
2
:
1
2: 1
2
:
1
. What percent of the rectangle's area is inside the square?
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12.5
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75
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87.5
<span class='latex-bold'>(A)</span>\ 12.5 \qquad <span class='latex-bold'>(B)</span>\ 25 \qquad <span class='latex-bold'>(C)</span>\ 50 \qquad <span class='latex-bold'>(D)</span>\ 75 \qquad <span class='latex-bold'>(E)</span>\ 87.5
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87.5
Calendar
A
4
×
4
4\times 4
4
×
4
block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? \setlength{\unitlength}{5mm} \begin{picture}(4,4)(0,0) \multiput(0,0)(0,1){5}{\line(1,0){4}} \multiput(0,0)(1,0){5}{\line(0,1){4}} \put(0,3){\makebox(1,1){\footnotesize{1}}} \put(1,3){\makebox(1,1){\footnotesize{2}}} \put(2,3){\makebox(1,1){\footnotesize{3}}} \put(3,3){\makebox(1,1){\footnotesize{4}}} \put(0,2){\makebox(1,1){\footnotesize{8}}} \put(1,2){\makebox(1,1){\footnotesize{9}}} \put(2,2){\makebox(1,1){\footnotesize{10}}} \put(3,2){\makebox(1,1){\footnotesize{11}}} \put(0,1){\makebox(1,1){\footnotesize{15}}} \put(1,1){\makebox(1,1){\footnotesize{16}}} \put(2,1){\makebox(1,1){\footnotesize{17}}} \put(3,1){\makebox(1,1){\footnotesize{18}}} \put(0,0){\makebox(1,1){\footnotesize{22}}} \put(1,0){\makebox(1,1){\footnotesize{23}}} \put(2,0){\makebox(1,1){\footnotesize{24}}} \put(3,0){\makebox(1,1){\footnotesize{25}}} \end{picture}
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<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ 10
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Table and Placemats
A round table has radius
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. Six rectangular place mats are placed on the table. Each place mat has width
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and length
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as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length
x
x
x
. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is
x
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? [asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("
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(A)
\ 2\sqrt {5} \minus{} \sqrt {3} \qquad
(B)
\ 3 \qquad
(C)
\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad
(D)
\ 2\sqrt {3} \qquad
(E)
\ \frac {5 \plus{} 2\sqrt {3}}{2}
Michael and the Garbage Truck
Michael walks at the rate of
5
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feet per second on a long straight path. Trash pails are located every
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feet along the path. A garbage truck travels at
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feet per second in the same direction as Michael and stops for
30
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seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?
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<span class='latex-bold'>(A)</span>\ 4\qquad <span class='latex-bold'>(B)</span>\ 5\qquad <span class='latex-bold'>(C)</span>\ 6\qquad <span class='latex-bold'>(D)</span>\ 7\qquad <span class='latex-bold'>(E)</span>\ 8
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Hide problems
Units Digit of Monomial/Exponent
Let k\equal{}2008^2\plus{}2^{2008}. What is the units digit of k^2\plus{}2^k?
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<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 6 \qquad <span class='latex-bold'>(E)</span>\ 8
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Quadrilateral
Quadrilateral
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ABCD
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has
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AB=BC=CD
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,
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=
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∘
\angle ABC=70^\circ
∠
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=
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∘
, and
∠
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C
D
=
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∘
\angle BCD=170^\circ
∠
BC
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=
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∘
. What is the degree measure of
∠
B
A
D
\angle BAD
∠
B
A
D
?
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<span class='latex-bold'>(A)</span>\ 75\qquad <span class='latex-bold'>(B)</span>\ 80\qquad <span class='latex-bold'>(C)</span>\ 85\qquad <span class='latex-bold'>(D)</span>\ 90\qquad <span class='latex-bold'>(E)</span>\ 95
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2
Hide problems
Ratio of Tangent Circle Areas
Points
A
A
A
and
B
B
B
lie on a circle centered at
O
O
O
, and \angle AOB\equal{}60^\circ. A second circle is internally tangent to the first and tangent to both
O
A
‾
\overline{OA}
O
A
and
O
B
‾
\overline{OB}
OB
. What is the ratio of the area of the smaller circle to that of the larger circle?
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<span class='latex-bold'>(A)</span>\ \frac{1}{16} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{9} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{8} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{4}
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Coins and Dice
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, their sum is
0
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.)
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<span class='latex-bold'>(A)</span>\ \frac{3}{8} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{2} \qquad <span class='latex-bold'>(C)</span>\ \frac{43}{72} \qquad <span class='latex-bold'>(D)</span>\ \frac{5}{8} \qquad <span class='latex-bold'>(E)</span>\ \frac{2}{3}
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13
2
Hide problems
Doug + Dave's Painting Team
Doug can paint a room in
5
5
5
hours. Dave can paint the same room in
7
7
7
hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let
t
t
t
be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by
t
t
t
?
(A)
\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\plus{}1)\equal{}1 \qquad
(B)
\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\plus{}1\equal{}1 \qquad
(C)
\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)t\equal{}1 \\
(D)
\ \left(\frac{1}{5}\plus{}\frac{1}{7}\right)(t\minus{}1)\equal{}1 \qquad
(E)
\ (5\plus{}7)t\equal{}1
Sequence Mean
For each positive integer
n
n
n
, the mean of the first
n
n
n
terms of a sequence is
n
n
n
. What is the
2008
2008
2008
th term of the sequence?
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056
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064
<span class='latex-bold'>(A)</span>\ 2008 \qquad <span class='latex-bold'>(B)</span>\ 4015 \qquad <span class='latex-bold'>(C)</span>\ 4016 \qquad <span class='latex-bold'>(D)</span>\ 4,030,056 \qquad <span class='latex-bold'>(E)</span>\ 4,032,064
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4
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14
2
Hide problems
Letterboxing
Older television screens have an aspect ratio of
4
:
3
4: 3
4
:
3
. That is, the ratio of the width to the height is
4
:
3
4: 3
4
:
3
. The aspect ratio of many movies is not
4
:
3
4: 3
4
:
3
, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of
2
:
1
2: 1
2
:
1
and is shown on an older television screen with a
27
27
27
-inch diagonal. What is the height, in inches, of each darkened strip? [asy]unitsize(1mm); defaultpen(linewidth(.8pt)); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,2.7)--(0,13.5)); draw((21.6,2.7)--(21.6,13.5));[/asy]
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<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 2.25 \qquad <span class='latex-bold'>(C)</span>\ 2.5 \qquad <span class='latex-bold'>(D)</span>\ 2.7 \qquad <span class='latex-bold'>(E)</span>\ 3
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Rotating a Right Triangle
Triangle
O
A
B
OAB
O
A
B
has O \equal{} (0,0), B \equal{} (5,0), and
A
A
A
in the first quadrant. In addition, \angle{ABO} \equal{} 90^\circ and \angle{AOB} \equal{} 30^\circ. Suppose that
O
A
‾
\overline{OA}
O
A
is rotated
9
0
∘
90^\circ
9
0
∘
counterclockwise about
O
O
O
. What are the coordinates of the image of
A
A
A
?
(A)
\ \left( \minus{} \frac {10}{3}\sqrt {3},5\right) \qquad
(B)
\ \left( \minus{} \frac {5}{3}\sqrt {3},5\right) \qquad
(C)
\ \left(\sqrt {3},5\right) \qquad
(D)
\ \left(\frac {5}{3}\sqrt {3},5\right) \\
(E)
\ \left(\frac {10}{3}\sqrt {3},5\right)
11
2
Hide problems
Steve and LeRoy's Leaky Boat
While Steve and LeRoy are fishing
1
1
1
mile from shore, their boat springs a leak, and water comes in at a constant rate of
10
10
10
gallons per minute. The boat will sink if it takes in more than
30
30
30
gallons of water. Steve starts rowing toward the shore at a constant rate of
4
4
4
miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
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<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ 10
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Recursive Sequence
Suppose that
(
u
n
)
\left(u_n\right)
(
u
n
)
is a sequence of real numbers satisfying u_{n \plus{} 2} \equal{} 2u_{n \plus{} 1} \plus{} u_{n}, and that u_3 \equal{} 9 and u_6 \equal{} 128. What is
u
5
u_5
u
5
?
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<span class='latex-bold'>(A)</span>\ 40 \qquad <span class='latex-bold'>(B)</span>\ 53 \qquad <span class='latex-bold'>(C)</span>\ 68 \qquad <span class='latex-bold'>(D)</span>\ 88 \qquad <span class='latex-bold'>(E)</span>\ 104
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8
2
Hide problems
Heather's Comparison Shopping
Heather compares the price of a new computer at two different stores. Store A offers
15
%
15\%
15%
off the sticker price followed by a
$
90
\$90
$90
rebate, and store B offers
25
%
25\%
25%
off the same sticker price with no rebate. Heather saves
$
15
\$15
$15
by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?
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<span class='latex-bold'>(A)</span>\ 750 \qquad <span class='latex-bold'>(B)</span>\ 900 \qquad <span class='latex-bold'>(C)</span>\ 1000 \qquad <span class='latex-bold'>(D)</span>\ 1050 \qquad <span class='latex-bold'>(E)</span>\ 1500
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Class Flowers
A class collects
$
50
\$50
$50
to buy flowers for a classmate who is in the hospital. Roses cost
$
3
\$3
$3
each, and carnations cost
$
2
\$2
$2
each. No other flowers are to be used. How many different bouquets could be purchased for exactly
$
50
\$50
$50
?
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<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ 7 \qquad <span class='latex-bold'>(C)</span>\ 9 \qquad <span class='latex-bold'>(D)</span>\ 16 \qquad <span class='latex-bold'>(E)</span>\ 17
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9
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Integer Fraction Conditions
Suppose that \frac {2x}{3} \minus{} \frac {x}{6} is an integer. Which of the following statements must be true about
x
x
x
?
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It is negative.
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It is even, but not necessarily a multiple of
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<span class='latex-bold'>(A)</span>\ \text{It is negative.} \qquad <span class='latex-bold'>(B)</span>\ \text{It is even, but not necessarily a multiple of }3\text{.}
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It is negative.
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It is a multiple of
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<span class='latex-bold'>(C)</span>\ \text{It is a multiple of }3\text{, but not necessarily even.}
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It is a multiple of
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<span class='latex-bold'>(D)</span>\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}
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It is a multiple of
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<span class='latex-bold'>(E)</span>\ \text{It is a multiple of }12\text{.}
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It is a multiple of
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.
Quadratic
A quadratic equation ax^2\minus{}2ax\plus{}b\equal{}0 has two real solutions. What is the average of the solutions?
(A)
\ 1 \qquad
(B)
\ 2 \qquad
(C)
\ \frac{b}{a} \qquad
(D)
\ \frac{2b}{a} \qquad
(E)
\ \sqrt{2b\minus{}a}
5
2
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Product of Fractions
Which of the following is equal to the product \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}?
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1004
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2008
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4016
<span class='latex-bold'>(A)</span>\ 251 \qquad <span class='latex-bold'>(B)</span>\ 502 \qquad <span class='latex-bold'>(C)</span>\ 1004 \qquad <span class='latex-bold'>(D)</span>\ 2008 \qquad <span class='latex-bold'>(E)</span>\ 4016
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2008
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4016
$ Function
For real numbers
a
a
a
and
b
b
b
, define a\$b\equal{}(a\minus{}b)^2. What is (x\minus{}y)^2\$(y\minus{}x)^2?
(A)
\ 0 \qquad
(B)
\ x^2\plus{}y^2 \qquad
(C)
\ 2x^2 \qquad
(D)
\ 2y^2 \qquad
(E)
\ 4xy
4
2
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Banana + Orange Prices
Suppose that
2
3
\frac{2}{3}
3
2
of
10
10
10
bananas are worth as much as
8
8
8
oranges. How many oranges are worth as much is
1
2
\frac{1}{2}
2
1
of
5
5
5
bananas?
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<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ \frac{5}{2} \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ \frac{7}{2} \qquad <span class='latex-bold'>(E)</span>\ 4
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4
Baseball
A semipro baseball league has teams with
21
21
21
players each. League rules state that a player must be paid at least
$
15
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000
\$15,000
$15
,
000
, and that the total of all players' salaries for each team cannot exceed
$
700
,
000
\$700,000
$700
,
000
. What is the maximum possiblle salary, in dollars, for a single player?
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<span class='latex-bold'>(A)</span>\ 270,000 \qquad <span class='latex-bold'>(B)</span>\ 385,000 \qquad <span class='latex-bold'>(C)</span>\ 400,000 \qquad <span class='latex-bold'>(D)</span>\ 430,000 \qquad <span class='latex-bold'>(E)</span>\ 700,000
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1
2
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Basketball
A basketball player made
5
5
5
baskets during a game. Each basket was worth either
2
2
2
or
3
3
3
points. How many different numbers could represent the total points scored by the player?
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<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ 6
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6
Doughnut Machine
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
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1:50 PM
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3:00 PM
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3:30 PM
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\qquad <span class='latex-bold'>(D)</span>
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4:30 PM
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\qquad <span class='latex-bold'>(E)</span>
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5:50 PM