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Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 10
2006 AMC 10
2006 AMC 10
Part of
AMC 10
Subcontests
(25)
17
2
Hide problems
2006 AMC 10 #17
In rectangle
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ADEH
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trisect
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\overline{HE}
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. In addition, AH \equal{} AC \equal{} 2. What is the area of quadrilateral
W
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WXYZ
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shown in the figure?[asy]defaultpen(linewidth(0.7));pointpen=black; pathpen=black; size(7cm); pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2); H=(0,0); G=(1,0); F=(2,0); E=(3,0); D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,NE); D('F',F,NE); D('G',G,NW); D('H',H,NW); D(A--F); D(B--E); D(D--G); D(C--H); Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H); D('W',W,N); D('X',X,plain.E); D('Y',Y,S); D('Z',Z,plain.W); D(A--D--E--H--cycle);[/asy]
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<span class='latex-bold'>(A) </span> \frac 12 \qquad <span class='latex-bold'>(B) </span> \frac {\sqrt {2}}2\qquad <span class='latex-bold'>(C) </span> \frac {\sqrt {3}}2 \qquad <span class='latex-bold'>(D) </span> \frac {2\sqrt {2}}3 \qquad <span class='latex-bold'>(E) </span> \frac {2\sqrt {3}}3
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Choosing balls
Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the contents of the two bags are the same?
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<span class='latex-bold'>(A) </span> \frac 1{10} \qquad <span class='latex-bold'>(B) </span> \frac 16 \qquad <span class='latex-bold'>(C) </span> \frac 15 \qquad <span class='latex-bold'>(D) </span> \frac 13 \qquad <span class='latex-bold'>(E) </span> \frac 12
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1
16
2
Hide problems
2006 AMC 10 #16
A circle of radius 1 is tangent to a circle of radius 2. The sides of
△
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\triangle ABC
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are tangent to the circles as shown, and the sides
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and
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\overline{AC}
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are congruent. What is the area of
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\triangle ABC
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BC
?[asy]defaultpen(black+linewidth(0.7)); size(7cm); real t=2^0.5; D((0,0)--(4*t,0)--(2*t,8)--cycle, black); D(CR((2*t,2),2), black); D(CR((2*t,5),1), black); dot(origin^^(4t,0)^^(2t,8)); label("B", (0,0), SW); label("C", (4*t,0), SE); label("A", (2*t,8), N); D((2*t,2)--(2*t,4), black); D((2*t,5)--(2*t,6), black); MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);[/asy]
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<span class='latex-bold'>(A) </span> \frac {35}2 \qquad <span class='latex-bold'>(B) </span> 15\sqrt {2} \qquad <span class='latex-bold'>(C) </span> \frac {64}3 \qquad <span class='latex-bold'>(D) </span> 16\sqrt {2} \qquad <span class='latex-bold'>(E) </span> 24
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24
Leap days
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?
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<span class='latex-bold'>(A) </span> \text{Tuesday} \qquad <span class='latex-bold'>(B) </span> \text{Wednesday} \qquad <span class='latex-bold'>(C) </span> \text{Thursday} \qquad <span class='latex-bold'>(D) </span> \text{Friday} \qquad <span class='latex-bold'>(E) </span> \text{Saturday}
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12
2
Hide problems
2006 AMC 10 #12
Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet?[asy]defaultpen(linewidth(0.7)); size(7cm); D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle, black); D((16,-8)--(24,-8), black); label('Dog', (24, -8), SE); label('I', (8,-8), (0,0)); MP('8', (16,-4), W); MP('8', (16,-12),W); MP('8', (20,-8), N); label('Rope', (20,-8),S); D((0,-20)--(16,-20)--(16,-36)--(0,-36)--cycle, black); D((16,-24)--(24,-24), black); label("II", (8,-28), (0,0)); MP('4', (16,-22), W); MP('8', (20,-24), N); label("Dog",(24,-24),SE); label("Rope", (20,-24), S); dot((24,-24)^^(24,-8));[/asy]
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<span class='latex-bold'>(A)</span>\text{ I, by }8\pi\qquad<span class='latex-bold'>(B)</span>\text{ I, by }6\pi\qquad<span class='latex-bold'>(C)</span>\text{ II, by }4\pi\qquad<span class='latex-bold'>(D) </span>\text{II, by }8\pi\qquad<span class='latex-bold'>(E)</span>\text{ II, by }10\pi
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Intersecting Lines
The lines x \equal{} \frac 14y \plus{} a and y \equal{} \frac 14x \plus{} b intersect at the point
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<span class='latex-bold'>(A) </span> 0 \qquad <span class='latex-bold'>(B) </span> \frac 34 \qquad <span class='latex-bold'>(C) </span> 1 \qquad <span class='latex-bold'>(D) </span> 2 \qquad <span class='latex-bold'>(E) </span> \frac 94
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Hide problems
Volume of octahedron.
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
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<span class='latex-bold'>(A) </span> \frac 18 \qquad <span class='latex-bold'>(B) </span> \frac 16 \qquad <span class='latex-bold'>(C) </span> \frac 14 \qquad <span class='latex-bold'>(D) </span> \frac 13 \qquad <span class='latex-bold'>(E) </span> \frac 12
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1
2006 AMC 10 #24
Circles with centers
O
O
O
and
P
P
P
have radii 2 and 4, respectively, and are externally tangent. Points
A
A
A
and
B
B
B
are on the circle centered at
O
O
O
, and points
C
C
C
and
D
D
D
are on the circle centered at
P
P
P
, such that
A
D
‾
\overline{AD}
A
D
and
B
C
‾
\overline{BC}
BC
are common external tangents to the circles. What is the area of hexagon
A
O
B
C
P
D
AOBCPD
A
OBCP
D
? [asy] size(250);defaultpen(linewidth(0.8)); pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2); pair top=X+15*dir(X--A), bottom=X+15*dir(X--B); draw(Circle(O, 2)^^Circle(P, 4)); draw(bottom--X--top); draw(A--O--B^^O--P^^D--P--C); pair point=X; label("
2
2
2
", midpoint(O--A), dir(point--midpoint(O--A))); label("
4
4
4
", midpoint(P--D), dir(point--midpoint(P--D))); label("
O
O
O
", O, SE); label("
P
P
P
", P, dir(point--P)); pair point=O; label("
A
A
A
", A, dir(point--A)); label("
B
B
B
", B, dir(point--B)); pair point=P; label("
C
C
C
", C, dir(point--C)); label("
D
D
D
", D, dir(point--D)); fill((-3,7)--(-3,-7)--(-7,-7)--(-7,7)--cycle, white);[/asy]
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<span class='latex-bold'>(A) </span> 18\sqrt {3} \qquad <span class='latex-bold'>(B) </span> 24\sqrt {2} \qquad <span class='latex-bold'>(C) </span> 36 \qquad <span class='latex-bold'>(D) </span> 24\sqrt {3} \qquad <span class='latex-bold'>(E) </span> 32\sqrt {2}
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Counting integers.
How many four-digit positive integers have at least one digit that is a 2 or a 3?
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<span class='latex-bold'>(A) </span> 2439 \qquad <span class='latex-bold'>(B) </span> 4096 \qquad <span class='latex-bold'>(C) </span> 4903 \qquad <span class='latex-bold'>(D) </span> 4904 \qquad <span class='latex-bold'>(E) </span> 5416
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Unfair Dice
For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio
1
:
2
:
3
:
4
:
5
:
6
1: 2: 3: 4: 5: 6
1
:
2
:
3
:
4
:
5
:
6
. What is the probability of rolling a total of 7 on the two dice?
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63
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<span class='latex-bold'>(A) </span> \frac 4{63} \qquad <span class='latex-bold'>(B) </span> \frac 18 \qquad <span class='latex-bold'>(C) </span> \frac 8{63} \qquad <span class='latex-bold'>(D) </span> \frac 16 \qquad <span class='latex-bold'>(E) </span> \frac 27
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20
2
Hide problems
Fun probability.
Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?
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<span class='latex-bold'>(A) </span> \frac 12 \qquad <span class='latex-bold'>(B) </span> \frac 35 \qquad <span class='latex-bold'>(C) </span> \frac 23 \qquad <span class='latex-bold'>(D) </span> \frac 45 \qquad <span class='latex-bold'>(E) </span> 1
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Rectangle by 2 vertices
In rectangle
A
B
C
D
ABCD
A
BC
D
, we have A \equal{} (6, \minus{} 22), B \equal{} (2006,178), and D \equal{} (8,y), for some integer
y
y
y
. What is the area of rectangle
A
B
C
D
ABCD
A
BC
D
?
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<span class='latex-bold'>(A) </span> 4000 \qquad <span class='latex-bold'>(B) </span> 4040 \qquad <span class='latex-bold'>(C) </span> 4400 \qquad <span class='latex-bold'>(D) </span> 40,000 \qquad <span class='latex-bold'>(E) </span> 40,400
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19
2
Hide problems
Angles in progression.
How many non-similar triangle have angles whose degree measures are distinct positive integers in arithmetic progression?
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<span class='latex-bold'>(A) </span> 0 \qquad <span class='latex-bold'>(B) </span> 1 \qquad <span class='latex-bold'>(C) </span> 59 \qquad <span class='latex-bold'>(D) </span> 89 \qquad <span class='latex-bold'>(E) </span> 178
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2006 AMC 10 #19
A circle of radius 2 is centered at
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OABC
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A
BC
has side length 1. Sides
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B
‾
\overline{AB}
A
B
and
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B
‾
\overline{CB}
CB
are extended past
b
b
b
to meet the circle at
D
D
D
and
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, respectively. What is the area of the shaded region in the figure, which is bounded by
B
D
‾
\overline{BD}
B
D
,
B
E
‾
\overline{BE}
BE
, and the minor arc connecting
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and
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?[asy] defaultpen(linewidth(0.8)); pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B; fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray); clip(B--Arc(O, 2, 30, 60)--cycle); draw(Circle(origin, 2)); draw((-2,0)--(2,0)^^(0,-2)--(0,2)); draw(A--D^^C--E); label("
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A
A
", A, dir(point--A)); label("
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C
C
", C, dir(point--C)); label("
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O
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", O, dir(point--O)); label("
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D
D
", D, dir(point--D)); label("
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E
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", E, dir(point--E)); label("
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B
B
", B, SW);[/asy]
(A)
\frac {\pi}3 \plus{} 1 \minus{} \sqrt {3} \qquad
(B)
\frac {\pi}2\left( 2 \minus{} \sqrt {3}\right) \qquad
(C)
\pi\left(2 \minus{} \sqrt {3}\right) \qquad
(D)
\frac {\pi}{6} \plus{} \frac {\sqrt {3} \minus{} 1}{2} \\ \qquad \indent
(E)
\frac {\pi}{3} \minus{} 1 \plus{} \sqrt {3}
18
2
Hide problems
License plates.
A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?
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<span class='latex-bold'>(A) </span> 10^4\cdot 26^2 \qquad <span class='latex-bold'>(B) </span> 10^3\cdot 26^3 \qquad <span class='latex-bold'>(C) </span> 5\cdot 10^4\cdot 26^2 \qquad <span class='latex-bold'>(D) </span> 10^2\cdot 26^4\\ <span class='latex-bold'>(E) </span> 5\cdot 10^3\cdot 26^3
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3
Sequences
Let
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a_1, a_2, ...
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be a sequence for which a_1 \equal{} 2\,\hspace{.2in}a_2 \equal{} 3\, \hspace{.2in}\text{and}\hspace{.2in}a_n \equal{} \frac {a_{n \minus{} 1}}{a_{n \minus{} 2}} \text{ for each positive integer } n \ge 3.What is
a
2006
a_{2006}
a
2006
?
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<span class='latex-bold'>(A) </span> \frac 12 \qquad <span class='latex-bold'>(B) </span> \frac 23 \qquad <span class='latex-bold'>(C) </span> \frac 32 \qquad <span class='latex-bold'>(D) </span> 2 \qquad <span class='latex-bold'>(E) </span> 3
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15
2
Hide problems
Who runs around a circular track anyway?
Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
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<span class='latex-bold'>(A) </span> 29 \qquad <span class='latex-bold'>(B) </span> 42 \qquad <span class='latex-bold'>(C) </span> 45 \qquad <span class='latex-bold'>(D) </span> 47 \qquad <span class='latex-bold'>(E) </span> 50
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Two Similar Rhombi
Rhombus
A
B
C
D
ABCD
A
BC
D
is similar to rhombus
B
F
D
E
BFDE
BF
D
E
. The area of rhombus
A
B
C
D
ABCD
A
BC
D
is 24, and \angle BAD \equal{} 60^\circ. What is the area of rhombus
B
F
D
E
BFDE
BF
D
E
?[asy] size(180); defaultpen(linewidth(0.7)+fontsize(11)); pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3)); pair point=(3/2, sqrt(3)/2); draw(B--C--D--A--B--F--D--E--B); label("
A
A
A
", A, dir(point--A)); label("
B
B
B
", B, dir(point--B)); label("
C
C
C
", C, dir(point--C)); label("
D
D
D
", D, dir(point--D)); label("
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", E, dir(point--E)); label("
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", F, dir(point--F));[/asy]
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<span class='latex-bold'>(A) </span> 6 \qquad <span class='latex-bold'>(B) </span> 4\sqrt {3} \qquad <span class='latex-bold'>(C) </span> 8 \qquad <span class='latex-bold'>(D) </span> 9 \qquad <span class='latex-bold'>(E) </span> 6\sqrt {3}
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3
13
2
Hide problems
Shooting dice
A player pays
$
5
\$ 5
$5
to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)
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<span class='latex-bold'>(A) </span> \$ 12 \qquad <span class='latex-bold'>(B) </span> \$ 30 \qquad <span class='latex-bold'>(C) </span> \$ 50\qquad <span class='latex-bold'>(D) </span> \$ 60 \qquad <span class='latex-bold'>(E) </span> \$ 100
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$100
Coffee and Cream
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
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<span class='latex-bold'>(A) </span> \frac 67 \qquad <span class='latex-bold'>(B) </span> \frac {13}{14} \qquad <span class='latex-bold'>(C) </span> 1 \qquad <span class='latex-bold'>(D) </span> \frac {14}{13} \qquad <span class='latex-bold'>(E) </span> \frac 76
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2
Hide problems
Quadratic
A parabola with equation y \equal{} x^2 \plus{} bx \plus{} c passes through the points
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(2,3)
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(4,3)
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. What is
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<span class='latex-bold'>(A) </span> 2 \qquad <span class='latex-bold'>(B) </span> 5 \qquad <span class='latex-bold'>(C) </span> 7 \qquad <span class='latex-bold'>(D) </span> 10 \qquad <span class='latex-bold'>(E) </span> 11
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2006 AMC 10 #8
A square of area
40
40
40
is inscribed in a semicircle as shown. What is the area of the semicircle? [asy] defaultpen(linewidth(0.8)); real r=sqrt(50), s=sqrt(10); draw(Arc(origin, r, 0, 180)); draw((r,0)--(-r,0), dashed); draw((s,0)--(s,2*s)--(-s,2*s)--(-s,0));[/asy]
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<span class='latex-bold'>(A) </span>20\pi\qquad<span class='latex-bold'>(B) </span>25\pi\qquad<span class='latex-bold'>(C) </span>30\pi\qquad<span class='latex-bold'>(D) </span>40\pi\qquad<span class='latex-bold'>(E) </span>50\pi
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Exponents
What non-zero real value for
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x
satisfies (7x)^{14} \equal{} (14x)^7?
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<span class='latex-bold'>(A) </span> \frac 17 \qquad <span class='latex-bold'>(B) </span> \frac 27 \qquad <span class='latex-bold'>(C) </span> 1 \qquad <span class='latex-bold'>(D) </span> 7 \qquad <span class='latex-bold'>(E) </span> 14
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2006 AMC 10 #6
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure
2
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2/\pi
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π
, as shown. What is the perimeter of this region? [asy] size(90); defaultpen(linewidth(0.7)); filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray(0.5)); filldraw(arc((1,0),1,180,0, CCW)--cycle,gray(0.7)); filldraw(arc((0,1),1,90,270)--cycle,gray(0.7)); filldraw(arc((1,2),1,0,180)--cycle,gray(0.7)); filldraw(arc((2,1),1,270,90, CCW)--cycle,gray(0.7));[/asy]
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<span class='latex-bold'>(A) </span>\frac {4}\pi\qquad<span class='latex-bold'>(B) </span>2\qquad<span class='latex-bold'>(C) </span>\frac {8}\pi\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>\frac{16}{\pi}
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Hide problems
Just posting it for the resources section ;)
Sandwiches at Joe's Fast Food cost
$
3
\$3
$3
each and sodas cost
$
2
\$2
$2
each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
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<span class='latex-bold'>(A) </span> 31\qquad <span class='latex-bold'>(B) </span> 32\qquad <span class='latex-bold'>(C) </span> 33\qquad <span class='latex-bold'>(D) </span> 34\qquad <span class='latex-bold'>(E) </span> 35
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Exponents
What is ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}?
(A)
\minus{} 2006 \qquad
(B)
\minus{} 1 \qquad
(C)
0 \qquad
(D)
1 \qquad
(E)
2006
2
2
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New operation
Define x\otimes y \equal{} x^3 \minus{} y. What is
h
⊗
(
h
⊗
h
)
h\otimes (h\otimes h)
h
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(
h
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h
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?
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\minus{} h\qquad
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0\qquad
(C)
h\qquad
(D)
2h\qquad
(E)
h^3
Spade Operation
For real numbers
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x
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y
, define x\spadesuit y \equal{} (x \plus{} y)(x \minus{} y). What is
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♠
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4
♠
5
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3\spadesuit(4\spadesuit 5)
3♠
(
4♠5
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?
(A)
\minus{} 72 \qquad
(B)
\minus{} 27 \qquad
(C)
\minus{} 24 \qquad
(D)
24 \qquad
(E)
72
3
2
Hide problems
Mary and alice
The ratio of Mary's age to Alice's age is
3
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3: 5
3
:
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. Alice is
30
30
30
years old. How old is Mary?
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<span class='latex-bold'>(A) </span> 15\qquad <span class='latex-bold'>(B) </span> 18\qquad <span class='latex-bold'>(C) </span> 20\qquad <span class='latex-bold'>(D) </span> 24\qquad <span class='latex-bold'>(E) </span> 50
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Simple Algebra
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
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<span class='latex-bold'>(A) </span> 10 \qquad <span class='latex-bold'>(B) </span> 14 \qquad <span class='latex-bold'>(C) </span> 17 \qquad <span class='latex-bold'>(D) </span> 20 \qquad <span class='latex-bold'>(E) </span> 24
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4
2
Hide problems
Digital watch
A digital watch displays hours and minutes with
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A
M
\text c{AM}
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and
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\text c{PM}
c
PM
. What is the largest possible sum of the digits in the display?
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<span class='latex-bold'>(A) </span> 17\qquad <span class='latex-bold'>(B) </span> 19\qquad <span class='latex-bold'>(C) </span> 21\qquad <span class='latex-bold'>(D) </span> 22\qquad <span class='latex-bold'>(E) </span> 23
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Circular ratios
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
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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> 6 \qquad <span class='latex-bold'>(D) </span> 8 \qquad <span class='latex-bold'>(E) </span> 9
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5
2
Hide problems
Anchovy pizza
Doug and Dave shared a pizza with
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equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was
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$8
, and there was an additional cost of
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$2
for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
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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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Rectangles and squares
A 2 x 3 rectangle and a 3 x 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
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<span class='latex-bold'>(A) </span> 16 \qquad <span class='latex-bold'>(B) </span> 25 \qquad <span class='latex-bold'>(C) </span> 36 \qquad <span class='latex-bold'>(D) </span> 49 \qquad <span class='latex-bold'>(E) </span> 64
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7
2
Hide problems
Rectangle cut into two hexagons
The
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18
8\times 18
8
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18
rectangle
A
B
C
D
ABCD
A
BC
D
is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is
y
y
y
? [asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("
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D
D
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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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Square root of fraction
Which of the following is equivalent to \displaystyle \sqrt {\frac {x}{1 \minus{} \frac {x \minus{} 1}{x}}} when
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\minus{} x \qquad
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x \qquad
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1 \qquad
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\sqrt {\frac x2} \qquad
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x\sqrt { \minus{} 1}
9
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Consecutive integers sum to 15
How many sets of two or more consecutive positive integers have a sum of 15?
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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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Counting Calories
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
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<span class='latex-bold'>(A) </span> 129 \qquad <span class='latex-bold'>(B) </span> 137 \qquad <span class='latex-bold'>(C) </span> 174 \qquad <span class='latex-bold'>(D) </span> 223 \qquad <span class='latex-bold'>(E) </span> 411
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10
2
Hide problems
Square root is an integer
For how many real values of
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is \sqrt {120 \minus{} \sqrt {x}} an integer?
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<span class='latex-bold'>(A) </span> 3\qquad <span class='latex-bold'>(B) </span> 6\qquad <span class='latex-bold'>(C) </span> 9\qquad <span class='latex-bold'>(D) </span> 10\qquad <span class='latex-bold'>(E) </span> 11
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Maximum Perimeter
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
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<span class='latex-bold'>(A) </span> 43 \qquad <span class='latex-bold'>(B) </span> 44 \qquad <span class='latex-bold'>(C) </span> 45 \qquad <span class='latex-bold'>(D) </span> 46 \qquad <span class='latex-bold'>(E) </span> 47
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11
2
Hide problems
Graph of equation
Which of the following describes the graph of the equation (x \plus{} y)^2 \equal{} x^2 \plus{} y^2?
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the empty set
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<span class='latex-bold'>(A)</span>\text{ the empty set}\qquad <span class='latex-bold'>(B)</span>\text{ one point}\qquad <span class='latex-bold'>(C)</span>\text{ two lines}
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<span class='latex-bold'>(D)</span>\text{ a circle}\qquad <span class='latex-bold'>(E)</span>\text{ the entire plane}
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Factorial sums
What is the tens digit in the sum 7! \plus{} 8! \plus{} 9! \plus{} \cdots \plus{} 2006!?
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<span class='latex-bold'>(A) </span> 1 \qquad <span class='latex-bold'>(B) </span> 3 \qquad <span class='latex-bold'>(C) </span> 4 \qquad <span class='latex-bold'>(D) </span> 6 \qquad <span class='latex-bold'>(E) </span> 9
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14
2
Hide problems
Linked rings
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring? [asy] size(200); defaultpen(linewidth(3)); real[] inrad = {40,34,28,21}; real[] outrad = {55,49,37,30}; real[] center; path[][] quad = new path[4][4]; center[0] = 0; for(int i=0;i<=3;i=i+1) { if(i != 0) { center = center[i-1] - inrad[i-1] - inrad+3.5; } quad[0] = arc((0,center),inrad,0,90)--arc((0,center),outrad,90,0)--cycle; quad[1] = arc((0,center),inrad,90,180)--arc((0,center),outrad,180,90)--cycle; quad[2] = arc((0,center),inrad,180,270)--arc((0,center),outrad,270,180)--cycle; quad[3] = arc((0,center),inrad,270,360)--arc((0,center),outrad,360,270)--cycle; draw(circle((0,center),inrad)^^circle((0,center),outrad)); } void fillring(int i,int j) { if ((j % 2) == 0) { fill(quad[j],white); } else { filldraw(quad[j],black); } } for(int i=0;i<=3;i=i+1) { for(int j=0;j<=3;j=j+1) { fillring(((2-i) % 4),j); } } for(int k=0;k<=2;k=k+1) { filldraw(circle((0,-228 - 25 * k),3),black); } real r = 130, s = -90; draw((0,57)--(r,57)^^(0,-57)--(r,-57),linewidth(0.7)); draw((2*r/3,56)--(2*r/3,-56),linewidth(0.7),Arrows(size=3)); label("
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",(2*r/3,-10),E); draw((0,39)--(s,39)^^(0,-39)--(s,-39),linewidth(0.7)); draw((9*s/10,38)--(9*s/10,-38),linewidth(0.7),Arrows(size=3)); label("
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",(9*s/10,0),W); [/asy]
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<span class='latex-bold'>(A) </span> 171\qquad <span class='latex-bold'>(B) </span> 173\qquad <span class='latex-bold'>(C) </span> 182\qquad <span class='latex-bold'>(D) </span> 188\qquad <span class='latex-bold'>(E) </span> 210
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Roots
Let
a
a
a
and
b
b
b
be the roots of the equation x^2 \minus{} mx \plus{} 2 \equal{} 0. Suppose that a \plus{} (1/b) and b \plus{} (1/a) are the roots of the equation x^2 \minus{} px \plus{} q \equal{} 0. What is
q
q
q
?
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<span class='latex-bold'>(A) </span> \frac 52 \qquad <span class='latex-bold'>(B) </span> \frac 72 \qquad <span class='latex-bold'>(C) </span> 4 \qquad <span class='latex-bold'>(D) </span> \frac 92 \qquad <span class='latex-bold'>(E) </span> 8
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22
2
Hide problems
Pig and goat currency
Two farmers agree that pigs are worth
$
300
\$300
$300
and that goats are worth
$
210
\$210
$210
. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a
$
390
\$390
$390
debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
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$
5
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<span class='latex-bold'>(A) </span> \$5\qquad <span class='latex-bold'>(B) </span> \$10\qquad <span class='latex-bold'>(C) </span> \$30\qquad <span class='latex-bold'>(D) </span> \$90\qquad <span class='latex-bold'>(E) </span> \$210
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Elmo's Sandwiches
Elmo makes
N
N
N
sandwiches for a fundraiser. For each sandwich he uses
B
B
B
globs of peanut butter at 4 cents per glob and
J
J
J
blobs of jam at 5 cents per glob. The cost of the peanut butter and jam to make all the sandwiches is
$
\$
$
2.53. Assume that
B
,
J
,
B, J,
B
,
J
,
and
N
N
N
are all positive integers with
N
>
1
N > 1
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1
. What is the cost of the jam Elmo uses to make the sandwiches?
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<span class='latex-bold'>(A) </span> \$1.05 \qquad <span class='latex-bold'>(B) </span> \$1.25 \qquad <span class='latex-bold'>(C) </span> \$1.45 \qquad <span class='latex-bold'>(D) </span> \$1.65 \qquad <span class='latex-bold'>(E) </span> \$1.85
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23
2
Hide problems
Internal tangent of circles
Circles with centers
A
A
A
and
B
B
B
have radii 3 and 8, respectively. A common internal tangent intersects the circles at
C
C
C
and
D
D
D
, respectively. Lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at
E
E
E
, and AE \equal{} 5. What is
C
D
CD
C
D
? [asy]unitsize(2.5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3;pair A=(0,0), Ep=(5,0), B=(5+40/3,0); pair M=midpoint(A--Ep); pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1]; pair D=B+8*dir(180+degrees(C));dot(A); dot(C); dot(B); dot(D); draw(C--D); draw(A--B); draw(Circle(A,3)); draw(Circle(B,8));label("
A
A
A
",A,W); label("
B
B
B
",B,E); label("
C
C
C
",C,SE); label("
E
E
E
",Ep,SSE); label("
D
D
D
",D,NW);[/asy]
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3
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3
<span class='latex-bold'>(A) </span> 13\qquad <span class='latex-bold'>(B) </span> \frac {44}{3}\qquad <span class='latex-bold'>(C) </span> \sqrt {221}\qquad <span class='latex-bold'>(D) </span> \sqrt {255}\qquad <span class='latex-bold'>(E) </span> \frac {55}{3}
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3
55
2006 AMC 10 #23
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? [asy] unitsize(1.5cm); defaultpen(.8); pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A); pair F = intersectionpoint( A--D, B--Ep ); draw( A -- B -- C -- cycle ); draw( A -- D ); draw( B -- Ep ); filldraw( D -- F -- Ep -- C -- cycle, mediumgray, black ); label("
7
7
7
",(1.25,0.2)); label("
7
7
7
",(2.2,0.45)); label("
3
3
3
",(0.45,0.35));[/asy]
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<span class='latex-bold'>(A) </span>15\qquad<span class='latex-bold'>(B) </span>17\qquad<span class='latex-bold'>(C) </span>\frac{35}{2}\qquad<span class='latex-bold'>(D) </span>18\qquad<span class='latex-bold'>(E) </span>\frac{55}{3}
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25
2
Hide problems
Why would a bug be moving along the edges of a cube?
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
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1
2187
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<span class='latex-bold'>(A) </span> \frac {1}{2187} \qquad <span class='latex-bold'>(B) </span> \frac {1}{729} \qquad <span class='latex-bold'>(C) </span> \frac {2}{243} \qquad <span class='latex-bold'>(D) </span> \frac {1}{81} \qquad <span class='latex-bold'>(E) </span> \frac {5}{243}
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2187
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729
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243
2
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81
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243
5
License Plates
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
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4
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5
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6
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7
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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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4
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8