Difference between revisions of "2006 AMC 12B Problems"
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== Problem 4 == | == Problem 4 == | ||
− | Mary is about to pay for five items at the grocery store. The prices of the items are <math> | + | Mary is about to pay for five items at the grocery store. The prices of the items are <math>$ </math>7.99<math>, </math>$ <math>4.99</math>, <math>$ </math>2.99<math>, </math>$ <math>1.99</math>, and <math>$ </math>0.99<math>. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the </math>$ <math>20.00</math> that she will receive in change? |
<math> | <math> | ||
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== Problem 14 == | == Problem 14 == | ||
− | Elmo makes <math>N</math> sandwiches for a fundraiser. For each sandwich he uses <math>B</math> globs of peanut butter at <math>4</math> cents per glob and <math>J</math> blobs of jam at <math>5</math> cents per glob. The cost of the peanut butter and jam to make all the sandwiches is <math> | + | Elmo makes <math>N</math> sandwiches for a fundraiser. For each sandwich he uses <math>B</math> globs of peanut butter at <math>4</math> cents per glob and <math>J</math> blobs of jam at <math>5</math> cents per glob. The cost of the peanut butter and jam to make all the sandwiches is <math>$</math> 2.53<math>. Assume that </math>B<math>, </math>J<math> and </math>N<math> are all positive integers with </math>N>1<math>. What is the cost of the jam Elmo uses to make the sandwiches? |
− | <math> | + | </math> |
\mathrm{(A)}\ 1.05 | \mathrm{(A)}\ 1.05 | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ 1.85 | \mathrm{(E)}\ 1.85 | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 14|Solution]] | [[2006 AMC 12B Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
− | {{ | + | Circles with centers </math> O<math> and </math> P<math> have radii 2 and 4, respectively, and are externally tangent. Points </math> A<math> and </math> B<math> are on the circle centered at </math> O<math>, and points </math> C<math> and </math> D<math> are on the circle centered at </math> P<math>, such that </math> \overline{AD}<math> and </math> \overline{BC}<math> are common external tangents to the circles. What is the area of hexagon </math> AOBCPD<math>? |
+ | |||
+ | {{image}} | ||
+ | |||
+ | </math> \textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}<math> | ||
[[2006 AMC 12B Problems/Problem 15|Solution]] | [[2006 AMC 12B Problems/Problem 15|Solution]] | ||
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== Problem 16 == | == Problem 16 == | ||
− | Regular hexagon <math>ABCDEF< | + | Regular hexagon </math>ABCDEF<math> has vertices </math>A<math> and </math>C<math> at </math>(0,0)<math> and </math>(7,1)<math>, respectively. What is its area? |
− | <math> | + | </math> |
\mathrm{(A)}\ 20\sqrt {3} | \mathrm{(A)}\ 20\sqrt {3} | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ 50 | \mathrm{(E)}\ 50 | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 16|Solution]] | [[2006 AMC 12B Problems/Problem 16|Solution]] | ||
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== Problem 17 == | == Problem 17 == | ||
− | For a particular peculiar pair of dice, the probabilities of rolling <math>1< | + | For a particular peculiar pair of dice, the probabilities of rolling </math>1<math>, </math>2<math>, </math>3<math>, </math>4<math>, </math>5<math> and </math>6<math> on each die are in the ratio </math>1:2:3:4:5:6<math>. What is the probability of rolling a total of </math>7<math> on the two dice? |
− | <math> | + | </math> |
\mathrm{(A)}\ \frac 4{63} | \mathrm{(A)}\ \frac 4{63} | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ \frac 27 | \mathrm{(E)}\ \frac 27 | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 17|Solution]] | [[2006 AMC 12B Problems/Problem 17|Solution]] | ||
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An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? | An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? | ||
− | <math> | + | </math> |
\mathrm{(A)}\ 120 | \mathrm{(A)}\ 120 | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ 231 | \mathrm{(E)}\ 231 | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 18|Solution]] | [[2006 AMC 12B Problems/Problem 18|Solution]] | ||
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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? | 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? | ||
− | <math> | + | </math> |
\mathrm{(A)}\ 4 | \mathrm{(A)}\ 4 | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ 8 | \mathrm{(E)}\ 8 | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 19|Solution]] | [[2006 AMC 12B Problems/Problem 19|Solution]] | ||
== Problem 20 == | == Problem 20 == | ||
− | Let <math>x< | + | Let </math>x<math> be chosen at random from the interval </math>(0,1)<math>. What is the probability that |
− | <math>\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0< | + | </math>\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0<math>? |
− | Here <math>\lfloor x\rfloor< | + | Here </math>\lfloor x\rfloor<math> denotes the greatest integer that is less than or equal to </math>x<math>. |
− | <math> | + | </math> |
\mathrm{(A)}\ \frac 18 | \mathrm{(A)}\ \frac 18 | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ \frac 14 | \mathrm{(E)}\ \frac 14 | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 20|Solution]] | [[2006 AMC 12B Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
− | Rectange <math>ABCD< | + | Rectange </math>ABCD<math> has area </math>2006<math>. An ellipse with area </math>2006\pi<math> passes through </math>A<math> and </math>C<math> and has foci at </math>B<math> and </math>D<math>. What is the perimeter of the rectangle? (The area of an ellipse is </math>ab\pi<math> where </math>2a<math> and </math>2b<math> are the lengths of the axes.) |
− | <math> | + | </math> |
\mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi} | \mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi} | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ \frac {32\sqrt {1003}}\pi | \mathrm{(E)}\ \frac {32\sqrt {1003}}\pi | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 21|Solution]] | [[2006 AMC 12B Problems/Problem 21|Solution]] | ||
== Problem 22 == | == Problem 22 == | ||
− | Suppose <math>a< | + | Suppose </math>a<math>, </math>b<math> and </math>c<math> are positive integers with </math>a+b+c=2006<math>, and </math>a!b!c!=m\cdot 10^n<math>, where </math>m<math> and </math>n<math> are integers and </math>m<math> is not divisible by </math>10<math>. What is the smallest possible value of </math>n<math>? |
− | <math> | + | </math> |
\mathrm{(A)}\ 489 | \mathrm{(A)}\ 489 | ||
\qquad | \qquad | ||
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\qquad | \qquad | ||
\mathrm{(E)}\ 501 | \mathrm{(E)}\ 501 | ||
− | < | + | <math> |
[[2006 AMC 12B Problems/Problem 22|Solution]] | [[2006 AMC 12B Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
− | Isosceles <math>\triangle ABC< | + | Isosceles </math>\triangle ABC<math> has a right angle at </math>C<math>. Point </math>P<math> is inside </math>\triangle ABC<math>, such that </math>PA=11<math>, </math>PB=7<math>, and </math>PC=6<math>. Legs </math>\overline{AC}<math> and </math>\overline{BC}<math> have length </math>s=\sqrt{a+b\sqrt{2}{<math>, where </math>a<math> and </math>b<math> are positive integers. What is </math>a+b$? |
<asy> | <asy> |
Revision as of 23:20, 25 August 2011
Contents
Problem 1
What is ?
Problem 2
For real numbers and
, define
. What is
?
Problem 3
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?
Problem 4
Mary is about to pay for five items at the grocery store. The prices of the items are 7.99
$
,
2.99
$
, and
0.99
$
that she will receive in change?
Problem 5
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
Problem 6
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.
Problem 7
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?
Problem 8
The lines and
intersect at the point
. What is
?
Problem 9
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?
Problem 10
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?
Problem 11
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?
Problem 12
The parabola has vertex
and
-intercept
, where
. What is
?
Problem 13
Rhombus is similar to rhombus
. The area of rhombus
is 24, and $\angle BAD \equal{} 60^\circ$ (Error compiling LaTeX. Unknown error_msg). What is the area of rhombus
?
Problem 14
Elmo makes sandwiches for a fundraiser. For each sandwich he uses
globs of peanut butter at
cents per glob and
blobs of jam at
cents per glob. The cost of the peanut butter and jam to make all the sandwiches is
2.53
B
J
N
N>1
\mathrm{(A)}\ 1.05
\qquad
\mathrm{(B)}\ 1.25
\qquad
\mathrm{(C)}\ 1.45
\qquad
\mathrm{(D)}\ 1.65
\qquad
\mathrm{(E)}\ 1.85
$[[2006 AMC 12B Problems/Problem 14|Solution]]
== Problem 15 ==
Circles with centers$ (Error compiling LaTeX. Unknown error_msg) O P
A
B
O
C
D
P
\overline{AD}
\overline{BC}
AOBCPD$?
{{image}}$ (Error compiling LaTeX. Unknown error_msg) \textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}$[[2006 AMC 12B Problems/Problem 15|Solution]]
== Problem 16 ==
Regular hexagon$ (Error compiling LaTeX. Unknown error_msg)ABCDEFA
C
(0,0)
(7,1)
\mathrm{(A)}\ 20\sqrt {3}
\qquad
\mathrm{(B)}\ 22\sqrt {3}
\qquad
\mathrm{(C)}\ 25\sqrt {3}
\qquad
\mathrm{(D)}\ 27\sqrt {3}
\qquad
\mathrm{(E)}\ 50
$[[2006 AMC 12B Problems/Problem 16|Solution]]
== Problem 17 ==
For a particular peculiar pair of dice, the probabilities of rolling$ (Error compiling LaTeX. Unknown error_msg)12
3
4
5
6
1:2:3:4:5:6
7
\mathrm{(A)}\ \frac 4{63}
\qquad
\mathrm{(B)}\ \frac 18
\qquad
\mathrm{(C)}\ \frac 8{63}
\qquad
\mathrm{(D)}\ \frac 16
\qquad
\mathrm{(E)}\ \frac 27
$[[2006 AMC 12B Problems/Problem 17|Solution]]
== Problem 18 == An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A)}\ 120 \qquad \mathrm{(B)}\ 121 \qquad \mathrm{(C)}\ 221 \qquad \mathrm{(D)}\ 230 \qquad \mathrm{(E)}\ 231 $[[2006 AMC 12B Problems/Problem 18|Solution]]
== Problem 19 == Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A)}\ 4 \qquad \mathrm{(B)}\ 5 \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(D)}\ 7 \qquad \mathrm{(E)}\ 8 $[[2006 AMC 12B Problems/Problem 19|Solution]]
== Problem 20 ==
Let$ (Error compiling LaTeX. Unknown error_msg)x(0,1)
\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0
\lfloor x\rfloor
x
\mathrm{(A)}\ \frac 18
\qquad
\mathrm{(B)}\ \frac 3{20}
\qquad
\mathrm{(C)}\ \frac 16
\qquad
\mathrm{(D)}\ \frac 15
\qquad
\mathrm{(E)}\ \frac 14
$[[2006 AMC 12B Problems/Problem 20|Solution]]
== Problem 21 ==
Rectange$ (Error compiling LaTeX. Unknown error_msg)ABCD2006
2006\pi
A
C
B
D
ab\pi
2a
2b
\mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi}
\qquad
\mathrm{(B)}\ \frac {1003}4
\qquad
\mathrm{(C)}\ 8\sqrt {1003}
\qquad
\mathrm{(D)}\ 6\sqrt {2006}
\qquad
\mathrm{(E)}\ \frac {32\sqrt {1003}}\pi
$[[2006 AMC 12B Problems/Problem 21|Solution]]
== Problem 22 ==
Suppose$ (Error compiling LaTeX. Unknown error_msg)ab
c
a+b+c=2006
a!b!c!=m\cdot 10^n
m
n
m
10
n
\mathrm{(A)}\ 489
\qquad
\mathrm{(B)}\ 492
\qquad
\mathrm{(C)}\ 495
\qquad
\mathrm{(D)}\ 498
\qquad
\mathrm{(E)}\ 501
$[[2006 AMC 12B Problems/Problem 22|Solution]]
== Problem 23 ==
Isosceles$ (Error compiling LaTeX. Unknown error_msg)\triangle ABCC
P
\triangle ABC
PA=11
PB=7
PC=6
\overline{AC}
\overline{BC}
s=\sqrt{a+b\sqrt{2}{
a
b
a+b$?
Problem 24
Let be the set of all points
in the coordinate plane such that
and
. What is the area of the subset of
for which
?
Problem 25
A sequence of non-negative integers is defined by the rule
for
. If
,
and
, how many different values of
are possible?