Difference between revisions of "2005 AMC 10B Problems"
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+ | {{AMC10 Problems|year=2005|ab=B}} | ||
== Problem 1 == | == Problem 1 == | ||
− | A scout troop buys <math>1000</math> candy bars at a price of five for | + | A scout troop buys <math>1000</math> candy bars at a price of five for $<math>2</math>. They sell all the candy bars at a price of two for $<math>1</math>. What was the profit, in dollars? |
<math>\mathrm{(A)} 100 \qquad \mathrm{(B)} 200 \qquad \mathrm{(C)} 300 \qquad \mathrm{(D)} 400 \qquad \mathrm{(E)} 500</math> | <math>\mathrm{(A)} 100 \qquad \mathrm{(B)} 200 \qquad \mathrm{(C)} 300 \qquad \mathrm{(D)} 400 \qquad \mathrm{(E)} 500</math> | ||
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== Problem 8 == | == Problem 8 == | ||
+ | An <math> 8</math>-foot by <math> 10</math>-foot floor is tiled with square tiles of size <math> 1</math> foot by <math> 1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math> 1/2</math> foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? | ||
+ | <asy>unitsize(2cm); | ||
+ | defaultpen(linewidth(.8pt)); | ||
+ | |||
+ | fill(unitsquare,gray); | ||
+ | filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black); | ||
+ | filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black); | ||
+ | filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black); | ||
+ | filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);</asy><math> \textrm{(A)}\ 80-20\pi \qquad | ||
+ | \textrm{(B)}\ 60-10\pi \qquad | ||
+ | \textrm{(C)}\ 80-10\pi \qquad | ||
+ | \textrm{(D)}\ 60+10\pi \qquad | ||
+ | \textrm{(E)}\ 80+10\pi</math> | ||
[[2005 AMC 10B Problems/Problem 8|Solution]] | [[2005 AMC 10B Problems/Problem 8|Solution]] | ||
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== Problem 11 == | == Problem 11 == | ||
− | The first term of a sequence is <math>2005</math>. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the <math>{2005}^{th}</math> term of the sequence? | + | The first term of a sequence is <math>2005</math>. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the <math>{2005}^{\text{th}}</math> term of the sequence? |
<math>\mathrm{(A)} 29 \qquad \mathrm{(B)} 55 \qquad \mathrm{(C)} 85 \qquad \mathrm{(D)} 133 \qquad \mathrm{(E)} 250 </math> | <math>\mathrm{(A)} 29 \qquad \mathrm{(B)} 55 \qquad \mathrm{(C)} 85 \qquad \mathrm{(D)} 133 \qquad \mathrm{(E)} 250 </math> | ||
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== Problem 14 == | == Problem 14 == | ||
+ | Equilateral <math> \triangle ABC</math> has side length <math> 2</math>, <math> M</math> is the midpoint of <math> \overline{AC}</math>, and <math> C</math> is the midpoint of <math> \overline{BD}</math>. What is the area of <math> \triangle CDM</math>? | ||
+ | <asy>defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | |||
+ | pair B = (0,0); | ||
+ | pair A = 2*dir(60); | ||
+ | pair C = (2,0); | ||
+ | pair D = (4,0); | ||
+ | pair M = midpoint(A--C); | ||
+ | |||
+ | label("$A$",A,NW);label("$B$",B,SW);label("$C$",C, SE);label("$M$",M,NE);label("$D$",D,SE); | ||
+ | |||
+ | draw(A--B--C--cycle); | ||
+ | draw(C--D--M--cycle);</asy><math> \textrm{(A)}\ \frac {\sqrt {2}}{2}\qquad \textrm{(B)}\ \frac {3}{4}\qquad \textrm{(C)}\ \frac {\sqrt {3}}{2}\qquad \textrm{(D)}\ 1\qquad \textrm{(E)}\ \sqrt {2}</math> | ||
[[2005 AMC 10B Problems/Problem 14|Solution]] | [[2005 AMC 10B Problems/Problem 14|Solution]] | ||
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== Problem 15 == | == Problem 15 == | ||
− | An envelope contains eight bills: <math>2</math> ones, <math>2</math> fives, <math>2</math> tens, and <math>2</math> twenties. Two bills are drawn at random without replacement. What is the probability that their sum is | + | An envelope contains eight bills: <math>2</math> ones, <math>2</math> fives, <math>2</math> tens, and <math>2</math> twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $<math>20</math> or more? |
<math>\mathrm{(A)} \frac{1}{4} \qquad \mathrm{(B)} \frac{2}{5} \qquad \mathrm{(C)} \frac{3}{7} \qquad \mathrm{(D)} \frac{1}{2} \qquad \mathrm{(E)} \frac{2}{3} </math> | <math>\mathrm{(A)} \frac{1}{4} \qquad \mathrm{(B)} \frac{2}{5} \qquad \mathrm{(C)} \frac{3}{7} \qquad \mathrm{(D)} \frac{1}{2} \qquad \mathrm{(E)} \frac{2}{3} </math> | ||
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== Problem 17 == | == Problem 17 == | ||
− | Suppose that <math>4^a = 5</math>, <math>5^b = 6</math>, <math>6^c = 7</math>, and <math>7^d = 8</math>. What is <math>a | + | Suppose that <math>4^a = 5</math>, <math>5^b = 6</math>, <math>6^c = 7</math>, and <math>7^d = 8</math>. What is <math>a \cdot b \cdot c \cdot d</math>? |
<math>\mathrm{(A)} 1 \qquad \mathrm{(B)} \frac{3}{2} \qquad \mathrm{(C)} 2 \qquad \mathrm{(D)} \frac{5}{2} \qquad \mathrm{(E)} 3 </math> | <math>\mathrm{(A)} 1 \qquad \mathrm{(B)} \frac{3}{2} \qquad \mathrm{(C)} 2 \qquad \mathrm{(D)} \frac{5}{2} \qquad \mathrm{(E)} 3 </math> | ||
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On a certain math exam, <math>10\%</math> of the students got <math>70</math> points, <math>25\%</math> got <math>80</math> points, <math>20\%</math> got <math>85</math> points, <math>15\%</math> got <math>90</math> points, and the rest got <math>95</math> points. What is the difference between the mean and the median score on this exam? | On a certain math exam, <math>10\%</math> of the students got <math>70</math> points, <math>25\%</math> got <math>80</math> points, <math>20\%</math> got <math>85</math> points, <math>15\%</math> got <math>90</math> points, and the rest got <math>95</math> points. What is the difference between the mean and the median score on this exam? | ||
− | <math>\mathrm{(A)} 0 \qquad \mathrm{(B)} 1 \qquad \mathrm{(C)} 2 \qquad \mathrm{(D)} 4 \qquad \mathrm{(E)} 5 </math> | + | <math>\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5 </math> |
[[2005 AMC 10B Problems/Problem 19|Solution]] | [[2005 AMC 10B Problems/Problem 19|Solution]] | ||
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== Problem 21 == | == Problem 21 == | ||
− | Forty slips are placed into a hat, each bearing a number <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, <math>9</math>, or <math>10</math>, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let <math>p</math> be the probability that all four slips bear the same number. Let <math>q</math> be the probability that two of the slips bear a number <math>a</math> and the other two bear a number <math>b \neq a</math>. What is the value of <math>q | + | Forty slips are placed into a hat, each bearing a number <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, <math>6</math>, <math>7</math>, <math>8</math>, <math>9</math>, or <math>10</math>, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let <math>p</math> be the probability that all four slips bear the same number. Let <math>q</math> be the probability that two of the slips bear a number <math>a</math> and the other two bear a number <math>b \neq a</math>. What is the value of <math>\frac{q}{p}</math>? |
<math>\mathrm{(A)} 162 \qquad \mathrm{(B)} 180 \qquad \mathrm{(C)} 324 \qquad \mathrm{(D)} 360 \qquad \mathrm{(E)} 720 </math> | <math>\mathrm{(A)} 162 \qquad \mathrm{(B)} 180 \qquad \mathrm{(C)} 324 \qquad \mathrm{(D)} 360 \qquad \mathrm{(E)} 720 </math> | ||
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== See also == | == See also == | ||
+ | {{AMC10 box|year=2005|ab=B|before=[[2005 AMC 10A Problems]]|after=[[2006 AMC 10A Problems]]}} | ||
* [[AMC 10]] | * [[AMC 10]] | ||
* [[AMC 10 Problems and Solutions]] | * [[AMC 10 Problems and Solutions]] | ||
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* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript] | * [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 21:20, 23 December 2020
2005 AMC 10B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
A scout troop buys candy bars at a price of five for $
. They sell all the candy bars at a price of two for $
. What was the profit, in dollars?
Problem 2
A positive number has the property that
of
is
. What is
?
Problem 3
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?
Problem 4
For real numbers and
, define
. What is the value of
?
Problem 5
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
Problem 6
At the beginning of the school year, Lisa's goal was to earn an A on at least of her
quizzes for the year. She earned an A on
of the first
quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?
Problem 7
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?
Problem 8
An -foot by
-foot floor is tiled with square tiles of size
foot by
foot. Each tile has a pattern consisting of four white quarter circles of radius
foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
Problem 9
One fair die has faces ,
,
,
,
,
and another has faces
,
,
,
,
,
. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?
Problem 10
In , we have
and
. Suppose that
is a point on line
such that
lies between
and
and
. What is
?
Problem 11
The first term of a sequence is . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the
term of the sequence?
Problem 12
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
Problem 13
How many numbers between and
are integer multiples of
or
but not
?
Problem 14
Equilateral has side length
,
is the midpoint of
, and
is the midpoint of
. What is the area of
?
Problem 15
An envelope contains eight bills: ones,
fives,
tens, and
twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $
or more?
Problem 16
The quadratic equation has roots that are twice those of
, and none of
,
, and
is zero. What is the value of
?
Problem 17
Suppose that ,
,
, and
. What is
?
Problem 18
All of David's telephone numbers have the form , where
,
,
,
,
,
, and
are distinct digits and in increasing order, and none is either
or
. How many different telephone numbers can David have?
Problem 19
On a certain math exam, of the students got
points,
got
points,
got
points,
got
points, and the rest got
points. What is the difference between the mean and the median score on this exam?
Problem 20
What is the average (mean) of all -digit numbers that can be formed by using each of the digits
,
,
,
, and
exactly once?
Problem 21
Forty slips are placed into a hat, each bearing a number ,
,
,
,
,
,
,
,
, or
, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let
be the probability that all four slips bear the same number. Let
be the probability that two of the slips bear a number
and the other two bear a number
. What is the value of
?
Problem 22
For how many positive integers less than or equal to
is
evenly divisible by
?
Problem 23
In trapezoid we have
parallel to
,
as the midpoint of
, and
as the midpoint of
. The area of
is twice the area of
. What is
?
Problem 24
Let and
be two-digit integers such that
is obtained by reversing the digits
of
. The integers
and
satisfy
for some positive integer
.
What is
?
Problem 25
A subset of the set of integers from
to
, inclusive, has the property that no two elements of
sum to
. What is the maximum possible number of elements in
?
See also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2005 AMC 10A Problems |
Followed by 2006 AMC 10A Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
- AMC 10
- AMC 10 Problems and Solutions
- 2005 AMC 10B
- 2005 AMC B Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.