Difference between revisions of "2005 AMC 12B Problems"
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+ | {{AMC12 Problems|year=2005|ab=B}} | ||
== Problem 1 == | == Problem 1 == | ||
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars? | A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars? | ||
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== Problem 2 == | == Problem 2 == | ||
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? | A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? | ||
+ | |||
<math> | <math> | ||
\mathrm{(A)}\ 2 \qquad | \mathrm{(A)}\ 2 \qquad | ||
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== Problem 5 == | == Problem 5 == | ||
− | An <math>8</math>-foot by <math>10</math>-foot floor is | + | 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> | <asy> | ||
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== Problem 9 == | == Problem 9 == | ||
− | On a certain math exam, <math>10</math> | + | 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> | ||
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== Problem 10 == | == Problem 10 == | ||
− | The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the | + | 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 terms. 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 11 == | == Problem 11 == | ||
− | 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? | + | 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}{7}}}} \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}{7}}}} \qquad \mathrm{(C)}\ {{{\frac{3}{7}}}} \qquad \mathrm{(D)}\ {{{\frac{1}{2}}}} \qquad \mathrm{(E)}\ {{{\frac{2}{3}}}}</math> | ||
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== Problem 13 == | == Problem 13 == | ||
− | Suppose that <math>4^{x_1}=5</math>, <math>5 | + | Suppose that <math>4^{x_1}=5</math>, <math>5^{x_2}=6</math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math>. What is <math>x_1x_2...x_{124}</math>? |
<math>\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</math> | <math>\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</math> | ||
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== Problem 15 == | == Problem 15 == | ||
− | The sum of four two-digit numbers is <math>221</math>. | + | The sum of four two-digit numbers is <math>221</math>. None of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits? |
<math> | <math> | ||
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<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath> | <cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath> | ||
− | There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y | + | There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y,z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math> |
<math> | <math> | ||
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== See also == | == See also == | ||
+ | |||
+ | {{AMC12 box|year=2005|ab=B|before=[[2005 AMC 12A Problems]]|after=[[2006 AMC 12A Problems]]}} | ||
+ | |||
* [[AMC 12]] | * [[AMC 12]] | ||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 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 12:02, 19 February 2020
2005 AMC 12B (Answer Key) Printable version: | 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 dollars. They sell all the candy bars at the price of two for dollar. What was their profit, in dollars?
Problem 2
A positive number has the property that of is . What is ?
Problem 3
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 4
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 5
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 6
In , we have and . Suppose that is a point on line such that lies between and and . What is ?
Problem 7
What is the area enclosed by the graph of ?
Problem 8
For how many values of is it true that the line passes through the vertex of the parabola ?
Problem 9
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 10
The first term of a sequence is . Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the term of the sequence?
Problem 11
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 12
The quadratic equation has roots twice those of , and none of and is zero. What is the value of ?
Problem 13
Suppose that , , , ... , . What is ?
Problem 14
A circle having center , with ,is tangent to the lines , and . What is the radius of this circle?
Problem 15
The sum of four two-digit numbers is . None of the eight digits is and no two of them are the same. Which of the following is not included among the eight digits?
Problem 16
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
Problem 17
How many distinct four-tuples of rational numbers are there with
?
Problem 18
Let and be points in the plane. Define as the region in the first quadrant consisting of those points such that is an acute triangle. What is the closest integer to the area of the region ?
Problem 19
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 20
Let and be distinct elements in the set
What is the minimum possible value of
Problem 21
A positive integer has divisors and has divisors. What is the greatest integer such that divides ?
Problem 22
A sequence of complex numbers is defined by the rule
where is the complex conjugate of and . Suppose that and . How many possible values are there for ?
Problem 23
Let be the set of ordered triples of real numbers for which
There are real numbers and such that for all ordered triples in we have What is the value of
Problem 24
All three vertices of an equilateral triangle are on the parabola , and one of its sides has a slope of . The -coordinates of the three vertices have a sum of , where and are relatively prime positive integers. What is the value of ?
Problem 25
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
See also
2005 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2005 AMC 12A Problems |
Followed by 2006 AMC 12A 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 12 Problems and Solutions |
- AMC 12
- AMC 12 Problems and Solutions
- 2005 AMC 12B
- 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.