Difference between revisions of "2007 AMC 12A Problems"
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+ | {{AMC12 Problems|year=2007|ab=A}} | ||
==Problem 1== | ==Problem 1== | ||
− | One ticket to a show costs <math>20</math> at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan? | + | One ticket to a show costs <math>\$20</math> at full price. Susan buys <math>4</math> tickets using a coupon that gives her a <math>25\%</math> discount. Pam buys <math>5</math> tickets using a coupon that gives her a <math>30\%</math> discount. How many more dollars does Pam pay than Susan? |
− | <math>\ | + | <math>\textbf{(A)} \ 2 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 10 \qquad \textbf{(D)} \ 15 \qquad \textbf{(E)} \ 20</math> |
[[2007 AMC 12A Problems/Problem 1 | Solution]] | [[2007 AMC 12A Problems/Problem 1 | Solution]] | ||
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==Problem 5== | ==Problem 5== | ||
− | Last year Mr. Jon Q. Public received an inheritance. He paid <math>20\%</math> in federal taxes on the inheritance, and paid <math>10\%</math> of what he had left in state taxes. He paid a total of <math>\</math> | + | Last year Mr. Jon Q. Public received an inheritance. He paid <math>20\%</math> in federal taxes on the inheritance, and paid <math>10\%</math> of what he had left in state taxes. He paid a total of <math>\textdollar10,500</math> for both taxes. How many dollars was his inheritance? |
− | <math>(\mathrm {A}) | + | <math>(\mathrm {A})\ 30\,000 \qquad (\mathrm {B})\ 32\,500 \qquad (\mathrm {C})\ 35\,000 \qquad (\mathrm {D})\ 37\,500 \qquad (\mathrm {E})\ 40\,000</math> |
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==Problem 10== | ==Problem 10== | ||
− | A [[triangle]] with side lengths in the [[ratio]] <math>3 : 4 : 5</math> is inscribed in a [[circle]] with [[radius]] 3. What | + | A [[triangle]] with side lengths in the [[ratio]] <math>3 : 4 : 5</math> is inscribed in a [[circle]] with [[radius]] 3. What is the area of the triangle? |
<math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math> | <math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math> | ||
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Let a, b, c, d, and e be distinct [[integer]]s such that | Let a, b, c, d, and e be distinct [[integer]]s such that | ||
− | < | + | <center><math>(6-a)(6-b)(6-c)(6-d)(6-e)=45</math></center> |
What is <math>a+b+c+d+e</math>? | What is <math>a+b+c+d+e</math>? | ||
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[[Triangle]]s <math>ABC</math> and <math>ADE</math> have [[area]]s <math>2007</math> and <math>7002,</math> respectively, with <math>B = (0,0),</math> <math>C = (223,0),</math> <math>D = (680,380),</math> and <math>E = (689,389).</math> What is the sum of all possible x-coordinates of <math>A</math>? | [[Triangle]]s <math>ABC</math> and <math>ADE</math> have [[area]]s <math>2007</math> and <math>7002,</math> respectively, with <math>B = (0,0),</math> <math>C = (223,0),</math> <math>D = (680,380),</math> and <math>E = (689,389).</math> What is the sum of all possible x-coordinates of <math>A</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ 282 \qquad \mathrm{(B)}\ 300 \qquad \mathrm{(C)}\ 600 \qquad \mathrm{(D)}\ 900 \qquad \mathrm{(E)}\ 1200</math> |
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==Problem 22== | ==Problem 22== | ||
+ | For each positive integer <math>n</math>, let <math>S(n)</math> denote the sum of the digits of <math>n.</math> For how many values of <math>n</math> is <math>n + S(n) + S(S(n)) = 2007?</math> | ||
+ | |||
+ | <math>\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5</math> | ||
[[2007 AMC 12A Problems/Problem 22 | Solution]] | [[2007 AMC 12A Problems/Problem 22 | Solution]] | ||
+ | |||
==Problem 23== | ==Problem 23== | ||
+ | [[Square]] <math>ABCD</math> has area <math>36,</math> and <math>\overline{AB}</math> is [[parallel]] to the [[x-axis]]. Vertices <math>A,</math> <math>B</math>, and <math>C</math> are on the graphs of <math>y = \log_{a}x,</math> <math>y = 2\log_{a}x,</math> and <math>y = 3\log_{a}x,</math> respectively. What is <math>a?</math> | ||
+ | |||
+ | <math>\mathrm{(A)}\ \sqrt [6]{3}\qquad \mathrm{(B)}\ \sqrt {3}\qquad \mathrm{(C)}\ \sqrt [3]{6}\qquad \mathrm{(D)}\ \sqrt {6}\qquad \mathrm{(E)}\ 6</math> | ||
[[2007 AMC 12A Problems/Problem 23 | Solution]] | [[2007 AMC 12A Problems/Problem 23 | Solution]] | ||
+ | |||
==Problem 24== | ==Problem 24== | ||
+ | For each integer <math>n>1</math>, let <math>F(n)</math> be the number of solutions to the equation <math>\sin{x}=\sin{(nx)}</math> on the interval <math>[0,\pi]</math>. What is <math>\sum_{n=2}^{2007} F(n)</math>? | ||
+ | |||
+ | <math>\mathrm{(A)}\ 2014524</math> <math>\mathrm{(B)}\ 2015028</math> <math>\mathrm{(C)}\ 2015033</math> <math>\mathrm{(D)}\ 2016532</math> <math>\mathrm{(E)}\ 2017033</math> | ||
+ | [[2007 AMC 12A Problems/Problem 24 | Solution]] | ||
− | |||
==Problem 25== | ==Problem 25== | ||
+ | Call a set of integers ''spacy'' if it contains no more than one out of any three consecutive integers. How many [[subset]]s of <math>\{1,2,3,\ldots,12\},</math> including the [[empty set]], are spacy? | ||
+ | |||
+ | <math>\mathrm{(A)}\ 121 \qquad \mathrm{(B)}\ 123 \qquad \mathrm{(C)}\ 125 \qquad \mathrm{(D)}\ 127 \qquad \mathrm{(E)}\ 129</math> | ||
[[2007 AMC 12A Problems/Problem 25 | Solution]] | [[2007 AMC 12A Problems/Problem 25 | Solution]] | ||
+ | |||
+ | == See also == | ||
+ | {{AMC12 box|year=2007|ab=A|before=[[2006 AMC 12B Problems|2006 AMC 12B]]|after=[[2007 AMC 12B Problems|2007 AMC 12B]]}} | ||
+ | * [[AMC 12]] | ||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=142 2007 AMC A Math Jam Transcript] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 21:47, 22 March 2021
2007 AMC 12A (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
One ticket to a show costs at full price. Susan buys tickets using a coupon that gives her a discount. Pam buys tickets using a coupon that gives her a discount. How many more dollars does Pam pay than Susan?
Problem 2
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?
Problem 3
The larger of two consecutive odd integers is three times the smaller. What is their sum?
Problem 4
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?
Problem 5
Last year Mr. Jon Q. Public received an inheritance. He paid in federal taxes on the inheritance, and paid of what he had left in state taxes. He paid a total of for both taxes. How many dollars was his inheritance?
Problem 6
Triangles and are isosceles with and . Point is inside triangle , angle measures 40 degrees, and angle measures 140 degrees. What is the degree measure of angle ?
Problem 7
Let , and be five consecutive terms in an arithmetic sequence, and suppose that . Which of or can be found?
Problem 8
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?
Problem 9
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
Problem 10
A triangle with side lengths in the ratio is inscribed in a circle with radius 3. What is the area of the triangle?
Problem 11
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let be the sum of all the terms in the sequence. What is the largest prime factor that always divides ?
Problem 12
Integers and , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that is even?
Problem 13
A piece of cheese is located at in a coordinate plane. A mouse is at and is running up the line . At the point the mouse starts getting farther from the cheese rather than closer to it. What is ?
Problem 14
Let a, b, c, d, and e be distinct integers such that
What is ?
Problem 15
The set is augmented by a fifth element , not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of ?
Problem 16
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
Problem 17
Suppose that and . What is ?
Problem 18
The polynomial has real coefficients, and What is
Problem 19
Triangles and have areas and respectively, with and What is the sum of all possible x-coordinates of ?
Problem 20
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
Problem 21
The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function are equal. Their common value must also be which of the following?
Problem 22
For each positive integer , let denote the sum of the digits of For how many values of is
Problem 23
Square has area and is parallel to the x-axis. Vertices , and are on the graphs of and respectively. What is
Problem 24
For each integer , let be the number of solutions to the equation on the interval . What is ?
Problem 25
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of including the empty set, are spacy?
See also
2007 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2006 AMC 12B |
Followed by 2007 AMC 12B |
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
- 2007 AMC A Math Jam Transcript
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.