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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>\mathrm{(A)}\ 2\qquad \mathrm{(B)}\ 5\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 15\qquad \mathrm{(E)}\ 20</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==
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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>10500 for both taxes. How many dollars was his inheritance?
+
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})30000</math>  <math>(\mathrm {B})32500</math>  <math>(\mathrm {C})35000</math>  <math>(\mathrm {D})37500</math>  <math>(\mathrm {E})40000</math>
+
<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==
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A [[triangle]] with side lengths in the [[ratio]] <math>3 : 4 : 5</math> is inscribed in a [[circle]] with [[radius]] 3. What s the area of the triangle?
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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
  
−
<div style="text-align:center;"><math>(6-a)(6-b)(6-c)(6-d)(6-e)=45</math></div>
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<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>\matrhm{(A)}\ 282 \qquad \mathrm{(B)}\ 300 \qquad \mathrm{(C)}\ 600 \qquad \mathrm{(D)}\ 900 \qquad \mathrm{(E)}\ 1200</math>
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<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==
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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 \qquad \mathrm{(B)}\ 2015028 \qquad \mathrm{(C)}\ 2015033 \qquad \mathrm{(D)}\ 2016532 \qquad \mathrm{(E)}\ 2017033</math>
  
 +
[[2007 AMC 12A Problems/Problem 24 | Solution]]
  
−
[[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 12:35, 15 April 2022

2007 AMC 12A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
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

Problem 1

One ticket to a show costs $$20$ 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?

$\textbf{(A)} \ 2 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 10 \qquad \textbf{(D)} \ 15 \qquad \textbf{(E)} \ 20$

Solution

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?

$\mathrm{(A)}\ 0.5\qquad \mathrm{(B)}\ 1\qquad \mathrm{(C)}\ 1.5\qquad \mathrm{(D)}\ 2\qquad \mathrm{(E)}\ 2.5$


Solution

Problem 3

The larger of two consecutive odd integers is three times the smaller. What is their sum?

$\mathrm{(A)}\ 4\qquad \mathrm{(B)}\ 8\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 16\qquad \mathrm{(E)}\ 20$

Solution

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?

$\mathrm{(A)}\ 7\qquad \mathrm{(B)}\ 9\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 12\qquad \mathrm{(E)}\ 14$

Solution

Problem 5

Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\textdollar10,500$ for both taxes. How many dollars was his inheritance?

$(\mathrm {A})\ 30\,000 \qquad (\mathrm {B})\ 32\,500 \qquad (\mathrm {C})\ 35\,000 \qquad (\mathrm {D})\ 37\,500 \qquad (\mathrm {E})\ 40\,000$


Solution

Problem 6

Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside triangle $ABC$, angle $ABC$ measures 40 degrees, and angle $ADC$ measures 140 degrees. What is the degree measure of angle $BAD$?

$\mathrm{(A)}\ 20\qquad \mathrm{(B)}\ 30\qquad \mathrm{(C)}\ 40\qquad \mathrm{(D)}\ 50\qquad \mathrm{(E)}\ 60$

Solution

Problem 7

Let $a, b, c, d$, and $e$ be five consecutive terms in an arithmetic sequence, and suppose that $a+b+c+d+e=30$. Which of $a, b, c, d,$ or $e$ can be found?

$\textrm{(A)} \ a\qquad \textrm{(B)}\ b\qquad \textrm{(C)}\ c\qquad \textrm{(D)}\ d\qquad \textrm{(E)}\ e$


Solution

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?

$\mathrm{(A)}\ 20\qquad \mathrm{(B)}\ 24\qquad \mathrm{(C)}\ 30\qquad \mathrm{(D)}\ 36\qquad \mathrm{(E)}\ 60$

Solution

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?

$\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad \mathrm{(D)}\ \frac 56\qquad \mathrm{(E)}\ \frac 78$


Solution

Problem 10

A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius 3. What is the area of the triangle?

$\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18$

Solution

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 $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?

$\mathrm{(A)}\ 3\qquad \mathrm{(B)}\ 7\qquad \mathrm{(C)}\ 13\qquad \mathrm{(D)}\ 37\qquad \mathrm{(E)}\ 43$

Solution

Problem 12

Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?

$\mathrm{(A)}\ \frac 38\qquad \mathrm{(B)}\ \frac 7{16}\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 9{16}\qquad \mathrm{(E)}\ \frac 58$

Solution

Problem 13

A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a+b$?

$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 14\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 22$

Solution

Problem 14

Let a, b, c, d, and e be distinct integers such that

$(6-a)(6-b)(6-c)(6-d)(6-e)=45$

What is $a+b+c+d+e$?

$\mathrm{(A)}\ 5\qquad \mathrm{(B)}\ 17\qquad \mathrm{(C)}\ 25\qquad \mathrm{(D)}\ 27\qquad \mathrm{(E)}\ 30$


Solution

Problem 15

The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, 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 $n$?

$\mathrm{(A)}\ 7\qquad \mathrm{(B)}\ 9\qquad \mathrm{(C)}\ 19\qquad \mathrm{(D)}\ 24\qquad \mathrm{(E)}\ 26$

Solution

Problem 16

How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?

$\mathrm{(A)}\ 96\qquad \mathrm{(B)}\ 104\qquad \mathrm{(C)}\ 112\qquad \mathrm{(D)}\ 120\qquad \mathrm{(E)}\ 256$

Solution

Problem 17

Suppose that $\sin a + \sin b = \sqrt{\frac{5}{3}}$ and $\cos a + \cos b = 1$. What is $\cos (a - b)$?

$\mathrm{(A)}\ \sqrt{\frac{5}{3}} - 1\qquad \mathrm{(B)}\ \frac 13\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 23\qquad \mathrm{(E)}\ 1$


Solution

Problem 18

The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ What is $a + b + c + d?$

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 9 \qquad \mathrm{(E)}\ 16$

Solution

Problem 19

Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B = (0,0),$ $C = (223,0),$ $D = (680,380),$ and $E = (689,389).$ What is the sum of all possible x-coordinates of $A$?

$\mathrm{(A)}\ 282 \qquad \mathrm{(B)}\ 300 \qquad \mathrm{(C)}\ 600 \qquad \mathrm{(D)}\ 900 \qquad \mathrm{(E)}\ 1200$


Solution

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?

$\mathrm{(A)}\ \frac{5\sqrt{2}-7}{3}\qquad \mathrm{(B)}\ \frac{10-7\sqrt{2}}{3}\qquad \mathrm{(C)}\ \frac{3-2\sqrt{2}}{3}\qquad \mathrm{(D)}\ \frac{8\sqrt{2}-11}{3}\qquad \mathrm{(E)}\ \frac{6-4\sqrt{2}}{3}$


Solution

Problem 21

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $f(x)=ax^{2}+bx+c$ are equal. Their common value must also be which of the following?

$\textrm{(A)}\ \textrm{the\ coefficient\ of\ }x^{2}~~~ \textrm{(B)}\ \textrm{the\ coefficient\ of\ }x$ $\textrm{(C)}\ \textrm{the\ y-intercept\ of\ the\ graph\ of\ }y=f(x)$ $\textrm{(D)}\ \textrm{one\ of\ the\ x-intercepts\ of\ the\ graph\ of\ }y=f(x)$ $\textrm{(E)}\ \textrm{the\ mean\ of\ the\ x-intercepts\ of\ the\ graph\ of\ }y=f(x)$


Solution

Problem 22

For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ For how many values of $n$ is $n + S(n) + S(S(n)) = 2007?$

$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$


Solution

Problem 23

Square $ABCD$ has area $36,$ and $\overline{AB}$ is parallel to the x-axis. Vertices $A,$ $B$, and $C$ are on the graphs of $y = \log_{a}x,$ $y = 2\log_{a}x,$ and $y = 3\log_{a}x,$ respectively. What is $a?$

$\mathrm{(A)}\ \sqrt [6]{3}\qquad \mathrm{(B)}\ \sqrt {3}\qquad \mathrm{(C)}\ \sqrt [3]{6}\qquad \mathrm{(D)}\ \sqrt {6}\qquad \mathrm{(E)}\ 6$


Solution

Problem 24

For each integer $n>1$, let $F(n)$ be the number of solutions to the equation $\sin{x}=\sin{(nx)}$ on the interval $[0,\pi]$. What is $\sum_{n=2}^{2007} F(n)$?

$\mathrm{(A)}\ 2014524 \qquad \mathrm{(B)}\ 2015028 \qquad \mathrm{(C)}\ 2015033 \qquad \mathrm{(D)}\ 2016532 \qquad \mathrm{(E)}\ 2017033$

Solution

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 $\{1,2,3,\ldots,12\},$ including the empty set, are spacy?

$\mathrm{(A)}\ 121 \qquad \mathrm{(B)}\ 123 \qquad \mathrm{(C)}\ 125 \qquad \mathrm{(D)}\ 127 \qquad \mathrm{(E)}\ 129$

Solution

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

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