Difference between revisions of "2008 AMC 12A Problems"
(→Problem 23) |
(→Problem 7) |
||
(38 intermediate revisions by 8 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AMC12 Problems|year=2008|ab=A}} | ||
==Problem 1== | ==Problem 1== | ||
− | A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? | + | A bakery owner turns on his doughnut machine at <math>\text{8:30}\ {\small\text{AM}}</math>. At <math>\text{11:10}\ {\small\text{AM}}</math> the machine has completed one third of the day's job. At what time will the doughnut machine complete the job? |
− | <math>\ | + | <math>\mathrm{(A)}\ \text{1:50}\ {\small\text{PM}}\qquad\mathrm{(B)}\ \text{3:00}\ {\small\text{PM}}\qquad\mathrm{(C)}\ \text{3:30}\ {\small\text{PM}}\qquad\mathrm{(D)}\ \text{4:30}\ {\small\text{PM}}\qquad\mathrm{(E)}\ \text{5:50}\ {\small\text{PM}}</math> |
− | + | [[2008 AMC 12A Problems/Problem 1|Solution]] | |
==Problem 2== | ==Problem 2== | ||
− | What is the reciprocal of <math>\frac{1}{2}+\frac{2}{3}</math>? | + | What is the [[reciprocal]] of <math>\frac{1}{2}+\frac{2}{3}</math>? |
− | <math>\ | + | <math>\mathrm{(A)}\ \frac{6}{7}\qquad\mathrm{(B)}\ \frac{7}{6}\qquad\mathrm{(C)}\ \frac{5}{3}\qquad\mathrm{(D)}\ 3\qquad\mathrm{(E)}\ \frac{7}{2}</math> |
− | + | [[2008 AMC 12A Problems/Problem 2|Solution]] | |
==Problem 3== | ==Problem 3== | ||
− | Suppose that <math>\ | + | Suppose that <math>\tfrac{2}{3}</math> of <math>10</math> bananas are worth as much as <math>8</math> oranges. How many oranges are worth as much as <math>\tfrac{1}{2}</math> of <math>5</math> bananas? |
− | <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> |
− | + | [[2008 AMC 12A Problems/Problem 3|Solution]] | |
==Problem 4== | ==Problem 4== | ||
− | Which of the following is equal to the product | + | Which of the following is equal to the [[product]] |
+ | <cmath>\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?</cmath> | ||
− | + | <math>\mathrm{(A)}\ 251\qquad\mathrm{(B)}\ 502\qquad\mathrm{(C)}\ 1004\qquad\mathrm{(D)}\ 2008\qquad\mathrm{(E)}\ 4016</math> | |
− | <math>\ | ||
− | |||
− | + | [[2008 AMC 12A Problems/Problem 4|Solution]] | |
− | |||
− | |||
==Problem 5== | ==Problem 5== | ||
Suppose that | Suppose that | ||
− | + | <cmath>\frac{2x}{3}-\frac{x}{6}</cmath> | |
− | < | ||
− | |||
− | |||
− | |||
is an integer. Which of the following statements must be true about <math>x</math>? | is an integer. Which of the following statements must be true about <math>x</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ \text{It is negative.}\qquad\mathrm{(B)}\ \text{It is even, but not necessarily a multiple of 3.}\\\qquad\mathrm{(C)}\ \text{It is a multiple of 3, but not necessarily even.}\\\qquad\mathrm{(D)}\ \text{It is a multiple of 6, but not necessarily a multiple of 12.}\\\qquad\mathrm{(E)}\ \text{It is a multiple of 12.}</math> |
− | \ | ||
− | \ | ||
− | \ | ||
− | + | [[2008 AMC 12A Problems/Problem 5|Solution]] | |
==Problem 6== | ==Problem 6== | ||
− | Heather compares the price of a new computer at two different stores. Store A offers <math>15\%</math> off the sticker price followed by a | + | Heather compares the price of a new computer at two different stores. Store <math>A</math> offers <math>15\%</math> off the sticker price followed by a <math>\textdollar 90</math> rebate, and store <math>B</math> offers <math>25\%</math> off the same sticker price with no rebate. Heather saves <math> \textdollar 15</math> by buying the computer at store <math>A</math> instead of store <math>B</math>. What is the sticker price of the computer, in dollars? |
− | <math>\ | + | <math>\mathrm{(A)}\ 750\qquad\mathrm{(B)}\ 900\qquad\mathrm{(C)}\ 1000\qquad\mathrm{(D)}\ 1050\qquad\mathrm{(E)}\ 1500</math> |
− | + | [[2008 AMC 12A Problems/Problem 6|Solution]] | |
==Problem 7== | ==Problem 7== | ||
− | While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing | + | While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? |
− | <math>\ | + | <math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 8\qquad\mathrm{(E)}\ 10</math> |
− | + | [[2008 AMC 12A Problems/Problem 7|Solution]] | |
==Problem 8== | ==Problem 8== | ||
− | What is the volume of a cube whose surface area is twice that of a cube with volume 1? | + | What is the [[volume]] of a [[cube]] whose [[surface area]] is twice that of a cube with volume 1? |
− | <math>\ | + | <math>\mathrm{(A)}\ \sqrt{2}\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 2\sqrt{2}\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 8</math> |
− | + | [[2008 AMC 12A Problems/Problem 8|Solution]] | |
==Problem 9== | ==Problem 9== | ||
+ | Older television screens have an aspect ratio of <math>4: 3</math>. That is, the ratio of the width to the height is <math>4: 3</math>. The aspect ratio of many movies is not <math>4: 3</math>, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of <math>2: 1</math> and is shown on an older television screen with a <math>27</math>-inch diagonal. What is the height, in inches, of each darkened strip? | ||
+ | <asy>unitsize(1mm); | ||
+ | filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); | ||
+ | filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); | ||
+ | draw((0,0)--(21.6,0)--(21.6,16.2)--(0,16.2)--cycle);</asy> | ||
+ | <math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 2.25\qquad\mathrm{(C)}\ 2.5\qquad\mathrm{(D)}\ 2.7\qquad\mathrm{(E)}\ 3</math> | ||
+ | |||
+ | [[2008 AMC 12A Problems/Problem 9|Solution]] | ||
− | |||
==Problem 10== | ==Problem 10== | ||
Doug can paint a room in <math>5</math> hours. Dave can paint the same room in <math>7</math> hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let <math>t</math> be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by <math>t</math>? | Doug can paint a room in <math>5</math> hours. Dave can paint the same room in <math>7</math> hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let <math>t</math> be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by <math>t</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ \left(\frac{1}{5}+\frac{1}{7}\right)\left(t+1\right)=1\qquad\mathrm{(B)}\ \left(\frac{1}{5}+\frac{1}{7}\right)t+1=1\qquad\mathrm{(C)}\ \left(\frac{1}{5}+\frac{1}{7}\right)t=1\\\mathrm{(D)}\ \left(\frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1\qquad\mathrm{(E)}\ \left(5+7\right)t=1</math> |
− | \\ | + | |
+ | [[2008 AMC 12A Problems/Problem 10|Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the <math>13</math> visible numbers have the greatest possible sum. What is that sum? | ||
− | + | <asy> | |
+ | unitsize(.8cm); | ||
− | ( | + | pen p = linewidth(1); |
+ | draw(shift(-2,0)*unitsquare,p); | ||
+ | label("1",(-1.5,0.5)); | ||
+ | draw(shift(-1,0)*unitsquare,p); | ||
+ | label("2",(-0.5,0.5)); | ||
+ | draw(unitsquare,p); | ||
+ | label("32",(0.5,0.5)); | ||
+ | draw(shift(1,0)*unitsquare,p); | ||
+ | label("16",(1.5,0.5)); | ||
+ | draw(shift(0,1)*unitsquare,p); | ||
+ | label("4",(0.5,1.5)); | ||
+ | draw(shift(0,-1)*unitsquare,p); | ||
+ | label("8",(0.5,-0.5)); | ||
+ | </asy> | ||
− | + | <math>\mathrm{(A)}\ 154\qquad\mathrm{(B)}\ 159\qquad\mathrm{(C)}\ 164\qquad\mathrm{(D)}\ 167\qquad\mathrm{(E)}\ 189</math> | |
− | + | [[2008 AMC 12A Problems/Problem 11|Solution]] | |
==Problem 12== | ==Problem 12== | ||
− | A function <math>f</math> has domain <math>[0,2]</math> and range <math>[0,1]</math>. (The notation <math>[a,b]</math> denotes <math>\{x:a \le x \le b \}</math>.) What are the domain and range, respectively, of the function <math>g</math> defined by <math>g(x)=1-f(x+1)</math>? | + | A [[function]] <math>f</math> has [[domain]] <math>[0,2]</math> and [[range]] <math>[0,1]</math>. (The notation <math>[a,b]</math> denotes <math>\{x:a \le x \le b \}</math>.) What are the domain and range, respectively, of the function <math>g</math> defined by <math>g(x)=1-f(x+1)</math>? |
− | <math>\ | + | <math>\mathrm{(A)}\ [-1,1],[-1,0]\qquad\mathrm{(B)}\ [-1,1],[0,1]\qquad\textbf{(C)}\ [0,2],[-1,0]\qquad\mathrm{(D)}\ [1,3],[-1,0]\qquad\mathrm{(E)}\ [1,3],[0,1]</math> |
+ | |||
+ | [[2008 AMC 12A Problems/Problem 12|Solution]] | ||
− | |||
==Problem 13== | ==Problem 13== | ||
Points <math>A</math> and <math>B</math> lie on a circle centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internally tangent to the first and tangent to both <math>\overline{OA}</math> and <math>\overline{OB}</math>. What is the ratio of the area of the smaller circle to that of the larger circle? | Points <math>A</math> and <math>B</math> lie on a circle centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internally tangent to the first and tangent to both <math>\overline{OA}</math> and <math>\overline{OB}</math>. What is the ratio of the area of the smaller circle to that of the larger circle? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ \frac{1}{16}\qquad\mathrm{(B)}\ \frac{1}{9}\qquad\mathrm{(C)}\ \frac{1}{8}\qquad\mathrm{(D)}\ \frac{1}{6}\qquad\mathrm{(E)}\ \frac{1}{4}</math> |
− | + | [[2008 AMC 12A Problems/Problem 13|Solution]] | |
==Problem 14== | ==Problem 14== | ||
− | What is the area of the region defined by the inequality <math>|3x-18|+|2y+7|\ | + | What is the area of the region defined by the [[inequality]] <math>|3x-18|+|2y+7|\le3</math>? |
− | <math>\ | + | <math>\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ \frac {7}{2}\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ \frac{9}{2}\qquad\mathrm{(E)}\ 5</math> |
− | + | [[2008 AMC 12A Problems/Problem 14|Solution]] | |
==Problem 15== | ==Problem 15== | ||
Let <math>k={2008}^{2}+{2}^{2008}</math>. What is the units digit of <math>k^2+2^k</math>? | Let <math>k={2008}^{2}+{2}^{2008}</math>. What is the units digit of <math>k^2+2^k</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 6\qquad\mathrm{(E)}\ 8</math> |
− | + | [[2008 AMC 12A Problems/Problem 15|Solution]] | |
==Problem 16== | ==Problem 16== | ||
The numbers <math>\log(a^3b^7)</math>, <math>\log(a^5b^{12})</math>, and <math>\log(a^8b^{15})</math> are the first three terms of an arithmetic sequence, and the <math>12^\text{th}</math> term of the sequence is <math>\log(b^n)</math>. What is <math>n</math>? | The numbers <math>\log(a^3b^7)</math>, <math>\log(a^5b^{12})</math>, and <math>\log(a^8b^{15})</math> are the first three terms of an arithmetic sequence, and the <math>12^\text{th}</math> term of the sequence is <math>\log(b^n)</math>. What is <math>n</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ 40\qquad\mathrm{(B)}\ 56\qquad\mathrm{(C)}\ 76\qquad\mathrm{(D)}\ 112\qquad\mathrm{(E)}\ 143</math> |
− | + | [[2008 AMC 12A Problems/Problem 16|Solution]] | |
==Problem 17== | ==Problem 17== | ||
Let <math>a_1,a_2,\ldots</math> be a sequence determined by the rule <math>a_n=a_{n-1}/2</math> if <math>a_{n-1}</math> is even and <math>a_n=3a_{n-1}+1</math> if <math>a_{n-1}</math> is odd. For how many positive integers <math>a_1 \le 2008</math> is it true that <math>a_1</math> is less than each of <math>a_2</math>, <math>a_3</math>, and <math>a_4</math>? | Let <math>a_1,a_2,\ldots</math> be a sequence determined by the rule <math>a_n=a_{n-1}/2</math> if <math>a_{n-1}</math> is even and <math>a_n=3a_{n-1}+1</math> if <math>a_{n-1}</math> is odd. For how many positive integers <math>a_1 \le 2008</math> is it true that <math>a_1</math> is less than each of <math>a_2</math>, <math>a_3</math>, and <math>a_4</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ 250\qquad\mathrm{(B)}\ 251\qquad\mathrm{(C)}\ 501\qquad\mathrm{(D)}\ 502\qquad\mathrm{(E)} 1004</math> |
− | + | [[2008 AMC 12A Problems/Problem 17|Solution]] | |
==Problem 18== | ==Problem 18== | ||
A triangle <math>\triangle ABC</math> with sides <math>5</math>, <math>6</math>, <math>7</math> is placed in the three-dimensional plane with one vertex on the positive <math>x</math> axis, one on the positive <math>y</math> axis, and one on the positive <math>z</math> axis. Let <math>O</math> be the origin. What is the volume of <math>OABC</math>? | A triangle <math>\triangle ABC</math> with sides <math>5</math>, <math>6</math>, <math>7</math> is placed in the three-dimensional plane with one vertex on the positive <math>x</math> axis, one on the positive <math>y</math> axis, and one on the positive <math>z</math> axis. Let <math>O</math> be the origin. What is the volume of <math>OABC</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ \sqrt{85}\qquad\mathrm{(B)}\ \sqrt{90}\qquad\mathrm{(C)}\ \sqrt{95}\qquad\mathrm{(D)}\ 10\qquad\mathrm{(E)}\ \sqrt{105}</math> |
− | + | [[2008 AMC 12A Problems/Problem 18|Solution]] | |
==Problem 19== | ==Problem 19== | ||
In the expansion of | In the expansion of | ||
+ | <cmath>\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,</cmath> | ||
+ | what is the [[coefficient]] of <math>x^{28}</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ 195\qquad\mathrm{(B)}\ 196\qquad\mathrm{(C)}\ 224\qquad\mathrm{(D)}\ 378\qquad\mathrm{(E)}\ 405</math> |
− | + | [[2008 AMC 12A Problems/Problem 19|Solution]] | |
− | |||
− | |||
− | |||
− | |||
==Problem 20== | ==Problem 20== | ||
Triangle <math>ABC</math> has <math>AC=3</math>, <math>BC=4</math>, and <math>AB=5</math>. Point <math>D</math> is on <math>\overline{AB}</math>, and <math>\overline{CD}</math> bisects the right angle. The inscribed circles of <math>\triangle ADC</math> and <math>\triangle BCD</math> have radii <math>r_a</math> and <math>r_b</math>, respectively. What is <math>r_a/r_b</math>? | Triangle <math>ABC</math> has <math>AC=3</math>, <math>BC=4</math>, and <math>AB=5</math>. Point <math>D</math> is on <math>\overline{AB}</math>, and <math>\overline{CD}</math> bisects the right angle. The inscribed circles of <math>\triangle ADC</math> and <math>\triangle BCD</math> have radii <math>r_a</math> and <math>r_b</math>, respectively. What is <math>r_a/r_b</math>? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \frac{3}{56}\left(10-\sqrt{2}\right)\qquad\mathrm{(C)}\ \frac{1}{14}\left(10-\sqrt{2}\right)\qquad\mathrm{(D)}\ \frac{5}{56}\left(10-\sqrt{2}\right)\\\mathrm{(E)}\ \frac{3}{28}\left(10-\sqrt{2}\right)</math> |
− | + | [[2008 AMC 12A Problems/Problem 20|Solution]] | |
==Problem 21== | ==Problem 21== | ||
− | + | A permutation <math>(a_1,a_2,a_3,a_4,a_5)</math> of <math>(1,2,3,4,5)</math> is <math>\textit{heavy-tailed}</math> if <math>a_1 + a_2 < a_4 + a_5</math>. What is the number of heavy-tailed permutations? | |
− | <math>\ | + | <math>\mathrm{(A)}\ 36\qquad\mathrm{(B)}\ 40\qquad\textbf{(C)}\ 44\qquad\mathrm{(D)}\ 48\qquad\mathrm{(E)}\ 52</math> |
− | |||
− | + | [[2008 AMC 12A Problems/Problem 21|Solution]] | |
==Problem 22== | ==Problem 22== | ||
+ | A round table has radius <math>4</math>. Six rectangular place mats are placed on the table. Each place mat has width <math>1</math> and length <math>x</math> as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length <math>x</math>. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is <math>x</math>? | ||
+ | |||
+ | <asy>unitsize(4mm); | ||
+ | defaultpen(linewidth(.8)+fontsize(8)); | ||
+ | draw(Circle((0,0),4)); | ||
+ | path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; | ||
+ | draw(mat); | ||
+ | draw(rotate(60)*mat); | ||
+ | draw(rotate(120)*mat); | ||
+ | draw(rotate(180)*mat); | ||
+ | draw(rotate(240)*mat); | ||
+ | draw(rotate(300)*mat); | ||
+ | label("\(x\)",(-1.55,2.1),E); | ||
+ | label("\(1\)",(-0.5,3.8),S);</asy> | ||
+ | |||
+ | <math>\mathrm{(A)}\ 2\sqrt{5}-\sqrt{3}\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ \frac{3\sqrt{7}-\sqrt{3}}{2}\qquad\mathrm{(D)}\ 2\sqrt{3}\qquad\mathrm{(E)}\ \frac{5+2\sqrt{3}}{2}</math> | ||
+ | |||
+ | [[2008 AMC 12A Problems/Problem 22|Solution]] | ||
− | |||
==Problem 23== | ==Problem 23== | ||
The solutions of the equation <math>z^4+4z^3i-6z^2-4zi-i=0</math> are the vertices of a convex polygon in the complex plane. What is the area of the polygon? | The solutions of the equation <math>z^4+4z^3i-6z^2-4zi-i=0</math> are the vertices of a convex polygon in the complex plane. What is the area of the polygon? | ||
− | <math>\ | + | <math>\mathrm{(A)}\ 2^{\frac{5}{8}}\qquad\mathrm{(B)}\ 2^{\frac{3}{4}}\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 2^{\frac{5}{4}}\qquad\mathrm{(E)}\ 2^{\frac{3}{2}}</math> |
− | + | [[2008 AMC 12A Problems/Problem 23|Solution]] | |
==Problem 24== | ==Problem 24== | ||
+ | Triangle <math>ABC</math> has <math>\angle C = 60^{\circ}</math> and <math>BC = 4</math>. Point <math>D</math> is the midpoint of <math>BC</math>. What is the largest possible value of <math>\tan{\angle BAD}</math>? | ||
+ | |||
+ | <math>\mathrm{(A)}\ \frac{\sqrt{3}}{6}\qquad\mathrm{(B)}\ \frac{\sqrt{3}}{3}\qquad\mathrm{(C)}\ \frac{\sqrt{3}}{2\sqrt{2}}\qquad\mathrm{(D)}\ \frac{\sqrt{3}}{4\sqrt{2}-3}\qquad\mathrm{(E)}\ 1</math> | ||
+ | |||
+ | [[2008 AMC 12A Problems/Problem 24|Solution]] | ||
− | |||
==Problem 25== | ==Problem 25== | ||
+ | A sequence <math>(a_1,b_1)</math>, <math>(a_2,b_2)</math>, <math>(a_3,b_3)</math>, <math>\ldots</math> of points in the coordinate plane satisfies | ||
+ | |||
+ | <math>(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)</math> for <math>n = 1,2,3,\ldots</math>. | ||
+ | |||
+ | Suppose that <math>(a_{100},b_{100}) = (2,4)</math>. What is <math>a_1 + b_1</math>? | ||
+ | |||
+ | <math>\mathrm{(A)}\ -\frac{1}{2^{97}}\qquad\mathrm{(B)}\ -\frac{1}{2^{99}}\qquad\mathrm{(C)}\ 0\qquad\mathrm{(D)}\ \frac{1}{2^{98}}\qquad\mathrm{(E)}\ \frac{1}{2^{96}}</math> | ||
− | + | [[2008 AMC 12A Problems/Problem 25|Solution]] | |
− | {{ | + | == See also == |
+ | {{AMC12 box|year=2008|ab=A|before=[[2007 AMC 12B Problems|2007 AMC 12B]]|after=[[2008 AMC 12B Problems|2008 AMC 12B]]}} | ||
+ | * [[AMC 12]] | ||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 13:56, 2 February 2021
2008 AMC 12A (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 bakery owner turns on his doughnut machine at . At
the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
Problem 2
What is the reciprocal of ?
Problem 3
Suppose that of
bananas are worth as much as
oranges. How many oranges are worth as much as
of
bananas?
Problem 4
Which of the following is equal to the product
Problem 5
Suppose that
is an integer. Which of the following statements must be true about
?
Problem 6
Heather compares the price of a new computer at two different stores. Store offers
off the sticker price followed by a
rebate, and store
offers
off the same sticker price with no rebate. Heather saves
by buying the computer at store
instead of store
. What is the sticker price of the computer, in dollars?
Problem 7
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
Problem 8
What is the volume of a cube whose surface area is twice that of a cube with volume 1?
Problem 9
Older television screens have an aspect ratio of . That is, the ratio of the width to the height is
. The aspect ratio of many movies is not
, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of
and is shown on an older television screen with a
-inch diagonal. What is the height, in inches, of each darkened strip?
Problem 10
Doug can paint a room in hours. Dave can paint the same room in
hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let
be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by
?
Problem 11
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the visible numbers have the greatest possible sum. What is that sum?
Problem 12
A function has domain
and range
. (The notation
denotes
.) What are the domain and range, respectively, of the function
defined by
?
Problem 13
Points and
lie on a circle centered at
, and
. A second circle is internally tangent to the first and tangent to both
and
. What is the ratio of the area of the smaller circle to that of the larger circle?
Problem 14
What is the area of the region defined by the inequality ?
Problem 15
Let . What is the units digit of
?
Problem 16
The numbers ,
, and
are the first three terms of an arithmetic sequence, and the
term of the sequence is
. What is
?
Problem 17
Let be a sequence determined by the rule
if
is even and
if
is odd. For how many positive integers
is it true that
is less than each of
,
, and
?
Problem 18
A triangle with sides
,
,
is placed in the three-dimensional plane with one vertex on the positive
axis, one on the positive
axis, and one on the positive
axis. Let
be the origin. What is the volume of
?
Problem 19
In the expansion of
what is the coefficient of
?
Problem 20
Triangle has
,
, and
. Point
is on
, and
bisects the right angle. The inscribed circles of
and
have radii
and
, respectively. What is
?
Problem 21
A permutation of
is
if
. What is the number of heavy-tailed permutations?
Problem 22
A round table has radius . Six rectangular place mats are placed on the table. Each place mat has width
and length
as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length
. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is
?
Problem 23
The solutions of the equation are the vertices of a convex polygon in the complex plane. What is the area of the polygon?
Problem 24
Triangle has
and
. Point
is the midpoint of
. What is the largest possible value of
?
Problem 25
A sequence ,
,
,
of points in the coordinate plane satisfies
for
.
Suppose that . What is
?
See also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2007 AMC 12B |
Followed by 2008 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.