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Difference between revisions of "2008 AMC 12A Problems"

(Fixed Asymptote on 22)
(Style, formatter answer choices for Problem 8, removed {{empty}})
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<math>\textbf{(A)} \text{ 1:50 PM } \qquad \textbf{(B)} \text{ 3:00 PM } \qquad \textbf{(C)} \text{ 3:30 PM } \qquad \textbf{(D)} \text{ 4:30 PM } \qquad \textbf{(E)} \text{ 5:50 PM }</math>
 
<math>\textbf{(A)} \text{ 1:50 PM } \qquad \textbf{(B)} \text{ 3:00 PM } \qquad \textbf{(C)} \text{ 3:30 PM } \qquad \textbf{(D)} \text{ 4:30 PM } \qquad \textbf{(E)} \text{ 5:50 PM }</math>
  
−
([[2008 AMC 12A Problems/Problem 1|Solution]])
+
[[2008 AMC 12A Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
Line 11: Line 11:
 
<math>\textbf{(A)} \frac{6}{7} \qquad \textbf{(B)} \frac{7}{6}  \qquad \textbf{(C)} \frac{5}{3}  \qquad \textbf{(D)}  3  \qquad \textbf{(E)}  \frac{7}{2} </math>
 
<math>\textbf{(A)} \frac{6}{7} \qquad \textbf{(B)} \frac{7}{6}  \qquad \textbf{(C)} \frac{5}{3}  \qquad \textbf{(D)}  3  \qquad \textbf{(E)}  \frac{7}{2} </math>
  
−
([[2008 AMC 12A Problems/Problem 2|Solution]])
+
[[2008 AMC 12A Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
Line 18: Line 18:
 
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac {5}{2} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac {7}{2} \qquad \textbf{(E)}\ 4</math>
 
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac {5}{2} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac {7}{2} \qquad \textbf{(E)}\ 4</math>
  
−
([[2008 AMC 12A Problems/Problem 3|Solution]])
+
[[2008 AMC 12A Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
Line 29: Line 29:
 
<math>\textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016</math>
 
<math>\textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016</math>
  
−
([[2008 AMC 12A Problems/Problem 4|Solution]])
+
[[2008 AMC 12A Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
Line 45: Line 45:
 
\textbf{(E)}\ \text{It is a multiple of }12\text{.}</math>
 
\textbf{(E)}\ \text{It is a multiple of }12\text{.}</math>
  
−
([[2008 AMC 12A Problems/Problem 5|Solution]])
+
[[2008 AMC 12A Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==Problem 6==
Line 52: Line 52:
 
<math>\textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500</math>
 
<math>\textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500</math>
  
−
([[2008 AMC 12A Problems/Problem 6|Solution]])
+
[[2008 AMC 12A Problems/Problem 6|Solution]]
  
 
==Problem 7==
 
==Problem 7==
Line 59: Line 59:
 
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math>
 
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10</math>
  
−
([[2008 AMC 12A Problems/Problem 7|Solution]])
+
[[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>\textbf{(A)} \sqrt{2} \qquad \textbf{(B)} 2 \qquad \textbf{(C)} 2\sqrt{2} \qquad \textbf{(D)} 4 \qquad \textbf{(E)} 8 </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]])
+
[[2008 AMC 12A Problems/Problem 8|Solution]]
  
 
==Problem 9==
 
==Problem 9==
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<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3</math>
 
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3</math>
  
−
([[2008 AMC 12A Problems/Problem 9|Solution]])
+
[[2008 AMC 12A Problems/Problem 9|Solution]]
  
 
==Problem 10==
 
==Problem 10==
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\textbf{(D)}\ \left( \frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1 \qquad \textbf{(E)}\ \left(5+7\right)t=1</math>
 
\textbf{(D)}\ \left( \frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1 \qquad \textbf{(E)}\ \left(5+7\right)t=1</math>
  
−
([[2008 AMC 12A Problems/Problem 10|Solution]])
+
[[2008 AMC 12A Problems/Problem 10|Solution]]
  
 
==Problem 11==
 
==Problem 11==
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<math>\textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189</math>
 
<math>\textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189</math>
  
−
([[2008 AMC 12A Problems/Problem 11|Solution]])
+
[[2008 AMC 12A Problems/Problem 11|Solution]]
  
 
==Problem 12==
 
==Problem 12==
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<math>\textbf{(A)}\ [ - 1,1],[ - 1,0] \qquad \textbf{(B)}\ [ - 1,1],[0,1] \qquad \textbf{(C)}\ [0,2],[ - 1,0] \qquad \textbf{(D)}\ [1,3],[ - 1,0] \qquad \textbf{(E)}\ [1,3],[0,1]</math>
 
<math>\textbf{(A)}\ [ - 1,1],[ - 1,0] \qquad \textbf{(B)}\ [ - 1,1],[0,1] \qquad \textbf{(C)}\ [0,2],[ - 1,0] \qquad \textbf{(D)}\ [1,3],[ - 1,0] \qquad \textbf{(E)}\ [1,3],[0,1]</math>
  
−
([[2008 AMC 12A Problems/Problem 12|Solution]])
+
[[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?
Line 128: Line 129:
 
<math>\textbf{(A)}\ \frac {1}{16} \qquad \textbf{(B)}\ \frac {1}{9} \qquad \textbf{(C)}\ \frac {1}{8} \qquad \textbf{(D)}\ \frac {1}{6} \qquad \textbf{(E)}\ \frac {1}{4}</math>
 
<math>\textbf{(A)}\ \frac {1}{16} \qquad \textbf{(B)}\ \frac {1}{9} \qquad \textbf{(C)}\ \frac {1}{8} \qquad \textbf{(D)}\ \frac {1}{6} \qquad \textbf{(E)}\ \frac {1}{4}</math>
  
−
([[2008 AMC 12A Problems/Problem 13|Solution]])
+
[[2008 AMC 12A Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
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<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{9}{2} \qquad \textbf{(E)}\ 5</math>
 
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{9}{2} \qquad \textbf{(E)}\ 5</math>
  
−
([[2008 AMC 12A Problems/Problem 14|Solution]])
+
[[2008 AMC 12A Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
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<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8</math>
 
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8</math>
  
−
([[2008 AMC 12A Problems/Problem 15|Solution]])
+
[[2008 AMC 12A Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
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<math>\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 112 \qquad \textbf{(E)}\ 143</math>
 
<math>\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 112 \qquad \textbf{(E)}\ 143</math>
  
−
([[2008 AMC 12A Problems/Problem 16|Solution]])
+
[[2008 AMC 12A Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
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<math>\textbf{(A)} 250 \qquad \textbf{(B)} 251 \qquad \textbf{(C)} 501 \qquad \textbf{(D)} 502 \qquad \textbf{(E)} 1004</math>
 
<math>\textbf{(A)} 250 \qquad \textbf{(B)} 251 \qquad \textbf{(C)} 501 \qquad \textbf{(D)} 502 \qquad \textbf{(E)} 1004</math>
  
−
([[2008 AMC 12A Problems/Problem 17|Solution]])
+
[[2008 AMC 12A Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
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<math>\textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad  \textbf{(E)}\ \sqrt{105}</math>
 
<math>\textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad  \textbf{(E)}\ \sqrt{105}</math>
  
−
([[2008 AMC 12A Problems/Problem 18|Solution]])
+
[[2008 AMC 12A Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
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<math>\textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405</math>
 
<math>\textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405</math>
  
−
([[2008 AMC 12A Problems/Problem 19|Solution]])
+
[[2008 AMC 12A Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
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<math>\textbf{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right) \qquad \textbf{(B)}\ \frac{3}{56}\left(10-\sqrt{2}\right) \qquad \textbf{(C)}\ \frac{1}{14}\left(10-\sqrt{2}\right) \qquad \textbf{(D)}\ \frac{5}{56}\left(10-\sqrt{2}\right) \\ \textbf{(E)}\ \frac{3}{28}\left(10-\sqrt{2}\right)</math>
 
<math>\textbf{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right) \qquad \textbf{(B)}\ \frac{3}{56}\left(10-\sqrt{2}\right) \qquad \textbf{(C)}\ \frac{1}{14}\left(10-\sqrt{2}\right) \qquad \textbf{(D)}\ \frac{5}{56}\left(10-\sqrt{2}\right) \\ \textbf{(E)}\ \frac{3}{28}\left(10-\sqrt{2}\right)</math>
  
−
([[2008 AMC 12A Problems/Problem 20|Solution]])
+
[[2008 AMC 12A Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==
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<math>\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 52</math>
 
<math>\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 52</math>
  
−
([[2008 AMC 12A Problems/Problem 21|Solution]])
+
[[2008 AMC 12A Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 22==
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<math>\textbf{(A)}\ 2\sqrt {5} - \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} - \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 + 2\sqrt {3}}{2}</math>
 
<math>\textbf{(A)}\ 2\sqrt {5} - \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} - \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 + 2\sqrt {3}}{2}</math>
  
−
([[2008 AMC 12A Problems/Problem 22|Solution]])
+
[[2008 AMC 12A Problems/Problem 22|Solution]]
  
 
==Problem 23==
 
==Problem 23==
Line 217: Line 218:
 
<math>\textbf{(A)}\ 2^{\frac{5}{8}} \qquad \textbf{(B)}\ 2^{\frac{3}{4}} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2^{\frac{5}{4}} \qquad \textbf{(E)}\ 2^{\frac{3}{2}}</math>
 
<math>\textbf{(A)}\ 2^{\frac{5}{8}} \qquad \textbf{(B)}\ 2^{\frac{3}{4}} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2^{\frac{5}{4}} \qquad \textbf{(E)}\ 2^{\frac{3}{2}}</math>
  
−
([[2008 AMC 12A Problems/Problem 23|Solution]])
+
[[2008 AMC 12A Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
Line 224: Line 225:
 
<math>\textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} - 3} \qquad \textbf{(E)}\ 1</math>
 
<math>\textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} - 3} \qquad \textbf{(E)}\ 1</math>
  
−
([[2008 AMC 12A Problems/Problem 24|Solution]])
+
[[2008 AMC 12A Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
Line 235: Line 236:
 
<math>\textbf{(A)}\ - \frac {1}{2^{97}} \qquad \textbf{(B)}\ - \frac {1}{2^{99}} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \frac {1}{2^{98}} \qquad \textbf{(E)}\ \frac {1}{2^{96}}</math>
 
<math>\textbf{(A)}\ - \frac {1}{2^{97}} \qquad \textbf{(B)}\ - \frac {1}{2^{99}} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \frac {1}{2^{98}} \qquad \textbf{(E)}\ \frac {1}{2^{96}}</math>
  
−
([[2008 AMC 12A Problems/Problem 25|Solution]])
+
[[2008 AMC 12A Problems/Problem 25|Solution]]
−
 
 
−
{{empty}}
 

Revision as of 00:36, 26 April 2008

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?

$\textbf{(A)} \text{ 1:50 PM } \qquad \textbf{(B)} \text{ 3:00 PM } \qquad \textbf{(C)} \text{ 3:30 PM } \qquad \textbf{(D)} \text{ 4:30 PM } \qquad \textbf{(E)} \text{ 5:50 PM }$

Solution

Problem 2

What is the reciprocal of $\frac{1}{2}+\frac{2}{3}$?

$\textbf{(A)} \frac{6}{7} \qquad \textbf{(B)} \frac{7}{6} \qquad \textbf{(C)} \frac{5}{3} \qquad \textbf{(D)} 3 \qquad \textbf{(E)} \frac{7}{2}$

Solution

Problem 3

Suppose that $\frac {2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much is $\frac {1}{2}$ of $5$ bananas?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ \frac {5}{2} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac {7}{2} \qquad \textbf{(E)}\ 4$

Solution

Problem 4

Which of the following is equal to the product

$\frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n + 4}{4n}\cdots\frac {2008}{2004}$?

$\textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$

Solution

Problem 5

Suppose that

$\frac {2x}{3} - \frac {x}{6}$

is an integer. Which of the following statements must be true about $x$?

$\textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.} \\ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.} \\ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.} \\ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$

Solution

Problem 6

Heather compares the price of a new computer at two different stores. Store A offers $15\%$ off the sticker price followed by a <dollar/>$90$ rebate, and store B offers $25\%$ off the same sticker price with no rebate. Heather saves <dollar/>$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?

$\textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$

Solution

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 toward 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?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

Solution

Problem 8

What is the volume of a cube whose surface area is twice that of a cube with volume 1?

$\mathrm{(A)}\ \sqrt{2}\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 2\sqrt{2}\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 8$

Solution

Problem 9

Older television screens have an aspect ratio of $4: 3$. That is, the ratio of the width to the height is $4: 3$. The aspect ratio of many movies is not $4: 3$, 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 $2: 1$ and is shown on an older television screen with a $27$-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]

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3$

Solution

Problem 10

Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ 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 $t$?

$\textbf{(A)}\ \left( \frac{1}{5}+\frac{1}{7}\right)\left( t+1 \right)=1 \qquad \textbf{(B)}\ \left( \frac{1}{5}+\frac{1}{7}\right)t+1=1 \qquad \textbf{(C)}\left( \frac{1}{5}+\frac{1}{7}\right)t=1 \\

\textbf{(D)}\ \left( \frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1 \qquad \textbf{(E)}\ \left(5+7\right)t=1$ (Error compiling LaTeX. Unknown error_msg)

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 $13$ 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]

$\textbf{(A)}\ 154 \qquad \textbf{(B)}\ 159 \qquad \textbf{(C)}\ 164 \qquad \textbf{(D)}\ 167 \qquad \textbf{(E)}\ 189$

Solution

Problem 12

A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\{x:a \le x \le b \}$.) What are the domain and range, respectively, of the function $g$ defined by $g(x)=1-f(x+1)$?

$\textbf{(A)}\ [ - 1,1],[ - 1,0] \qquad \textbf{(B)}\ [ - 1,1],[0,1] \qquad \textbf{(C)}\ [0,2],[ - 1,0] \qquad \textbf{(D)}\ [1,3],[ - 1,0] \qquad \textbf{(E)}\ [1,3],[0,1]$

Solution

Problem 13

Points $A$ and $B$ lie on a circle centered at $O$, and $\angle AOB = 60^\circ$. A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle?

$\textbf{(A)}\ \frac {1}{16} \qquad \textbf{(B)}\ \frac {1}{9} \qquad \textbf{(C)}\ \frac {1}{8} \qquad \textbf{(D)}\ \frac {1}{6} \qquad \textbf{(E)}\ \frac {1}{4}$

Solution

Problem 14

What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le 3$?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{9}{2} \qquad \textbf{(E)}\ 5$

Solution

Problem 15

Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Solution

Problem 16

The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^\text{th}$ term of the sequence is $\log(b^n)$. What is $n$?

$\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 112 \qquad \textbf{(E)}\ 143$

Solution

Problem 17

Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n=a_{n-1}/2$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$?

$\textbf{(A)} 250 \qquad \textbf{(B)} 251 \qquad \textbf{(C)} 501 \qquad \textbf{(D)} 502 \qquad \textbf{(E)} 1004$

Solution

Problem 18

A triangle $\triangle ABC$ with sides $5$, $6$, $7$ is placed in the three-dimensional plane with one vertex on the positive $x$ axis, one on the positive $y$ axis, and one on the positive $z$ axis. Let $O$ be the origin. What is the volume of $OABC$?

$\textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \sqrt{105}$

Solution

Problem 19

In the expansion of

$\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2$,

what is the coefficient of $x^{28}$?

$\textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405$

Solution

Problem 20

Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$?

$\textbf{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right) \qquad \textbf{(B)}\ \frac{3}{56}\left(10-\sqrt{2}\right) \qquad \textbf{(C)}\ \frac{1}{14}\left(10-\sqrt{2}\right) \qquad \textbf{(D)}\ \frac{5}{56}\left(10-\sqrt{2}\right) \\ \textbf{(E)}\ \frac{3}{28}\left(10-\sqrt{2}\right)$

Solution

Problem 21

A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?

$\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 52$

Solution

Problem 22

A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ 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 $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?

[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]

$\textbf{(A)}\ 2\sqrt {5} - \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} - \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 + 2\sqrt {3}}{2}$

Solution

Problem 23

The solutions of the equation $z^4+4z^3i-6z^2-4zi-i=0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon?

$\textbf{(A)}\ 2^{\frac{5}{8}} \qquad \textbf{(B)}\ 2^{\frac{3}{4}} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2^{\frac{5}{4}} \qquad \textbf{(E)}\ 2^{\frac{3}{2}}$

Solution

Problem 24

Triangle $ABC$ has $\angle C = 60^{\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\tan{\angle BAD}$?

$\textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} - 3} \qquad \textbf{(E)}\ 1$

Solution

Problem 25

A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\ldots$ of points in the coordinate plane satisfies

$(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)$ for $n = 1,2,3,\ldots$.

Suppose that $(a_{100},b_{100}) = (2,4)$. What is $a_1 + b_1$?

$\textbf{(A)}\ - \frac {1}{2^{97}} \qquad \textbf{(B)}\ - \frac {1}{2^{99}} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \frac {1}{2^{98}} \qquad \textbf{(E)}\ \frac {1}{2^{96}}$

Solution

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