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Difference between revisions of "2008 AMC 12A Problems"

(Problem 17)
(Problem 8)
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([[2008 AMC 12A Problems/Problem 7|Solution]])
 
([[2008 AMC 12A Problems/Problem 7|Solution]])
 
==Problem 8==
 
==Problem 8==
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What is the volume of a cube whose surface area is twice that of a cube with volume 1?
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<math>\textbf{(A)} \sqrt{2} \qquad \textbf{(B)} 2  \qquad \textbf{(C)} 2\sqrt{2}  \qquad \textbf{(D)}  4  \qquad \textbf{(E)}  8 </math>
  
 
([[2008 AMC 12A Problems/Problem 8|Solution]])
 
([[2008 AMC 12A Problems/Problem 8|Solution]])
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==Problem 9==
 
==Problem 9==
  

Revision as of 15:39, 17 February 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

(Solution)

Problem 4

(Solution)

Problem 5

(Solution)

Problem 6

Consider a function $f(x)$ with domain $[0,2]$ and range $[0,1]$. Let $g(x)=1-f(x+1)$. What are the domain and range, respectively, of $g(x)$?

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

(Solution)

Problem 7

(Solution)

Problem 8

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

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

(Solution)

Problem 9

(Solution)

Problem 10

(Solution)

Problem 11

(Solution)

Problem 12

(Solution)

Problem 13

(Solution)

Problem 14

(Solution)

Problem 15

(Solution)

Problem 16

(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

(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

(Solution)

Problem 21

(Solution)

Problem 22

(Solution)

Problem 23

(Solution)

Problem 24

(Solution)

Problem 25

(Solution)

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