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2008 AMC 12A Problems

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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?

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

Problem 2

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

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

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?

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

Problem 4

Which of the following is equal to the product

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

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

Problem 5

Suppose that

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

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

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

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?

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

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?

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

Problem 8

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

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

Problem 9

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)

Problem 11

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 $g(x)=1-f(x+1)$ ?

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

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?

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

Problem 14

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

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

Problem 15

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

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

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 $n$ ?

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

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 $a_4$ ?

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

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 $OABC$ ?

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

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 $x^{28}$ ?

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

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 $r_a/r_b$ ?

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

Problem 21

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 $r_a/r_b$ ?

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

Problem 22

Problem 23

Problem 24

Problem 25

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