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Difference between revisions of "2008 AMC 12A Problems"

(→‎Problem 18: fix)
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==Problem 5==
 
==Problem 5==
 +
Suppose that
 +
 +
<center>
 +
<math>\frac {2x}{3} - \frac {x}{6}</math>
 +
</center>
 +
 +
is an integer. Which of the following statements must be true about <math>x</math>?
 +
 +
<math>\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{.}</math>
  
 
([[2008 AMC 12A Problems/Problem 5|Solution]])
 
([[2008 AMC 12A Problems/Problem 5|Solution]])
 +
 
==Problem 6==
 
==Problem 6==
−
Consider a function <math>f(x)</math> with domain <math>[0,2]</math> and range <math>[0,1]</math>. Let <math>g(x)=1-f(x+1)</math>. What are the domain and range, respectively, of <math>g(x)</math>?
+
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 <dollar/><math>90</math> rebate, and store B offers <math>25\%</math> off the same sticker price with no rebate. Heather saves <dollar/><math>15</math> by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?
  
−
<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)}\ 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]])
Line 60: Line 73:
  
 
([[2008 AMC 12A Problems/Problem 11|Solution]])
 
([[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>?
 +
 +
<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]])

Revision as of 20:43, 18 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

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

(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

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

(Solution)

Problem 15

(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 if $OABC$?

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

(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

(Solution)

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 $r_a/r_b$?

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

(Solution)

Problem 22

(Solution)

Problem 23

(Solution)

Problem 24

(Solution)

Problem 25

(Solution)

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