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Difference between revisions of "2005 AMC 12B Problems"

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== Problem 10 ==
 
== Problem 10 ==
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The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?
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<math>\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}</math>
  
 
[[2005 AMC 12B Problems/Problem 10|Solution]]
 
[[2005 AMC 12B Problems/Problem 10|Solution]]

Revision as of 11:23, 26 March 2010

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Problem 1

A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?

$\mathrm{(A)}\ 100 \qquad \mathrm{(B)}\ 200 \qquad \mathrm{(C)}\ 300 \qquad \mathrm{(D)}\ 400 \qquad \mathrm{(E)}\ 500$

Solution

Problem 2

A positive number $x$ has the property that $x\%$ of $x$ is $4$. What is $x$? $\mathrm{(A)}\ 2 \qquad \mathrm{(B)}\ 4 \qquad \mathrm{(C)}\ 10 \qquad \mathrm{(D)}\ 20 \qquad \mathrm{(E)}\ 40$

Solution

Problem 3

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?

$\mathrm{(A)}\ \frac15 \qquad \mathrm{(B)}\ \frac13 \qquad \mathrm{(C)}\ \frac25 \qquad \mathrm{(D)}\ \frac23 \qquad \mathrm{(E)}\ \frac45$

Solution

Problem 4

At the beginning of the school year, Lisa's goal was to earn an A on at least $80\%$ of her $50$ quizzes for the year. She earned an A on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?

$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$

Solution

Problem 5

An $8$-foot by $10$-foot floor is tiles with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

[asy] unitsize(2cm); defaultpen(linewidth(.8pt)); fill(unitsquare,gray); filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black); filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black); filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black); filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black); [/asy]

$\mathrm{(A)}\ 80-20\pi \qquad \mathrm{(B)}\ 60-10\pi \qquad \mathrm{(C)}\ 80-10\pi \qquad \mathrm{(D)}\ 60+10\pi \qquad \mathrm{(E)}\ 80+10\pi$

Solution

Problem 6

In $\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$?

$\mathrm{(A)}\ 3 \qquad \mathrm{(B)}\ 2\sqrt{3} \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 5 \qquad \mathrm{(E)}\ 4\sqrt{2}$

Solution

Problem 7

What is the area enclosed by the graph of $|3x|+|4y|=12$?

$\mathrm{(A)}\ 6 \qquad \mathrm{(B)}\ 12 \qquad \mathrm{(C)}\ 16 \qquad \mathrm{(D)}\ 24 \qquad \mathrm{(E)}\ 25$

Solution

Problem 8

For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$ ?

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 10 \qquad \mathrm{(E)}\ \text{infinitely many}$

Solution

Problem 9

Solution

Problem 10

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?

$\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}$

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

The sum of four two-digit numbers is $221$. Non of the eight digits is $0$ and no two of them are the same. Which of the following is not included among the eight digits?

$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$

Solution

Problem 16

Solution

Problem 17

How many distinct four-tuples $(a, b, c, d)$ of rational numbers are there with

$a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005$?

$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 17 \qquad \mathrm{(D)}\ 2004 \qquad \mathrm{(E)}\ \text{infinitely many}$

Solution

Problem 18

Let $A(2,2)$ and $B(7,7)$ be points in the plane. Define $R$ as the region in the first quadrant consisting of those points $C$ such that $\triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $R$?

$\mathrm{(A)}\ 25 \qquad \mathrm{(B)}\ 39 \qquad \mathrm{(C)}\ 51 \qquad \mathrm{(D)}\ 60 \qquad \mathrm{(E)}\ 80 \qquad$

Solution

Problem 19

Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^{2}-y^{2}=m^{2}$ for some positive integer $m$. What is $x+y+m$?

$\mathrm{(A)}\ 88 \qquad \mathrm{(B)}\ 112 \qquad \mathrm{(C)}\ 116 \qquad \mathrm{(D)}\ 144 \qquad \mathrm{(E)}\ 154 \qquad$

Solution

Problem 20

Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set

\[\{-7,-5,-3,-2,2,4,6,13\}.\]

What is the minimum possible value of

\[(a+b+c+d)^{2}+(e+f+g+h)^{2}?\]

$\mathrm{(A)}\ 30 \qquad \mathrm{(B)}\ 32 \qquad \mathrm{(C)}\ 34 \qquad \mathrm{(D)}\ 40 \qquad \mathrm{(E)}\ 50$

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

See also

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