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Difference between revisions of "2005 AMC 12B Problems"

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{{AMC12 Problems|year=2005|ab=B}}
 
== Problem 1 ==
 
== Problem 1 ==
 
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar.  What was their profit, in dollars?
 
A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar.  What was their profit, in dollars?
Line 15: Line 15:
 
== Problem 2 ==
 
== Problem 2 ==
 
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>.  What is <math>x</math>?
 
A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>.  What is <math>x</math>?
 +
 
<math>
 
<math>
 
\mathrm{(A)}\ 2      \qquad
 
\mathrm{(A)}\ 2      \qquad
Line 26: Line 27:
  
 
== Problem 3 ==
 
== 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?  
+
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?  
  
 
<math>
 
<math>
Line 52: Line 53:
  
 
== Problem 5 ==
 
== Problem 5 ==
−
An <math>8</math>-foot by <math>10</math>-foot floor is tiles with square tiles of size <math>1</math> foot by <math>1</math> foot.  Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> 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?
+
An <math>8</math>-foot by <math>10</math>-foot floor is tiled with square tiles of size <math>1</math> foot by <math>1</math> foot.  Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> 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>
 
<asy>
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== Problem 9 ==
 
== Problem 9 ==
 +
On a certain math exam, <math>10\%</math> of the students got <math>70</math> points, <math>25\%</math> got <math>80</math> points, <math>20\%</math> got <math>85</math> points, <math>15\%</math> got <math>90</math> points, and the rest got <math>95</math> points.  What is the difference between the mean and the median score on this exam?
 +
 +
<math>\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{4}}} \qquad \mathrm{(E)}\ {{{5}}}</math>
  
 
[[2005 AMC 12B Problems/Problem 9|Solution]]
 
[[2005 AMC 12B Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
The first term of a sequence is <math>2005</math>. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the <math>2005^{\text{th}}</math> term of the sequence?
 +
 +
<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]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
An envelope contains eight bills: <math>2</math> ones, <math>2</math> fives, <math>2</math> tens, and <math>2</math> twenties.  Two bills are drawn at random without replacement.  What is the probability that their sum is &#36;<math>20</math> or more?
 +
 +
<math>\mathrm{(A)}\ {{{\frac{1}{4}}}} \qquad \mathrm{(B)}\ {{{\frac{2}{7}}}} \qquad \mathrm{(C)}\ {{{\frac{3}{7}}}} \qquad \mathrm{(D)}\ {{{\frac{1}{2}}}} \qquad \mathrm{(E)}\ {{{\frac{2}{3}}}}</math>
  
 
[[2005 AMC 12B Problems/Problem 11|Solution]]
 
[[2005 AMC 12B Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>?
 +
 +
<math>\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}</math>
  
 
[[2005 AMC 12B Problems/Problem 12|Solution]]
 
[[2005 AMC 12B Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
Suppose that <math>4^{x_1}=5</math>, <math>5^{x_2}=6</math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math>.  What is <math>x_1x_2...x_{124}</math>?
 +
 +
<math>\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</math>
  
 
[[2005 AMC 12B Problems/Problem 13|Solution]]
 
[[2005 AMC 12B Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
 +
A circle having center <math>(0,k)</math>, with <math>k>6</math>,is tangent to the lines <math>y=x</math>, <math>y=-x</math> and <math>y=6</math>. What is the radius of this circle?
 +
 +
<math>
 +
\mathrm{(A)}\ 6\sqrt{2}-6 \qquad
 +
\mathrm{(B)}\ 6 \qquad
 +
\mathrm{(C)}\ 6\sqrt{2} \qquad
 +
\mathrm{(D)}\ 12 \qquad
 +
\mathrm{(E)}\ 6+6\sqrt{2}
 +
</math>
  
 
[[2005 AMC 12B Problems/Problem 14|Solution]]
 
[[2005 AMC 12B Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
 +
The sum of four two-digit numbers is <math>221</math>. None of the eight digits is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits?
 +
 +
<math>
 +
\mathrm{(A)}\ 1      \qquad
 +
\mathrm{(B)}\ 2      \qquad
 +
\mathrm{(C)}\ 3      \qquad
 +
\mathrm{(D)}\ 4      \qquad
 +
\mathrm{(E)}\ 5
 +
</math>
  
 
[[2005 AMC 12B Problems/Problem 15|Solution]]
 
[[2005 AMC 12B Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
 +
 +
<math>
 +
\mathrm {(A)}\ \sqrt{2}  \qquad
 +
\mathrm {(B)}\ \sqrt{3}  \qquad
 +
\mathrm {(C)}\ 1+\sqrt{2}\qquad
 +
\mathrm {(D)}\ 1+\sqrt{3}\qquad
 +
\mathrm {(E)}\ 3
 +
</math>
  
 
[[2005 AMC 12B Problems/Problem 16|Solution]]
 
[[2005 AMC 12B Problems/Problem 16|Solution]]
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<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath>
 
<cmath>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</cmath>
 +
 +
<math>
 +
\mathrm{(A)}\ 30    \qquad
 +
\mathrm{(B)}\ 32    \qquad
 +
\mathrm{(C)}\ 34    \qquad
 +
\mathrm{(D)}\ 40    \qquad
 +
\mathrm{(E)}\ 50
 +
</math>
  
 
[[2005 AMC 12B Problems/Problem 20|Solution]]
 
[[2005 AMC 12B Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
A positive integer <math>n</math> has <math>60</math> divisors and <math>7n</math> has <math>80</math> divisors.  What is the greatest integer <math>k</math> such that <math>7^k</math> divides <math>n</math>?
 +
 +
<math>\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}</math>
  
 
[[2005 AMC 12B Problems/Problem 21|Solution]]
 
[[2005 AMC 12B Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
 +
 +
A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule
 +
 +
<cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath>
 +
 +
where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</math>. How many possible values are there for <math>z_{0}</math>?
 +
 +
<math>
 +
\textbf{(A)}\ 1 \qquad
 +
\textbf{(B)}\ 2 \qquad
 +
\textbf{(C)}\ 4 \qquad
 +
\textbf{(D)}\ 2005 \qquad
 +
\textbf{(E)}\ 2^{2005}
 +
</math>
  
 
[[2005 AMC 12B Problems/Problem 22|Solution]]
 
[[2005 AMC 12B Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
 +
Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers for which
 +
 +
<cmath>\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.</cmath>
 +
There are real numbers <math>a</math> and <math>b</math> such that for all ordered triples <math>(x,y,z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math>
 +
 +
<math>
 +
\textbf{(A)}\ \frac {15}{2} \qquad
 +
\textbf{(B)}\ \frac {29}{2} \qquad
 +
\textbf{(C)}\ 15 \qquad
 +
\textbf{(D)}\ \frac {39}{2} \qquad
 +
\textbf{(E)}\ 24
 +
</math>
  
 
[[2005 AMC 12B Problems/Problem 23|Solution]]
 
[[2005 AMC 12B Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
All three vertices of an equilateral triangle are on the parabola <math>y=x^2</math>, and one of its sides has a slope of <math>2</math>.  The <math>x</math>-coordinates of the three vertices have a sum of <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers.  What is the value of <math>m+n</math>?
 +
 +
<math>\mathrm{(A)}\ {{{14}}} \qquad \mathrm{(B)}\ {{{15}}} \qquad \mathrm{(C)}\ {{{16}}} \qquad \mathrm{(D)}\ {{{17}}} \qquad \mathrm{(E)}\ {{{18}}}</math>
  
 
[[2005 AMC 12B Problems/Problem 24|Solution]]
 
[[2005 AMC 12B Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== Problem 25 ==
 +
 +
Six ants simultaneously stand on the six [[vertex|vertices]] of a regular [[octahedron]], with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex?
 +
 +
<math>\mathrm{(A)}\ \frac {5}{256}
 +
\qquad\mathrm{(B)}\ \frac {21}{1024}
 +
\qquad\mathrm{(C)}\ \frac {11}{512}
 +
\qquad\mathrm{(D)}\ \frac {23}{1024}
 +
\qquad\mathrm{(E)}\ \frac {3}{128}</math>
  
 
[[2005 AMC 12B Problems/Problem 25|Solution]]
 
[[2005 AMC 12B Problems/Problem 25|Solution]]
  
 
== See also ==
 
== See also ==
 +
 +
{{AMC12 box|year=2005|ab=B|before=[[2005 AMC 12A Problems]]|after=[[2006 AMC 12A Problems]]}}
 +
 
* [[AMC 12]]
 
* [[AMC 12]]
 
* [[AMC 12 Problems and Solutions]]
 
* [[AMC 12 Problems and Solutions]]
Line 228: Line 325:
 
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]
 
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 13:59, 19 July 2024

2005 AMC 12B (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

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 tiled 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

On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?

$\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{4}}} \qquad \mathrm{(E)}\ {{{5}}}$

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 $2005^{\text{th}}$ 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

An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $$20$ or more?

$\mathrm{(A)}\ {{{\frac{1}{4}}}} \qquad \mathrm{(B)}\ {{{\frac{2}{7}}}} \qquad \mathrm{(C)}\ {{{\frac{3}{7}}}} \qquad \mathrm{(D)}\ {{{\frac{1}{2}}}} \qquad \mathrm{(E)}\ {{{\frac{2}{3}}}}$

Solution

Problem 12

The quadratic equation $x^2+mx+n$ has roots twice those of $x^2+px+m$, and none of $m,n,$ and $p$ is zero. What is the value of $n/p$?

$\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}$

Solution

Problem 13

Suppose that $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$. What is $x_1x_2...x_{124}$?

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

Solution

Problem 14

A circle having center $(0,k)$, with $k>6$,is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?

$\mathrm{(A)}\ 6\sqrt{2}-6 \qquad \mathrm{(B)}\ 6 \qquad \mathrm{(C)}\ 6\sqrt{2} \qquad \mathrm{(D)}\ 12 \qquad \mathrm{(E)}\ 6+6\sqrt{2}$

Solution

Problem 15

The sum of four two-digit numbers is $221$. None 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

Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?

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

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

A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?

$\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}$

Solution

Problem 22

A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule

\[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\]

where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 2005 \qquad \textbf{(E)}\ 2^{2005}$

Solution

Problem 23

Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which

\[\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.\] There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a+b?$

$\textbf{(A)}\ \frac {15}{2} \qquad \textbf{(B)}\ \frac {29}{2} \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ \frac {39}{2} \qquad \textbf{(E)}\ 24$

Solution

Problem 24

All three vertices of an equilateral triangle are on the parabola $y=x^2$, and one of its sides has a slope of $2$. The $x$-coordinates of the three vertices have a sum of $m/n$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m+n$?

$\mathrm{(A)}\ {{{14}}} \qquad \mathrm{(B)}\ {{{15}}} \qquad \mathrm{(C)}\ {{{16}}} \qquad \mathrm{(D)}\ {{{17}}} \qquad \mathrm{(E)}\ {{{18}}}$

Solution

Problem 25

Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?

$\mathrm{(A)}\ \frac {5}{256} \qquad\mathrm{(B)}\ \frac {21}{1024} \qquad\mathrm{(C)}\ \frac {11}{512} \qquad\mathrm{(D)}\ \frac {23}{1024} \qquad\mathrm{(E)}\ \frac {3}{128}$

Solution

See also

2005 AMC 12B (Problems • Answer Key • Resources)
Preceded by
2005 AMC 12A Problems 		Followed by
2006 AMC 12A Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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