Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
GET READY FOR THE AMC 12 WITH AoPS
Learn with outstanding instructors and top-scoring students from around the world in our AMC 12 Problem Series online course.
CHECK SCHEDULE

Difference between revisions of "2005 AMC 12A Problems"

m (→‎Problem 19)
(Changed to correct link for math jam)
 
(14 intermediate revisions by 9 users not shown)
Line 1: Line 1:
 +
{{AMC12 Problems|year=2005|ab=A}}
 
== Problem 1 ==
 
== Problem 1 ==
 
Two is <math>10 \%</math> of <math>x</math> and <math>20 \%</math> of <math>y</math>. What is <math>x - y</math>?
 
Two is <math>10 \%</math> of <math>x</math> and <math>20 \%</math> of <math>y</math>. What is <math>x - y</math>?
Line 27: Line 28:
  
 
== Problem 4 ==
 
== Problem 4 ==
−
A store normally sells windows at <dollar/><math>100</math> each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
+
A store normally sells windows at <math>\$100</math> each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
  
 
<math>
 
<math>
 
(\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 200 \qquad (\mathrm {C})\ 300 \qquad (\mathrm {D}) \ 400 \qquad (\mathrm {E})\ 500
 
(\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 200 \qquad (\mathrm {C})\ 300 \qquad (\mathrm {D}) \ 400 \qquad (\mathrm {E})\ 500
 
</math>
 
</math>
 +
 
[[2005 AMC 12A Problems/Problem 4|Solution]]
 
[[2005 AMC 12A Problems/Problem 4|Solution]]
  
Line 38: Line 40:
  
 
<math>
 
<math>
−
(\mathrm {A}) \ 23 \qquad (\mathrm {B}) \ 24 \qquad (\mathrm {C})\ 25 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 27
+
(\mathrm {A}) \ 23 \qquad (\mathrm {B}) \ 24 \qquad (\mathrm {C})\ 25 \qquad (\mathrm {D}) \ 26 \qquad (\mathrm {E})\ 27
 
</math>
 
</math>
  
Line 54: Line 56:
 
== Problem 7 ==
 
== Problem 7 ==
 
Square <math>EFGH</math> is inside the square <math>ABCD</math> so that each side of <math>EFGH</math> can be extended to pass through a vertex of <math>ABCD</math>. Square <math>ABCD</math> has side length <math>\sqrt {50}</math> and <math>BE = 1</math>. What is the area of the inner square <math>EFGH</math>?
 
Square <math>EFGH</math> is inside the square <math>ABCD</math> so that each side of <math>EFGH</math> can be extended to pass through a vertex of <math>ABCD</math>. Square <math>ABCD</math> has side length <math>\sqrt {50}</math> and <math>BE = 1</math>. What is the area of the inner square <math>EFGH</math>?
 +
<asy>
 +
unitsize(4cm);
 +
defaultpen(linewidth(.8pt)+fontsize(10pt));
 +
pair D=(0,0), C=(1,0), B=(1,1), A=(0,1);
 +
pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0];
 +
pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H);
 +
draw(A--B--C--D--cycle);
 +
draw(D--F);
 +
draw(C--E);
 +
draw(B--H);
 +
draw(A--G);
 +
label("$A$",A,NW);
 +
label("$B$",B,NE);
 +
label("$C$",C,SE);
 +
label("$D$",D,SW);
 +
label("$E$",E,NNW);
 +
label("$F$",F,ENE);
 +
label("$G$",G,SSE);
 +
label("$H$",H,WSW);</asy>
  
 
<math>
 
<math>
−
(\mathrm {A}) \ 25 \qquad (\mathrm {B}) \ 32 \qquad (\mathrm {C})\ 36 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 40
+
(\mathrm {A}) \ 25 \qquad (\mathrm {B}) \ 32 \qquad (\mathrm {C})\ 36 \qquad (\mathrm {D}) \ 40 \qquad (\mathrm {E})\ 42
 
</math>
 
</math>
  
Line 108: Line 129:
  
 
== Problem 13 ==
 
== Problem 13 ==
−
The regular 5-point star <math>ABCDE</math> is drawn and in each vertex, there is a number. Each <math>A,B,C,D,</math> and <math>E</math> are chosen such that all 5 of them came from set <math>\{3,5,6,7,9\}</math>. Each letter is a different number (so one possible way is <math>A = 3, B = 5, C = 6, D = 7, E = 9</math>). Let <math>AB</math> be the sum of the numbers on <math>A</math> and <math>B</math>, and so forth. If <math>AB, BC, CD, DE,</math> and <math>EA</math> form an arithmetic sequence (not necessarily in increasing order), find the value of <math>CD</math>.
+
In the five-sided star shown, the letters <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math> and <math>E</math> are replaced by the
 +
numbers 3, 5, 6, 7 and 9, although not necessarily in that order. The sums of the
 +
numbers at the ends of the line segments <math>\overline{AB}</math>, <math>\overline{BC}</math>, <math>\overline{CD}</math>, <math>\overline{DE}</math>, and <math>\overline{EA}</math> form an
 +
arithmetic sequence, although not necessarily in that order. What is the middle
 +
term of the arithmetic sequence?
 +
 
 +
<asy>
 +
draw((0,0)--(0.5,1.54)--(1,0)--(-0.31,0.95)--(1.31,0.95)--cycle);
 +
label("$A$",(0.5,1.54),N);
 +
label("$B$",(1,0),SE);
 +
label("$C$",(-0.31,0.95),W);
 +
label("$D$",(1.31,0.95),E);
 +
label("$E$",(0,0),SW);
 +
</asy>
  
 
<math>
 
<math>
Line 127: Line 161:
 
== Problem 15 ==
 
== Problem 15 ==
 
Let <math>\overline{AB}</math> be a diameter of a circle and <math>C</math> be a point on <math>\overline{AB}</math> with <math>2 \cdot AC = BC</math>. Let <math>D</math> and <math>E</math> be points on the circle such that <math>\overline{DC} \perp \overline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math> to the area of <math>\triangle ABD</math>?
 
Let <math>\overline{AB}</math> be a diameter of a circle and <math>C</math> be a point on <math>\overline{AB}</math> with <math>2 \cdot AC = BC</math>. Let <math>D</math> and <math>E</math> be points on the circle such that <math>\overline{DC} \perp \overline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the ratio of the area of <math>\triangle DCE</math> to the area of <math>\triangle ABD</math>?
 +
 +
<asy>
 +
unitsize(2.5cm);
 +
defaultpen(fontsize(10pt)+linewidth(.8pt));
 +
dotfactor=3;
 +
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);
 +
pair D=dir(aCos(C.x)), E=(-D.x,-D.y);
 +
draw(A--B--D--cycle);
 +
draw(D--E--C);
 +
draw(unitcircle,white);
 +
drawline(D,C);
 +
dot(O);
 +
clip(unitcircle);
 +
draw(unitcircle);
 +
label("$E$",E,SSE);
 +
label("$B$",B,E);
 +
label("$A$",A,W);
 +
label("$D$",D,NNW);
 +
label("$C$",C,SW);
 +
draw(rightanglemark(D,C,B,2));</asy>
  
 
<math>(\text {A}) \ \frac {1}{6} \qquad (\text {B}) \ \frac {1}{4} \qquad (\text {C})\ \frac {1}{3} \qquad (\text {D}) \ \frac {1}{2} \qquad (\text {E})\ \frac {2}{3}</math>
 
<math>(\text {A}) \ \frac {1}{6} \qquad (\text {B}) \ \frac {1}{4} \qquad (\text {C})\ \frac {1}{3} \qquad (\text {D}) \ \frac {1}{2} \qquad (\text {E})\ \frac {2}{3}</math>
Line 133: Line 187:
  
 
== Problem 16 ==
 
== Problem 16 ==
−
[[Image:2005_12A_AMC-16.png]]
 
  
 
Three circles of radius <math>s</math> are drawn in the first quadrant of the <math>xy</math>-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the <math>x</math>-axis, and the third is tangent to the first circle and the <math>y</math>-axis. A circle of radius <math>r > s</math> is tangent to both axes and to the second and third circles. What is <math>r/s</math>?
 
Three circles of radius <math>s</math> are drawn in the first quadrant of the <math>xy</math>-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the <math>x</math>-axis, and the third is tangent to the first circle and the <math>y</math>-axis. A circle of radius <math>r > s</math> is tangent to both axes and to the second and third circles. What is <math>r/s</math>?
  
−
<math>
+
<asy>
−
(\mathrm {A}) \ 5 \qquad (\mathrm {B}) \ 6 \qquad (\mathrm {C})\ 8 \qquad (\mathrm {D}) \ 9 \qquad (\mathrm {E})\ 10
+
import graph;
−
</math>
+
unitsize(3mm);
 +
defaultpen(linewidth(.8pt)+fontsize(10pt));
 +
dotfactor=3;
 +
pair O0=(9,9), O1=(1,1), O2=(3,1), O3=(1,3);
 +
pair P0=O0+9*dir(-45), P3=O3+dir(70);
 +
pair[] ps={O0,O1,O2,O3};
 +
dot(ps);
 +
draw(Circle(O0,9));
 +
draw(Circle(O1,1));
 +
draw(Circle(O2,1));
 +
draw(Circle(O3,1));
 +
draw(O0--P0,linetype("3 3"));
 +
draw(O3--P3,linetype("2 2"));
 +
draw((0,0)--(18,0));
 +
draw((0,0)--(0,18));
 +
label("$r$",midpoint(O0--P0),NE);
 +
label("$s$",(-1.5,4));
 +
draw((-1,4)--midpoint(O3--P3));</asy>
 +
 
 +
<math>(\mathrm {A}) \ 5 \qquad (\mathrm {B}) \ 6 \qquad (\mathrm {C})\ 8 \qquad (\mathrm {D}) \ 9 \qquad (\mathrm {E})\ 10</math>
  
 
[[2005 AMC 12A Problems/Problem 16|Solution]]
 
[[2005 AMC 12A Problems/Problem 16|Solution]]
Line 181: Line 253:
  
 
Let <math>f^{[2]}(x) = f(f(x))</math>, and <math>f^{[n + 1]}(x) = f^{[n]}(f(x))</math> for each integer <math>n \geq 2</math>. For how many values of <math>x</math> in <math>[0,1]</math> is <math>f^{[2005]}(x) = \frac {1}{2}</math>?
 
Let <math>f^{[2]}(x) = f(f(x))</math>, and <math>f^{[n + 1]}(x) = f^{[n]}(f(x))</math> for each integer <math>n \geq 2</math>. For how many values of <math>x</math> in <math>[0,1]</math> is <math>f^{[2005]}(x) = \frac {1}{2}</math>?
−
<math>
+
 
−
(\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2005 \qquad (\mathrm {C})\ 4010 \qquad (\mathrm {D}) \ 2005^2 \qquad (\mathrm {E})\ 2^{2005}
+
<math> (\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2005 \qquad (\mathrm {C})\ 4010 \qquad (\mathrm {D}) \ 2005^2 \qquad (\mathrm {E})\ 2^{2005} </math>
−
</math>
 
  
 
[[2005 AMC 12A Problems/Problem 20|Solution]]
 
[[2005 AMC 12A Problems/Problem 20|Solution]]
Line 223: Line 294:
  
 
== See also ==
 
== See also ==
 +
 +
{{AMC12 box|year=2005|ab=A|before=[[2004 AMC 12B Problems]]|after=[[2005 AMC 12B Problems]]}}
 +
 
* [[AMC 12]]
 
* [[AMC 12]]
 
* [[AMC 12 Problems and Solutions]]
 
* [[AMC 12 Problems and Solutions]]
 
* [[2005 AMC 12A]]
 
* [[2005 AMC 12A]]
−
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=48 2005 AMC A Math Jam Transcript]
+
* [https://artofproblemsolving.com/school/mathjams-transcripts?id=48 2005 AMC A Math Jam Transcript]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 21:18, 7 September 2024

2005 AMC 12A (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

Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?

$(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20$

Solution

Problem 2

The equations $2x + 7 = 3$ and $bx - 10 = - 2$ have the same solution. What is the value of $b$?

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

Solution

Problem 3

A rectangle with diagonal length $x$ is twice as long as it is wide. What is the area of the rectangle?

$(\mathrm {A}) \ \frac 14x^2 \qquad (\mathrm {B}) \ \frac 25x^2 \qquad (\mathrm {C})\ \frac 12x^2 \qquad (\mathrm {D}) \ x^2 \qquad (\mathrm {E})\ \frac 32x^2$

Solution

Problem 4

A store normally sells windows at $$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?

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

Solution

Problem 5

The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers?

$(\mathrm {A}) \ 23 \qquad (\mathrm {B}) \ 24 \qquad (\mathrm {C})\ 25 \qquad (\mathrm {D}) \ 26 \qquad (\mathrm {E})\ 27$

Solution

Problem 6

Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?

$(\mathrm {A}) \ 4 \qquad (\mathrm {B}) \ 5 \qquad (\mathrm {C})\ 6 \qquad (\mathrm {D}) \ 7 \qquad (\mathrm {E})\ 8$

Solution

Problem 7

Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$? [asy] unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW);[/asy]

$(\mathrm {A}) \ 25 \qquad (\mathrm {B}) \ 32 \qquad (\mathrm {C})\ 36 \qquad (\mathrm {D}) \ 40 \qquad (\mathrm {E})\ 42$

Solution

Problem 8

Let $A,M$, and $C$ be digits with

\[(100A+10M+C)(A+M+C) = 2005\]

What is $A$?

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

Solution

Problem 9

There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of these values of $a$?

$(\mathrm {A}) \ -16 \qquad (\mathrm {B}) \ -8 \qquad (\mathrm {C})\ 0 \qquad (\mathrm {D}) \ 8 \qquad (\mathrm {E})\ 20$

Solution

Problem 10

A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?

$(\mathrm {A}) \ 3 \qquad (\mathrm {B}) \ 4 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 6 \qquad (\mathrm {E})\ 7$

Solution

Problem 11

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?

$(\mathrm {A}) \ 41 \qquad (\mathrm {B}) \ 42 \qquad (\mathrm {C})\ 43 \qquad (\mathrm {D}) \ 44 \qquad (\mathrm {E})\ 45$

Solution

Problem 12

A line passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?

$(\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 8 \qquad (\mathrm {E})\ 9$

Solution

Problem 13

In the five-sided star shown, the letters $A$, $B$, $C$, $D$ and $E$ are replaced by the numbers 3, 5, 6, 7 and 9, although not necessarily in that order. The sums of the numbers at the ends of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic sequence?

[asy] draw((0,0)--(0.5,1.54)--(1,0)--(-0.31,0.95)--(1.31,0.95)--cycle); label("$A$",(0.5,1.54),N); label("$B$",(1,0),SE); label("$C$",(-0.31,0.95),W); label("$D$",(1.31,0.95),E); label("$E$",(0,0),SW); [/asy]

$(\mathrm {A}) \ 9 \qquad (\mathrm {B}) \ 10 \qquad (\mathrm {C})\ 11 \qquad (\mathrm {D}) \ 12 \qquad (\mathrm {E})\ 13$

Solution

Problem 14

On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?

$(\mathrm {A}) \ \frac{5}{11} \qquad (\mathrm {B}) \ \frac{10}{21} \qquad (\mathrm {C})\ \frac{1}{2} \qquad (\mathrm {D}) \ \frac{11}{21} \qquad (\mathrm {E})\ \frac{6}{11}$

Solution

Problem 15

Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?

[asy] unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]

$(\text {A}) \ \frac {1}{6} \qquad (\text {B}) \ \frac {1}{4} \qquad (\text {C})\ \frac {1}{3} \qquad (\text {D}) \ \frac {1}{2} \qquad (\text {E})\ \frac {2}{3}$

Solution

Problem 16

Three circles of radius $s$ are drawn in the first quadrant of the $xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $x$-axis, and the third is tangent to the first circle and the $y$-axis. A circle of radius $r > s$ is tangent to both axes and to the second and third circles. What is $r/s$?

[asy] import graph; unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=3; pair O0=(9,9), O1=(1,1), O2=(3,1), O3=(1,3); pair P0=O0+9*dir(-45), P3=O3+dir(70); pair[] ps={O0,O1,O2,O3}; dot(ps); draw(Circle(O0,9)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(O0--P0,linetype("3 3")); draw(O3--P3,linetype("2 2")); draw((0,0)--(18,0)); draw((0,0)--(0,18)); label("$r$",midpoint(O0--P0),NE); label("$s$",(-1.5,4)); draw((-1,4)--midpoint(O3--P3));[/asy]

$(\mathrm {A}) \ 5 \qquad (\mathrm {B}) \ 6 \qquad (\mathrm {C})\ 8 \qquad (\mathrm {D}) \ 9 \qquad (\mathrm {E})\ 10$

Solution

Problem 17

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?

$(\mathrm {A}) \ \frac {1}{12} \qquad (\mathrm {B}) \ \frac {1}{9} \qquad (\mathrm {C})\ \frac {1}{8} \qquad (\mathrm {D}) \ \frac {1}{6} \qquad (\mathrm {E})\ \frac {1}{4}$

2005 AMC 12A Problem 17.png

Solution

Problem 18

Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000?

$(\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 102 \qquad (\mathrm {C})\ 104 \qquad (\mathrm {D}) \ 106 \qquad (\mathrm {E})\ 108$

Solution

Problem 19

A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?

$(\mathrm {A}) \ 1404 \qquad (\mathrm {B}) \ 1462 \qquad (\mathrm {C})\ 1604 \qquad (\mathrm {D}) \ 1605 \qquad (\mathrm {E})\ 1804$

Solution

Problem 20

For each $x$ in $[0,1]$, define

$\begin{cases} f(x) = 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2};\\ f(x) = 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1. \end{cases}$

Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. For how many values of $x$ in $[0,1]$ is $f^{[2005]}(x) = \frac {1}{2}$?

$(\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2005 \qquad (\mathrm {C})\ 4010 \qquad (\mathrm {D}) \ 2005^2 \qquad (\mathrm {E})\ 2^{2005}$

Solution

Problem 21

How many ordered triples of integers $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$?

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

Solution

Problem 22

A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$?

$\mathrm{(A) } 8 \qquad \mathrm{(B) } 10 \qquad \mathrm{(C) } 12 \qquad \mathrm{(D) } 14 \qquad \mathrm{(E) } 16$

Solution

Problem 23

Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{ 2, 2^2, 2^3, \ldots, 2^{25} \}$. What is the probability that $\log_{a} b$ is an integer?

$\mathrm {(A) } \frac{2}{25} \qquad \mathrm {(B) } \frac{31}{300} \qquad \mathrm {(C) } \frac{13}{100} \qquad \mathrm {(D) } \frac{7}{50} \qquad \mathrm {(E) } \frac{1}{2}$

Solution

Problem 24

Let $P(x) = (x - 1)(x - 2)(x - 3)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree 3 such that $P(Q(x)) = P(x) \cdot R(x)$?

$\mathrm {(A) } 19 \qquad \mathrm {(B) } 22 \qquad \mathrm {(C) } 24 \qquad \mathrm {(D) } 27 \qquad \mathrm {(E) } 32$

Solution

Problem 25

Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x, y,$ and $z$ are each chosen from the set $\{ 0, 1, 2\}$. How many equilateral triangles have all their vertices in $S$?

$\mathrm {(A) } 72 \qquad \mathrm {(B) } 76 \qquad \mathrm {(C) } 80 \qquad \mathrm {(D) } 84 \qquad \mathrm {(E) } 88$

Solution

See also

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by
2004 AMC 12B Problems 		Followed by
2005 AMC 12B 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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12A_Problems&oldid=226740"