Difference between revisions of "2005 AMC 12A Problems"

Latest revision as of 21:18, 7 September 2024

2005 AMC 12A (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions 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. 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. 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). Figures are not necessarily drawn to scale. 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 $W$ ?

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

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

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

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Difference between revisions of "2005 AMC 12A Problems"

Latest revision as of 21:18, 7 September 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 1: / Line 1: @@
+{{AMC12 Problems|year=2005|ab=A}}
 == 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>?
-[[2005 AMC 12A Problem 1|Solution]]
+<math>
+(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20
+</math>
+[[2005 AMC 12A Problems/Problem 1|Solution]]
 == Problem 2 ==
+The equations <math>2x + 7 = 3</math> and <math>bx - 10 = - 2</math> have the same solution. What is the value of <math>b</math>?
-[[2005 AMC 12A Problem 2|Solution]]
+<math>
+(\mathrm {A}) \ -8 \qquad (\mathrm {B}) \ -4 \qquad (\mathrm {C})\ 2 \qquad (\mathrm {D}) \ 4 \qquad (\mathrm {E})\ 8
+</math>
+[[2005 AMC 12A Problems/Problem 2|Solution]]
 == Problem 3 ==
+A rectangle with diagonal length <math>x</math> is twice as long as it is wide. What is the area of the rectangle?
-[[2005 AMC 12A Problem 3|Solution]]
+<math>
+(\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
+</math>
+[[2005 AMC 12A Problems/Problem 3|Solution]]
 == Problem 4 ==
+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?
-[[2005 AMC 12A Problem 4|Solution]]
+<math>
+(\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 200 \qquad (\mathrm {C})\ 300 \qquad (\mathrm {D}) \ 400 \qquad (\mathrm {E})\ 500
+</math>
+[[2005 AMC 12A Problems/Problem 4|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?
+<math>
+(\mathrm {A}) \ 23 \qquad (\mathrm {B}) \ 24 \qquad (\mathrm {C})\ 25 \qquad (\mathrm {D}) \ 26 \qquad (\mathrm {E})\ 27
+</math>
-[[2005 AMC 12A Problem 5|Solution]]
+[[2005 AMC 12A Problems/Problem 5|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?
+<math>
+(\mathrm {A}) \ 4 \qquad (\mathrm {B}) \ 5 \qquad (\mathrm {C})\ 6 \qquad (\mathrm {D}) \ 7 \qquad (\mathrm {E})\ 8
+</math>
-[[2005 AMC 12A Problem 6|Solution]]
+[[2005 AMC 12A Problems/Problem 6|Solution]]
 == 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>?
+<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>
-[[2005 AMC 12A Problem 7|Solution]]
+<math>
+(\mathrm {A}) \ 25 \qquad (\mathrm {B}) \ 32 \qquad (\mathrm {C})\ 36 \qquad (\mathrm {D}) \ 40 \qquad (\mathrm {E})\ 42
+</math>
+[[2005 AMC 12A Problems/Problem 7|Solution]]
 == Problem 8 ==
-[[2005 AMC 12A Problem 8|Solution]]
+Let <math>A,M</math>, and <math>C</math> be digits with
+<cmath>(100A+10M+C)(A+M+C) = 2005</cmath>
+What is <math>A</math>?
+<math>
+(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 4 \qquad (\mathrm {E})\ 5
+</math>
+[[2005 AMC 12A Problems/Problem 8|Solution]]
 == Problem 9 ==
+There are two values of <math>a</math> for which the equation <math>4x^2 + ax + 8x + 9 = 0</math> has only one solution for <math>x</math>. What is the sum of these values of <math>a</math>?
-[[2005 AMC 12A Problem 9|Solution]]
+<math>(\mathrm {A}) \ -16 \qquad (\mathrm {B}) \ -8 \qquad (\mathrm {C})\ 0 \qquad (\mathrm {D}) \ 8 \qquad (\mathrm {E})\ 20</math>
+[[2005 AMC 12A Problems/Problem 9|Solution]]
 == Problem 10 ==
+A wooden cube <math>n</math> units on a side is painted red on all six faces and then cut into <math>n^3</math> unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is <math>n</math>?
-[[2005 AMC 12A Problem 10|Solution]]
+<math>
+(\mathrm {A}) \ 3 \qquad (\mathrm {B}) \ 4 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 6 \qquad (\mathrm {E})\ 7
+</math>
+[[2005 AMC 12A Problems/Problem 10|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?
-[[2005 AMC 12A Problem 11|Solution]]
+<math>(\mathrm {A}) \ 41 \qquad (\mathrm {B}) \ 42 \qquad (\mathrm {C})\ 43 \qquad (\mathrm {D}) \ 44 \qquad (\mathrm {E})\ 45</math>
+[[2005 AMC 12A Problems/Problem 11|Solution]]
 == Problem 12 ==
+A line passes through <math>A\ (1,1)</math> and <math>B\ (100,1000)</math>. How many other points with integer coordinates are on the line and strictly between <math>A</math> and <math>B</math>?
-[[2005 AMC 12A Problem 12|Solution]]
+<math>
+(\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 3 \qquad (\mathrm {D}) \ 8 \qquad (\mathrm {E})\ 9
+</math>
+[[2005 AMC 12A Problems/Problem 12|Solution]]
 == Problem 13 ==
+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?
-[[2005 AMC 12A Problem 13|Solution]]
+<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>
+(\mathrm {A}) \ 9 \qquad (\mathrm {B}) \ 10 \qquad (\mathrm {C})\ 11 \qquad (\mathrm {D}) \ 12 \qquad (\mathrm {E})\ 13
+</math>
+[[2005 AMC 12A Problems/Problem 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?
+<math>
+(\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}
+</math>
-[[2005 AMC 12A Problem 14|Solution]]
+[[2005 AMC 12A Problems/Problem 14|Solution]]
 == 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>?
-[[2005 AMC 12A Problem 15|Solution]]
+<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>
+[[2005 AMC 12A Problems/Problem 15|Solution]]
 == Problem 16 ==
-[[2005 AMC 12A Problem 16|Solution]]
+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>?
+<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>
+<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]]
 == 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 <math>W</math>?
-[[2005 AMC 12A Problem 17|Solution]]
+<math>
+(\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}
+</math>
+[[Image:2005 AMC 12A Problem 17.png]]
+[[2005 AMC 12A Problems/Problem 17|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?
+<math>
+(\mathrm {A}) \ 100 \qquad (\mathrm {B}) \ 102 \qquad (\mathrm {C})\ 104 \qquad (\mathrm {D}) \ 106 \qquad (\mathrm {E})\ 108
+</math>
-[[2005 AMC 12A Problem 18|Solution]]
+[[2005 AMC 12A Problems/Problem 18|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?
+<math>
+(\mathrm {A}) \ 1404 \qquad (\mathrm {B}) \ 1462 \qquad (\mathrm {C})\ 1604 \qquad (\mathrm {D}) \ 1605 \qquad (\mathrm {E})\ 1804
+</math>
-[[2005 AMC 12A Problem 19|Solution]]
+[[2005 AMC 12A Problems/Problem 19|Solution]]
 == Problem 20 ==
+For each <math>x</math> in <math>[0,1]</math>, define
+<math>\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}</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>?
-[[2005 AMC 12A Problem 20|Solution]]
+<math> (\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2005 \qquad (\mathrm {C})\ 4010 \qquad (\mathrm {D}) \ 2005^2 \qquad (\mathrm {E})\ 2^{2005} </math>
+[[2005 AMC 12A Problems/Problem 20|Solution]]
 == Problem 21 ==
+How many ordered triples of [[integer]]s <math>(a,b,c)</math>, with <math>a \ge 2</math>, <math>b\ge 1</math>, and <math>c \ge 0</math>, satisfy both <math>\log_a b = c^{2005}</math> and <math>a + b + c = 2005</math>?
-[[2005 AMC 12A Problem 21|Solution]]
+<math>\mathrm{(A)} \ 0 \qquad \mathrm{(B)} \ 1 \qquad \mathrm{(C)} \ 2 \qquad \mathrm{(D)} \ 3 \qquad \mathrm{(E)} \ 4</math>
+[[2005 AMC 12A Problems/Problem 21|Solution]]
 == Problem 22 ==
+A rectangular box <math>P</math> is inscribed in a sphere of radius <math>r</math>. The surface area of <math>P</math> is 384, and the sum of the lengths of its 12 edges is 112. What is <math>r</math>?
-[[2005 AMC 12A Problem 22|Solution]]
+<math>\mathrm{(A) } 8 \qquad \mathrm{(B) } 10 \qquad \mathrm{(C) } 12 \qquad \mathrm{(D) } 14 \qquad \mathrm{(E) } 16</math>
+[[2005 AMC 12A Problems/Problem 22|Solution]]
 == Problem 23 ==
+Two distinct numbers <math>a</math> and <math>b</math> are chosen randomly from the set <math>\{ 2, 2^2, 2^3, \ldots, 2^{25} \}</math>. What is the probability that <math>\log_{a} b</math> is an integer?
-[[2005 AMC 12A Problem 23|Solution]]
+<math>\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}</math>
+[[2005 AMC 12A Problems/Problem 23|Solution]]
 == Problem 24 ==
+Let <math>P(x) = (x - 1)(x - 2)(x - 3)</math>. For how many polynomials <math>Q(x)</math> does there exist a polynomial <math>R(x)</math> of degree 3 such that <math>P(Q(x)) = P(x) \cdot R(x)</math>?
-[[2005 AMC 12A Problem 24|Solution]]
+<math>\mathrm {(A) } 19 \qquad \mathrm {(B) } 22 \qquad \mathrm {(C) } 24 \qquad \mathrm {(D) } 27 \qquad \mathrm {(E) } 32</math>
+[[2005 AMC 12A Problems/Problem 24|Solution]]
 == Problem 25 ==
+Let <math>S</math> be the set of all points with coordinates <math>(x,y,z)</math>, where <math>x, y,</math> and <math>z</math> are each chosen from the set <math>\{ 0, 1, 2\}</math>. How many equilateral triangles have all their vertices in <math>S</math>?
+<math>\mathrm {(A) } 72 \qquad \mathrm {(B) } 76 \qquad \mathrm {(C) } 80 \qquad \mathrm {(D) } 84 \qquad \mathrm {(E) } 88</math>
-[[2005 AMC 12A Problem 25|Solution]]
+[[2005 AMC 12A Problems/Problem 25|Solution]]
 == See also ==
+{{AMC12 box|year=2005|ab=A|before=[[2004 AMC 12B Problems]]|after=[[2005 AMC 12B Problems]]}}
 * [[AMC 12]]
 * [[AMC 12 Problems and Solutions]]
 * [[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]]
+{{MAA Notice}}