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Difference between revisions of "2005 AMC 12A Problems"

(22,23,25)
(add questions 16-25)
Line 60: Line 60:
  
 
== 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>?
 +
 +
<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]]
  
 
== Problem 17 ==
 
== 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>?
 +
 +
<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>
  
 
[[2005 AMC 12A Problems/Problem 17|Solution]]
 
[[2005 AMC 12A Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== 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 Problems/Problem 18|Solution]]
 
[[2005 AMC 12A Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== 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 Problems/Problem 19|Solution]]
 
[[2005 AMC 12A Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== 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>?
 +
<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]]
 
[[2005 AMC 12A Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
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>?
 +
 +
<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 21|Solution]]
 
[[2005 AMC 12A Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== 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 $r$?
+
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>?
−
\[
+
 
−
\text{(A) } 8 \qquad \text{(B) } 10 \qquad \text{(C) } 12 \qquad \text{(D) } 14 \qquad \text{(E) } 16
+
<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]]
 
[[2005 AMC 12A Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== 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?
+
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?
−
\[
+
 
−
\text {(A) } \frac{2}{25} \qquad \text {(B) } \frac{31}{300} \qquad \text {(C) } \frac{13}{100} \qquad \text {(D) } \frac{7}{50} \qquad \text {(E) } \frac{1}{2}
+
<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]]
 
[[2005 AMC 12A Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== 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>?
 +
 +
<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]]
 
[[2005 AMC 12A Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== 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 $S$?
+
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>?
−
\[
+
 
−
\text {(A) } 72 \qquad \text {(B) } 76 \qquad \text {(C) } 80 \qquad \text {(D) } 84 \qquad \text {(E) } 88
+
<math>\mathrm {(A) } 72 \qquad \mathrm {(B) } 76 \qquad \mathrm {(C) } 80 \qquad \mathrm {(D) } 84 \qquad \mathrm {(E) } 88</math>
−
\]
+
 
 
[[2005 AMC 12A Problems/Problem 25|Solution]]
 
[[2005 AMC 12A Problems/Problem 25|Solution]]
  

Revision as of 21:52, 21 September 2007

Problem 1

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

2005 12A AMC-16.png

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 $r/s$?

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

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

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 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

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