2005 AMC 12A Problems

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 <dollar/> $100$ 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 number is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers?

$(\mathrm {A}) \ 23 \qquad (\mathrm {B}) \ 24 \qquad (\mathrm {C})\ 25 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 27$

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

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 $S$ ?