2005 AMC 10B Problems

Problem 1

A scout troop buys $1000$ candy bars at a price of five for $ $2$ . They sell all the candy bars at a price of two for $ $1$ . What was the 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

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?

$\mathrm{(A)} \frac{1}{10} \qquad \mathrm{(B)} \frac{1}{9} \qquad \mathrm{(C)} \frac{1}{3} \qquad \mathrm{(D)} \frac{4}{9} \qquad \mathrm{(E)} \frac{5}{9}$

Solution

Problem 4

For real numbers $a$ and $b$ , define $a \diamond b = \sqrt{a^2 + b^2}$ . What is the value of

$(5 \diamond 12) \diamond ((-12) \diamond (-5))$ ?

$\mathrm{(A)} 0 \qquad \mathrm{(B)} \frac{17}{2} \qquad \mathrm{(C)} 13 \qquad \mathrm{(D)} 13\sqrt{2} \qquad \mathrm{(E)} 26$

Solution

Problem 5

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?

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

Solution

Problem 6

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 7

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

$\mathrm{(A)} \frac{\pi}{16} \qquad \mathrm{(B)} \frac{\pi}{8} \qquad \mathrm{(C)} \frac{3\pi}{16} \qquad \mathrm{(D)} \frac{\pi}{4} \qquad \mathrm{(E)} \frac{\pi}{2}$

Solution

Problem 8

Solution

Problem 9

One fair die has faces $1$ , $1$ , $2$ , $2$ , $3$ , $3$ and another has faces $4$ , $4$ , $5$ , $5$ , $6$ , $6$ . The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?

$\mathrm{(A)} \frac{1}{3} \qquad \mathrm{(B)} \frac{4}{9} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{5}{9} \qquad \mathrm{(E)} \frac{2}{3}$

Solution

Problem 10

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 11

The first term of a sequence is $2005$ . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the ${2005}^{th}$ 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 12

Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?

$\mathrm{(A)} \left(\frac{1}{12}\right)^{12} \qquad \mathrm{(B)} \left(\frac{1}{6}\right)^{12} \qquad \mathrm{(C)} 2\left(\frac{1}{6}\right)^{11} \qquad \mathrm{(D)} \frac{5}{2}\left(\frac{1}{6}\right)^{11} \qquad \mathrm{(E)} \left(\frac{1}{6}\right)^{10}$

Solution

Problem 13

How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$ ?

$\mathrm{(A)} 501 \qquad \mathrm{(B)} 668 \qquad \mathrm{(C)} 835 \qquad \mathrm{(D)} 1002 \qquad \mathrm{(E)} 1169$

Solution

Problem 14

Solution

Problem 15

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}{5} \qquad \mathrm{(C)} \frac{3}{7} \qquad \mathrm{(D)} \frac{1}{2} \qquad \mathrm{(E)} \frac{2}{3} $[[2005 AMC 10B Problems/Problem 15|Solution]]

== Problem 16 ==

The quadratic equation$ (Error compiling LaTeX. Unknown error_msg)x^2 + mx + n = 0 $has roots that are twice those of$ x^2 + px + m = 0 $, and none of$ 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 $[[2005 AMC 10B Problems/Problem 16|Solution]]

== Problem 17 ==

Suppose that$ (Error compiling LaTeX. Unknown error_msg)4^a = 5 $,$ 5^b = 6 $,$ 6^c = 7 $, and$ 7^d = 8 $. What is$ a * b * c * d $?$ \mathrm{(A)} 1 \qquad \mathrm{(B)} \frac{3}{2} \qquad \mathrm{(C)} 2 \qquad \mathrm{(D)} \frac{5}{2} \qquad \mathrm{(E)} 3 $[[2005 AMC 10B Problems/Problem 17|Solution]]

== Problem 18 ==

All of David's telephone numbers have the form$ (Error compiling LaTeX. Unknown error_msg)555-abc-defg $, where$ a $,$ b $,$ c $,$ d $,$ e $,$ f $, and$ g $are distinct digits and in increasing order, and none is either$ 0 $or$ 1 $. How many different telephone numbers can David have?$ \mathrm{(A)} 1 \qquad \mathrm{(B)} 2 \qquad \mathrm{(C)} 7 \qquad \mathrm{(D)} 8 \qquad \mathrm{(E)} 9 $[[2005 AMC 10B Problems/Problem 18|Solution]]

== Problem 19 ==

On a certain math exam,$ (Error compiling LaTeX. Unknown error_msg)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 $[[2005 AMC 10B Problems/Problem 19|Solution]]

== Problem 20 ==

What is the average (mean) of all$ (Error compiling LaTeX. Unknown error_msg)5 $-digit numbers that can be formed by using each of the digits$ 1 $,$ 3 $,$ 5 $,$ 7 $, and$ 8 $exactly once?$ \mathrm{(A)} 48000 \qquad \mathrm{(B)} 49999.5 \qquad \mathrm{(C)} 53332.8 \qquad \mathrm{(D)} 55555 \qquad \mathrm{(E)} 56432.8 $[[2005 AMC 10B Problems/Problem 20|Solution]]

== Problem 21 ==

Forty slips are placed into a hat, each bearing a number$ (Error compiling LaTeX. Unknown error_msg)1 $,$ 2 $,$ 3 $,$ 4 $,$ 5 $,$ 6 $,$ 7 $,$ 8 $,$ 9 $, or$ 10 $, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let$ p $be the probability that all four slips bear the same number. Let$ q $be the probability that two of the slips bear a number$ a $and the other two bear a number$ b \neq a $. What is the value of$ q/p $?$ \mathrm{(A)} 162 \qquad \mathrm{(B)} 180 \qquad \mathrm{(C)} 324 \qquad \mathrm{(D)} 360 \qquad \mathrm{(E)} 720 $[[2005 AMC 10B Problems/Problem 21|Solution]]

== Problem 22 ==

For how many positive integers$ (Error compiling LaTeX. Unknown error_msg)n $less than or equal to$ 24 $is$ n! $evenly divisible by$ 1 + 2 + \ldots + n $?$ \mathrm{(A)} 8 \qquad \mathrm{(B)} 12 \qquad \mathrm{(C)} 16 \qquad \mathrm{(D)} 17 \qquad \mathrm{(E)} 21 $[[2005 AMC 10B Problems/Problem 22|Solution]]

== Problem 23 ==

In trapezoid$ (Error compiling LaTeX. Unknown error_msg)ABCD $we have$ \overline{AB} $parallel to$ \overline{DC} $,$ E $as the midpoint of$ \overline{BC} $, and$ F $as the midpoint of$ \overline{DA} $. The area of$ ABEF $is twice the area of$ FECD $. What is$ AB/DC $?$ \mathrm{(A)} 2 \qquad \mathrm{(B)} 3 \qquad \mathrm{(C)} 5 \qquad \mathrm{(D)} 6 \qquad \mathrm{(E)} 8 $[[2005 AMC 10B Problems/Problem 23|Solution]]

== Problem 24 ==

Let$ (Error compiling LaTeX. Unknown error_msg)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 $[[2005 AMC 10B Problems/Problem 24|Solution]]

== Problem 25 ==

A subset$ (Error compiling LaTeX. Unknown error_msg)B $of the set of integers from$ 1 $to$ 100 $, inclusive, has the property that no two elements of$ B $sum to$ 125 $. What is the maximum possible number of elements in$ B $?$ \mathrm{(A)} 50 \qquad \mathrm{(B)} 51 \qquad \mathrm{(C)} 62 \qquad \mathrm{(D)} 65 \qquad \mathrm{(E)} 68 $