question string | answer string | difficulty int64 | problem_id string |
|---|---|---|---|
One ticket to a show costs $\$20$ at full price. Susan buys $4$ tickets using a coupon that gives her a $25\%$ discount. Pam buys $5$ tickets using a coupon that gives her a $30\%$ discount. How many more dollars does Pam pay than Susan?
\textbf{(A)} \ 2 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 10 \qquad \textbf{... | C | 2 | 2007A-P1 |
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?
$\mathrm{(A)}\ 0.5\qquad \... | D | 2 | 2007A-P2 |
The larger of two consecutive odd integers is three times the smaller. What is their sum?
$\mathrm{(A)}\ 4\qquad \mathrm{(B)}\ 8\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 16\qquad \mathrm{(E)}\ 20$ | A | 2 | 2007A-P3 |
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?
$\mathrm{(A)}\ 7\qquad \mathrm{(B)}\ 9\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 12\qquad \mathrm{(E)}\ 14$ | A | 2 | 2007A-P4 |
Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\textdollar10,500$ for both taxes. How many dollars was his inheritance?
$(\mathrm {A})\ 30\,000 \qquad (\mathrm {B})\ 32\,500 \qquad (\mathrm... | D | 2 | 2007A-P5 |
Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside triangle $ABC$, angle $ABC$ measures 40 degrees, and angle $ADC$ measures 140 degrees. What is the degree measure of angle $BAD$?
$\mathrm{(A)}\ 20\qquad \mathrm{(B)}\ 30\qquad \mathrm{(C)}\ 40\qquad \mathrm{(D)}\ 50\qquad \mathrm{(E... | D | 2 | 2007A-P6 |
Let $a, b, c, d$, and $e$ be five consecutive terms in an arithmetic sequence, and suppose that $a+b+c+d+e=30$. Which of $a, b, c, d,$ or $e$ can be found?
$\textrm{(A)} \ a\qquad \textrm{(B)}\ b\qquad \textrm{(C)}\ c\qquad \textrm{(D)}\ d\qquad \textrm{(E)}\ e$ | C | 2 | 2007A-P7 |
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?
$\mathrm{(A)}... | C | 2 | 2007A-P8 |
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to hi... | B | 2 | 2007A-P9 |
A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius 3. What is the area of the triangle?
$\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18$ | A | 2 | 2007A-P10 |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | D | 3 | 2007A-P11 |
Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?
$\mathrm{(A)}\ \frac 38\qquad \mathrm{(B)}\ \frac 7{16}\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 9{16}\qquad \mathrm{(E)}\ \frac 58$ | E | 3 | 2007A-P12 |
A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a+b$?
$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 14\qquad \mathrm{(D)}... | B | 3 | 2007A-P13 |
Let a, b, c, d, and e be distinct integers such that
<center>$(6-a)(6-b)(6-c)(6-d)(6-e)=45$</center>
What is $a+b+c+d+e$?
$\mathrm{(A)}\ 5\qquad \mathrm{(B)}\ 17\qquad \mathrm{(C)}\ 25\qquad \mathrm{(D)}\ 27\qquad \mathrm{(E)}\ 30$ | C | 3 | 2007A-P14 |
The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$?
$\mathrm{(A)}\ 7\qquad \mathrm{(B)}\ 9\qquad \mathrm{(C)}\ 19\qquad \mathrm{(D)}\ 24\qquad \mathrm{(E)}\ 26$ | E | 3 | 2007A-P15 |
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
$\mathrm{(A)}\ 96\qquad \mathrm{(B)}\ 104\qquad \mathrm{(C)}\ 112\qquad \mathrm{(D)}\ 120\qquad \mathrm{(E)}\ 256$ | C | 3 | 2007A-P16 |
Suppose that $\sin a + \sin b = \sqrt{\frac{5}{3}}$ and $\cos a + \cos b = 1$. What is $\cos (a - b)$?
$\mathrm{(A)}\ \sqrt{\frac{5}{3}} - 1\qquad \mathrm{(B)}\ \frac 13\qquad \mathrm{(C)}\ \frac 12\qquad \mathrm{(D)}\ \frac 23\qquad \mathrm{(E)}\ 1$ | B | 3 | 2007A-P17 |
The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ What is $a + b + c + d?$
$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 4 \qquad \mathrm{(D)}\ 9 \qquad \mathrm{(E)}\ 16$ | D | 3 | 2007A-P18 |
Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B = (0,0),$ $C = (223,0),$ $D = (680,380),$ and $E = (689,389).$ What is the sum of all possible x-coordinates of $A$?
$\mathrm{(A)}\ 282 \qquad \mathrm{(B)}\ 300 \qquad \mathrm{(C)}\ 600 \qquad \mathrm{(D)}\ 900 \qquad \mathrm{(E)}\ 1200$ | E | 3 | 2007A-P19 |
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
$\mathrm{(A)}\ \frac{5\sqrt{2}-7}{3}\qquad \mathrm{(B)}\ \frac{10-7\sqrt{2}}{3}\qquad \mathrm{(C)}\ \frac{3-2\sqrt{2}}{3}\qquad \mathrm{(D)}\ \frac{8\sqrt{2}-11}{3}\qquad \mathrm{(... | B | 3 | 2007A-P20 |
The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $f(x)=ax^{2}+bx+c$ are equal. Their common value must also be which of the following?
$\textrm{(A)}\ \textrm{the\ coefficient\ of\ }x^{2}~~~ \textrm{(B)}\ \textrm{the\ coefficient\ of\ }x$
$\textrm{(C)}\ \textrm{the\ y-inte... | A | 4 | 2007A-P21 |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ For how many values of $n$ is $n + S(n) + S(S(n)) = 2007?$
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$ | D | 4 | 2007A-P22 |
Square $ABCD$ has area $36,$ and $\overline{AB}$ is parallel to the x-axis. Vertices $A,$ $B$, and $C$ are on the graphs of $y = \log_{a}x,$ $y = 2\log_{a}x,$ and $y = 3\log_{a}x,$ respectively. What is $a?$
$\mathrm{(A)}\ \sqrt [6]{3}\qquad \mathrm{(B)}\ \sqrt {3}\qquad \mathrm{(C)}\ \sqrt [3]{6}\qquad \mathrm{(D)}\ ... | A | 4 | 2007A-P23 |
For each integer $n>1$, let $F(n)$ be the number of solutions to the equation $\sin{x}=\sin{(nx)}$ on the interval $[0,\pi]$. What is $\sum_{n=2}^{2007} F(n)$?
$\mathrm{(A)}\ 2014524 \qquad \mathrm{(B)}\ 2015028 \qquad \mathrm{(C)}\ 2015033 \qquad \mathrm{(D)}\ 2016532 \qquad \mathrm{(E)}\ 2017033$ | D | 4 | 2007A-P24 |
Call a set of integers ''spacy'' if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\ldots,12\},$ including the empty set, are spacy?
$\mathrm{(A)}\ 121 \qquad \mathrm{(B)}\ 123 \qquad \mathrm{(C)}\ 125 \qquad \mathrm{(D)}\ 127 \qquad \mathrm{(E)}\ 129$ | E | 4 | 2007A-P25 |
Isabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?
$\mathrm{(A) \ } 678\qquad \mathrm{(B) ... | E | 2 | 2007B-P1 |
A college student drove his compact car 120 miles home for the weekend and averaged 30 miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20 miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
$\mathrm {(A)} 22\qquad \mathrm {(B)} 24\qquad... | B | 2 | 2007B-P2 |
The point $O$ is the center of the circle circumscribed about triangle $ABC$, with $\angle BOC = 120^{\circ}$ and $\angle AOB = 140^{\circ}$, as shown. What is the degree measure of $\angle ABC$?
<center>Image:2007_12B_AMC-3.png</center>
$\mathrm {(A)} 35\qquad \mathrm {(B)} 40\qquad \mathrm {(C)} 45\qquad \mathrm {(... | D | 2 | 2007B-P3 |
At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?
$\mathrm {(A)} 6\qquad \mathrm {(B)} 8\qquad \mathrm {(C)} 9\qquad \mathrm {(D)} 12\qquad \mathrm {(E)} 18$ | B | 2 | 2007B-P4 |
The 2007 AMC 12 contests will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave the last 3 unanswered. How many of the first 22 problems ... | D | 2 | 2007B-P5 |
Triangle $ABC$ has side lengths $AB = 5$, $BC = 6$, and $AC = 7$. Two bugs start simultaneously from $A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$?
$\mathrm {(A)} 1\qquad \mathrm {(B)} 2\qquad \mathrm {(C)} 3\qquad \mathrm {(D)} 4\qquad \ma... | D | 2 | 2007B-P6 |
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. What is the degree measure of $\angle E$?
$\mathrm {(A)} 90\qquad \mathrm {(B)} 108\qquad \mathrm {(C)} 120\qquad \mathrm {(D)} 144\qquad \mathrm {(E)} 150$ | E | 2 | 2007B-P7 |
Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$ ?
$\mathrm {(A)} 2\qquad \mathrm {(B)} 3\qquad \mathrm {(C)} 4\qquad \mathrm {(D)} 5\qquad \mathrm {(E)} 6$ | D | 2 | 2007B-P8 |
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$?
$\mathrm {(A)} 7\qquad \mathrm {(B)} 13\qquad \mathrm {(C)} 31\qquad \mathrm {(D)} 111\qquad \mathrm {(E)} 211$ | A | 2 | 2007B-P9 |
Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40$% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30$% of the group are girls. How many girls were initially in the group?
$\mathrm {(A)} 4\qquad \mathrm {(B)} 6\qquad \mathrm {(C... | C | 2 | 2007B-P10 |
The angles of quadrilateral $ABCD$ satisfy $\angle A=2\angle B=3\angle C=4\angle D$. What is the degree measure of $\angle A$, rounded to the nearest whole number?
$\mathrm {(A)} 125\qquad \mathrm {(B)} 144\qquad \mathrm {(C)} 153\qquad \mathrm {(D)} 173\qquad \mathrm {(E)} 180$ | D | 3 | 2007B-P11 |
A teacher gave a test to a class in which $10\%$ of the students are juniors and $90\%$ are seniors. The average score on the test was $84$. The juniors all received the same score, and the average score of the seniors was $83$. What score did each of the juniors receive on the test?
$\mathrm {(A)} 85\qquad \mathrm {(... | C | 3 | 2007B-P12 |
A traffic light runs repeatedly through the following cycle: green for $30$ seconds, then yellow for $3$ seconds, and then red for $30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?
$\mathrm {(A)} \frac{1}{63}\qquad \ma... | D | 3 | 2007B-P13 |
Point $P$ is inside equilateral $\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$?
$\mathrm {(A)} 4\qquad \mathrm {(B)} 3\sqrt{3}\qquad \mathrm {(C)} 6\qquad \... | D | 3 | 2007B-P14 |
The geometric series $a+ar+ar^2\cdots$ has a sum of $7$, and the terms involving odd powers of $r$ have a sum of $3$. What is $a+r$?
$\mathrm {(A)} \frac{4}{3}\qquad \mathrm {(B)} \frac{12}{7}\qquad \mathrm {(C)} \frac{3}{2}\qquad \mathrm {(D)} \frac{7}{3}\qquad \mathrm {(E)} \frac{5}{2}$ | E | 3 | 2007B-P15 |
Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
$\mathrm {(A)} 15\qquad \mathrm {(B)} 18\qqu... | A | 3 | 2007B-P16 |
If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10}b$, what is the median of the set $\{0,1,a,b,1/b\}$?
$\mathrm {(A)} 0\qquad \mathrm {(B)} 1\qquad \mathrm {(C)} a\qquad \mathrm {(D)} b\qquad \mathrm {(E)} \frac{1}{b}$ | D | 3 | 2007B-P17 |
Let $a$, $b$, and $c$ be digits with $a\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. What is $a+b+c$?
$\mathrm {(A)} 10\qquad \mathrm {(B)} 13\qqu... | C | 3 | 2007B-P18 |
Rhombus $ABCD$, with side length $6$, is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin(\angle ABC)$?
$\mathrm {(A)} \frac{\pi}{9}\qquad \mathrm {(B)} \frac{1}{2}\qquad \mathrm {(C)} \frac{\pi}{6}\qquad \mathrm {(D)} \frac{\pi}{4}\qquad \mathrm {(E)} \frac{\sqrt{3}}... | A | 3 | 2007B-P19 |
The parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, what is the smallest possible value of $a+b+c+d$?
$\mathrm {(A)} 13\... | D | 3 | 2007B-P20 |
The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
$\mathrm {(A)} 100\qquad \mathrm {(B)} 101\qquad \mathrm {(C)} 102\qquad \mathrm {(D)} 103\qquad \mathrm {(E)} 104$ | A | 4 | 2007B-P21 |
Two particles move along the edges of equilateral $\triangle ABC$ in the direction $$
A\Rightarrow B\Rightarrow C\Rightarrow A,
$$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles ... | A | 4 | 2007B-P22 |
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?
$\mathrm {(A)} 6\qquad \mathrm {(B)} 7\qquad \mathrm {(C)} 8\qquad \mathrm {(D)} 10\qquad \mathrm {(E)} 12$ | A | 4 | 2007B-P23 |
How many pairs of positive integers $(a,b)$ are there such that $\gcd{(a,b)}=1$ and $$
\frac{a}{b}+\frac{14b}{9a}
$$ is an integer?
$\mathrm {(A)} 4\qquad \mathrm {(B)} 6\qquad \mathrm {(C)} 9\qquad \mathrm {(D)} 12\qquad \mathrm {(E)} \text{infinitely many}$ | A | 4 | 2007B-P24 |
Points $A,B,C,D$ and $E$ are located in 3-dimensional space with $AB=BC=CD=DE=EA=2$ and $\angle ABC=\angle CDE=\angle DEA=90^o$. The plane of $\triangle ABC$ is parallel to $\overline{DE}$. What is the area of $\triangle BDE$?
$\mathrm {(A)} \sqrt{2}\qquad \mathrm {(B)} \sqrt{3}\qquad \mathrm {(C)} 2\qquad \mathrm {(D... | C | 4 | 2007B-P25 |
A bakery owner turns on his doughnut machine at $\text{8:30}\ {\small\text{AM}}$. At $\text{11:10}\ {\small\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
$\mathrm{(A)}\ \text{1:50}\ {\small\text{PM}}\qquad\mathrm{(B)}\ \text{3:00}\ {\small\tex... | D | 2 | 2008A-P1 |
What is the reciprocal of $\frac{1}{2}+\frac{2}{3}$?
$\mathrm{(A)}\ \frac{6}{7}\qquad\mathrm{(B)}\ \frac{7}{6}\qquad\mathrm{(C)}\ \frac{5}{3}\qquad\mathrm{(D)}\ 3\qquad\mathrm{(E)}\ \frac{7}{2}$ | A | 2 | 2008A-P2 |
Suppose that $\tfrac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\tfrac{1}{2}$ of $5$ bananas?
$\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ \frac{5}{2}\qquad\mathrm{(C)}\ 3\qquad\mathrm{(D)}\ \frac{7}{2}\qquad\mathrm{(E)}\ 4$ | C | 2 | 2008A-P3 |
Which of the following is equal to the product
$$
\frac{8}{4}\cdot\frac{12}{8}\cdot\frac{16}{12}\cdot\cdots\cdot\frac{4n+4}{4n}\cdot\cdots\cdot\frac{2008}{2004}?
$$
$\mathrm{(A)}\ 251\qquad\mathrm{(B)}\ 502\qquad\mathrm{(C)}\ 1004\qquad\mathrm{(D)}\ 2008\qquad\mathrm{(E)}\ 4016$ | B | 2 | 2008A-P4 |
Suppose that
$$
\frac{2x}{3}-\frac{x}{6}
$$
is an integer. Which of the following statements must be true about $x$?
$\mathrm{(A)}\ \text{It is negative.}\qquad\mathrm{(B)}\ \text{It is even, but not necessarily a multiple of 3.}\\\qquad\mathrm{(C)}\ \text{It is a multiple of 3, but not necessarily even.}\\\qquad\math... | B | 2 | 2008A-P5 |
Heather compares the price of a new computer at two different stores. Store $A$ offers $15\%$ off the sticker price followed by a $\textdollar 90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves $\textdollar 15$ by buying the computer at store $A$ instead of store $B$. What ... | A | 2 | 2008A-P6 |
While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the... | D | 2 | 2008A-P7 |
What is the volume of a cube whose surface area is twice that of a cube with volume 1?
$\mathrm{(A)}\ \sqrt{2}\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 2\sqrt{2}\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 8$ | C | 2 | 2008A-P8 |
Older television screens have an aspect ratio of $4: 3$. That is, the ratio of the width to the height is $4: 3$. The aspect ratio of many movies is not $4: 3$, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose ... | D | 2 | 2008A-P9 |
Doug can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $t$?
$... | D | 2 | 2008A-P10 |
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum?
```asy
unitsize(.8cm);
pen p = linewidth(1);
draw(shift(-2,0)*unitsquare,p);
label("1",(-1.5,0.5));
draw(shift(-1,0)*unitsquare,... | C | 3 | 2008A-P11 |
A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\{x:a \le x \le b \}$.) What are the domain and range, respectively, of the function $g$ defined by $g(x)=1-f(x+1)$?
$\mathrm{(A)}\ [-1,1],[-1,0]\qquad\mathrm{(B)}\ [-1,1],[0,1]\qquad\textbf{(C)}\ [0,2],[-1,0]\qquad\mathrm{(D)}\ [1,3],... | B | 3 | 2008A-P12 |
Points $A$ and $B$ lie on a circle centered at $O$, and $\angle AOB = 60^\circ$. A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle?
$\mathrm{(A)}\ \frac{1}{16}\qquad\mathrm{(B)}\ \... | B | 3 | 2008A-P13 |
What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$?
$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ \frac {7}{2}\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ \frac{9}{2}\qquad\mathrm{(E)}\ 5$ | A | 3 | 2008A-P14 |
Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 6\qquad\mathrm{(E)}\ 8$ | D | 3 | 2008A-P15 |
The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^\text{th}$ term of the sequence is $\log(b^n)$. What is $n$?
$\mathrm{(A)}\ 40\qquad\mathrm{(B)}\ 56\qquad\mathrm{(C)}\ 76\qquad\mathrm{(D)}\ 112\qquad\mathrm{(E)}\ 143$ | D | 3 | 2008A-P16 |
Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n=a_{n-1}/2$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$?
$\mathrm{(A)}\ 250\qquad\mathrm{(B)}\ 251\qquad\mathrm{(C)}\ 501\qqu... | D | 3 | 2008A-P17 |
A triangle $\triangle ABC$ with sides $5$, $6$, $7$ is placed in the three-dimensional plane with one vertex on the positive $x$ axis, one on the positive $y$ axis, and one on the positive $z$ axis. Let $O$ be the origin. What is the volume of $OABC$?
$\mathrm{(A)}\ \sqrt{85}\qquad\mathrm{(B)}\ \sqrt{90}\qquad\mathrm{... | C | 3 | 2008A-P18 |
In the expansion of
$$
\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,
$$
what is the coefficient of $x^{28}$?
$\mathrm{(A)}\ 195\qquad\mathrm{(B)}\ 196\qquad\mathrm{(C)}\ 224\qquad\mathrm{(D)}\ 378\qquad\mathrm{(E)}\ 405$ | C | 3 | 2008A-P19 |
Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$?
$\mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \... | E | 3 | 2008A-P20 |
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is $\textit{heavy-tailed}$ if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?
$\mathrm{(A)}\ 36\qquad\mathrm{(B)}\ 40\qquad\textbf{(C)}\ 44\qquad\mathrm{(D)}\ 48\qquad\mathrm{(E)}\ 52$ | D | 4 | 2008A-P21 |
A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so th... | C | 4 | 2008A-P22 |
The solutions of the equation $z^4+4z^3i-6z^2-4zi-i=0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon?
$\mathrm{(A)}\ 2^{\frac{5}{8}}\qquad\mathrm{(B)}\ 2^{\frac{3}{4}}\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 2^{\frac{5}{4}}\qquad\mathrm{(E)}\ 2^{\frac{3}{2}}$ | D | 4 | 2008A-P23 |
Triangle $ABC$ has $\angle C = 60^{\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\tan{\angle BAD}$?
$\mathrm{(A)}\ \frac{\sqrt{3}}{6}\qquad\mathrm{(B)}\ \frac{\sqrt{3}}{3}\qquad\mathrm{(C)}\ \frac{\sqrt{3}}{2\sqrt{2}}\qquad\mathrm{(D)}\ \frac{\sqrt{3}}{4\sqrt{2}-3}\qqu... | D | 4 | 2008A-P24 |
A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\ldots$ of points in the coordinate plane satisfies
$(a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)$ for $n = 1,2,3,\ldots$.
Suppose that $(a_{100},b_{100}) = (2,4)$. What is $a_1 + b_1$?
$\mathrm{(A)}\ -\frac{1}{2^{97}}\qquad\mathrm{(B)}\ -\frac{... | D | 4 | 2008A-P25 |
A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?
\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6 | E | 2 | 2008B-P1 |
A $4\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?
$\begin{tabular}[t]... | B | 2 | 2008B-P2 |
A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possible salary, in dollars, for a single player?
\textbf{(A)}\ 270,000 \qq... | C | 2 | 2008B-P3 |
On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. What is the ratio of the area of the smaller sector $COD$ to the area of the circle?
```asy
unitsize(6mm);
defaultpen(linewidth(0.7)+fontsize(8pt));
pair C = 3*dir (30);
pair D = 3... | D | 2 | 2008B-P4 |
A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?
(Solution)
\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 7 \qquad \... | C | 2 | 2008B-P5 |
Postman Pete has a pedometer to count his steps. The pedometer records up to $99999$ steps, then flips over to $00000$ on the next step. Pete plans to determine his mileage for a year. On January $1$ Pete sets the pedometer to $00000$. During the year, the pedometer flips from $99999$ to $00000$ forty-four times. On De... | A | 2 | 2008B-P6 |
For real numbers $a$ and $b$, define $a$\$$b = (a - b)^2$. What is $(x - y)^2$\$$(y - x)^2$?
(Solution)
\textbf{(A)}\ 0 \qquad \textbf{(B)}\ x^2 + y^2 \qquad \textbf{(C)}\ 2x^2 \qquad \textbf{(D)}\ 2y^2 \qquad \textbf{(E)}\ 4xy | A | 2 | 2008B-P7 |
Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $4$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $9$ times the length of $\overline{CD}$. The length of $\overline{BC}$ is what fraction of the length of $\overline{AD}$?
(Solution)
\textbf{(A)}\ \frac {1}{36} \... | C | 2 | 2008B-P8 |
Points $A$ and $B$ are on a circle of radius $5$ and $AB = 6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$?
(Solution)
\textbf{(A)}\ \sqrt {10} \qquad \textbf{(B)}\ \frac {7}{2} \qquad \textbf{(C)}\ \sqrt {14} \qquad \textbf{(D)}\ \sqrt {15} \qquad \textbf{(E)}\ 4 | A | 2 | 2008B-P9 |
Bricklayer Brenda would take $9$ hours to build a chimney alone, and bricklayer Brandon would take $10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chim... | B | 2 | 2008B-P10 |
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
\textbf{(A)}\ 4000 \qquad \textbf{(B)}\ 2000(4-\sqrt{2}) \qquad \textbf{(C)}\ 6000 \qquad \textbf... | A | 3 | 2008B-P11 |
For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the $2008$th term of the sequence?
\textbf{(A)}\ 2008 \qquad \textbf{(B)}\ 4015 \qquad \textbf{(C)}\ 4016 \qquad \textbf{(D)}\ 4030056 \qquad \textbf{(E)}\ 4032064 | B | 3 | 2008B-P12 |
Vertex $E$ of equilateral triangle $\triangle ABE$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle ABE$ whose distance from $\overline{AD}$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$?
(Solution)
\textbf{(A)}\ \fra... | B | 3 | 2008B-P13 |
A circle has a radius of $\log_{10}{(a^2)}$ and a circumference of $\log_{10}{(b^4)}$. What is $\log_{a}{b}$?
(Solution)
\textbf{(A)}\ \frac{1}{4\pi} \qquad \textbf{(B)}\ \frac{1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi} | C | 3 | 2008B-P14 |
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the... | C | 3 | 2008B-P15 |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies hal... | B | 3 | 2008B-P16 |
Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$?
(Solution)
\textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 18\qquad\te... | C | 3 | 2008B-P17 |
A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $196$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$, respectively. What is the volume of the pyramid?
(Solution)
\textbf{(A)}\ 392 \qquad \textbf{(B)}\ 196\sqrt {6} \qquad \textbf{(C)}\ 392\sqrt {2} \qquad \te... | E | 3 | 2008B-P18 |
A function $f$ is defined by $f(z) = (4 + i) z^2 + \alpha z + \gamma$ for all complex numbers $z$, where $\alpha$ and $\gamma$ are complex numbers and $i^2 = - 1$. Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $| \alpha | + |\gamma |$?
(Solution)
\textbf{(A)} \; 1 \qquad \text... | B | 3 | 2008B-P19 |
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck travels at $10$ feet per second in the same direction as Michael and stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just le... | B | 3 | 2008B-P20 |
Two circles of radius 1 are to be constructed as follows. The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ and $(2,0)$. The center of circle $B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $(0,1)$ to $(2,1)$. Wha... | E | 4 | 2008B-P21 |
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?
(Solution)
\textbf{(A)}... | E | 4 | 2008B-P22 |
The sum of the base-$10$ logarithms of the divisors of $10^n$ is $792$. What is $n$?
(Solution)
\textbf{(A)}\ 11\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ 15 | A | 4 | 2008B-P23 |
Let $A_0 = (0,0)$. Distinct points $A_1,A_2,\ldots$ lie on the $x$-axis, and distinct points $B_1,B_2,\ldots$ lie on the graph of $y = \sqrt {x}$. For every positive integer $n$, $A_{n - 1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\ge100$?
(Solution)
\textbf{(A)}\ 13\qqua... | C | 4 | 2008B-P24 |
Let $ABCD$ be a trapezoid with $AB||CD$, $AB = 11$, $BC = 5$, $CD = 19$, and $DA = 7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$?
(Solution)
\textbf{(A)}\ 28\sqrt {3}\qquad \textbf{(B)}\ 30\sqrt {3}\qquad \textbf{... | B | 4 | 2008B-P25 |
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