View source for 2007 AMC 10B Problems

==Problem 1==

Isabella's house has <math>3</math> bedrooms. Each bedroom is <math>12</math> feet long, <math>10</math> feet wide, and <math>8</math> feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy <math>60</math> square feet in each bedroom. How many square feet of walls must be painted?

<math>\textbf{(A) } 678 \qquad\textbf{(B) } 768 \qquad\textbf{(C) } 786 \qquad\textbf{(D) } 867 \qquad\textbf{(E) } 876</math>

[[2007 AMC 10B Problems/Problem 1|Solution]]

==Problem 2==

Define the operation <math>\star</math> by <math>a \star b = (a+b)b.</math> What is <math>(3 \star 5) - (5 \star 3)?</math>

<math>\textbf{(A) } -16 \qquad\textbf{(B) } -8 \qquad\textbf{(C) } 0 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 16</math>

[[2007 AMC 10B Problems/Problem 2|Solution]]

==Problem 3==

A college student drove his compact car <math>120</math> miles home for the weekend and averaged <math>30</math> miles per gallon. On the return trip the student drove his parents' SUV and averaged only <math>20</math> miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?

<math>\textbf{(A) } 22 \qquad\textbf{(B) } 24 \qquad\textbf{(C) } 25 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 28</math>

[[2007 AMC 10B Problems/Problem 3|Solution]]

==Problem 4==

The point <math>O</math> is the center of the circle circumscribed about <math>\triangle ABC,</math> with <math>\angle BOC=120^\circ</math> and <math>\angle AOB=140^\circ,</math> as shown. What is the degree measure of <math>\angle ABC?</math>

<math>\textbf{(A) } 35 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 50 \qquad\textbf{(E) } 60</math>

[[2007 AMC 10B Problems/Problem 4|Solution]]

==Problem 5==

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?

<math>\textbf{(A) } \text{All Dramps are Brafs and are Crups.}\\
\textbf{(B) } \text{All Brafs are Crups and are Dramps.}\\
\textbf{(C) } \text{All Arogs are Crups and are Dramps.}\\
\textbf{(D) } \text{All Crups are Arogs and are Brafs.}\\
\textbf{(E) } \text{All Arogs are Dramps and some Arogs may not be Crumps.}</math>

[[2007 AMC 10B Problems/Problem 5|Solution]]

==Problem 6==

The 2007 AMC10 will be scored by awarding <math>6</math> points for each correct response, <math>0</math> points for each incorrect response, and <math>1.5</math> points for each problem left unanswered. After looking over the <math>25</math> problems, Sarah has decided to attempt the first <math>22</math> and leave only the last <math>3</math> unanswered. How many of the first <math>22</math> problems must she solve correctly in order to score at least <math>100</math> points?

<math>\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17</math>

[[2007 AMC 10B Problems/Problem 6|Solution]]

==Problem 7==

All sides of the convex pentagon <math>ABCDE</math> are of equal length, and <math>\angle A= \angle B = 90^\circ.</math> What is the degree measure of <math>\angle E?</math>

<math>\textbf{(A) } 90 \qquad\textbf{(B) } 108 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 144 \qquad\textbf{(E) } 150</math>

[[2007 AMC 10B Problems/Problem 7|Solution]]

==Problem 8==

On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form <math>bbcac,</math> where <math>0 \le a < b < c \le 9,</math> and <math>b</math> was the average of <math>a</math> and <math>c.</math> How many different five-digit numbers satisfy all these properties?

<math>\textbf{(A) } 12 \qquad\textbf{(B) } 16 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 20 \qquad\textbf{(E) } 25</math>

[[2007 AMC 10B Problems/Problem 8|Solution]]

==Problem 9==

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is <math>1</math> place to its right in the alphabet (asumming that the letter <math>A</math> is one place to the right of the letter <math>Z</math>). The second time this same letter appears in the given message, it is replaced by the letter that is <math>1+2</math> places to the right, the third time it is replaced by the letter that is <math>1+2+3</math> places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter <math>s</math> in the message
<cmath>\text{"Lee's sis is a Mississippi miss, Chriss!"?}</cmath>

<math>\textbf{(A) } g \qquad\textbf{(B) } h \qquad\textbf{(C) } o \qquad\textbf{(D) } s \qquad\textbf{(E) } t</math>

[[2007 AMC 10B Problems/Problem 9|Solution]]

==Problem 10==

Two points <math>B</math> and <math>C</math> are in a plane. Let <math>S</math> be the set of all points <math>A</math> in the plane for which <math>\triangle ABC</math> has area <math>1.</math> Which of the following describes <math>S?</math>

<math>\textbf{(A) } \text{two parallel lines} \qquad\textbf{(B) } \text{a parabola} \qquad\textbf{(C) } \text{a circle} \qquad\textbf{(D) } \text{a line segment} \qquad\textbf{(E) } \text{two points}</math>

[[2007 AMC 10B Problems/Problem 10|Solution]]

==Problem 11==

A circle passes through the three vertices of an isosceles triangle that has two sides of length <math>3</math> and a base of length <math>2.</math> What is the area of this circle?

<math>\textbf{(A) } 2\pi \qquad\textbf{(B) } \frac{5}{2}\pi \qquad\textbf{(C) } \frac{81}{32}\pi \qquad\textbf{(D) } 3\pi \qquad\textbf{(E) } \frac{7}{2}\pi</math>

[[2007 AMC 10B Problems/Problem 11|Solution]]

==Problem 12==

Tom's age is <math>T</math> years, which is also the sum of the ages of his three children. His age <math>N</math> years ago was twice the sum of their ages then. What is <math>T/N?</math>

<math>\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6</math>

[[2007 AMC 10B Problems/Problem 12|Solution]]

==Problem 13==

Two circles of radius <math>2</math> are centered at <math>(2,0)</math> and at <math>(0,2).</math> What is the area of the intersection of the interiors of the two circles?

<math>\textbf{(A) } \pi -2 \qquad\textbf{(B) } \frac{\pi}{2} \qquad\textbf{(C) } \frac{\pi \sqrt{3}}{3} \qquad\textbf{(D) } 2(\pi -2) \qquad\textbf{(E) } \pi</math>

[[2007 AMC 10B Problems/Problem 13|Solution]]

==Problem 14==

Some boys and girls are having a car wash to raise money for a class trip to China. Initially <math>40\%</math> of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then <math>30\%</math> of the group are girls. How many girls were initially in the group?

<math>\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 10 \qquad\textbf{(E) } 12</math>

[[2007 AMC 10B Problems/Problem 14|Solution]]

==Problem 15==

The angles of quadrilateral <math>ABCD</math> satisfy <math>\angle A=2 \angle B=3 \angle C=4 \angle D.</math> What is the degree measure of <math>\angle A,</math> rounded to the nearest whole number?

<math>\textbf{(A) } 125 \qquad\textbf{(B) } 144 \qquad\textbf{(C) } 153 \qquad\textbf{(D) } 173 \qquad\textbf{(E) } 180</math>

[[2007 AMC 10B Problems/Problem 15|Solution]]

==Problem 16==

A teacher gave a test to a class in which <math>10\%</math> of the students are juniors and <math>90\%</math> are seniors. The average score on the test was <math>84.</math> The juniors all received the same score, and the average score of the seniors was <math>83.</math> What score did each of the juniors receive on the test?

<math>\textbf{(A) } 85 \qquad\textbf{(B) } 88 \qquad\textbf{(C) } 93 \qquad\textbf{(D) } 94 \qquad\textbf{(E) } 98</math>

[[2007 AMC 10B Problems/Problem 16|Solution]]

==Problem 17==

Point <math>P</math> is inside equilateral <math>\triangle ABC.</math> Points <math>Q, R,</math> and <math>S</math> are the feet of the perpendiculars from <math>P</math> to <math>\overline{AB}, \overline{BC},</math> and <math>\overline{CA},</math> respectively. Given that <math>PQ=1, PR=2,</math> and <math>PS=3,</math> what is <math>AB?</math>

<math>\textbf{(A) } 4 \qquad\textbf{(B) } 3\sqrt{3} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 4\sqrt{3} \qquad\textbf{(E) } 9</math>

[[2007 AMC 10B Problems/Problem 17|Solution]]

==Problem 18==

A circle of radius <math>1</math> is surrounded by <math>4</math> circles of radius <math>r</math> as shown. What is <math>r</math>?

<center><asy>
unitsize(3mm);
defaultpen(linewidth(.8pt)+fontsize(7pt));
dotfactor=4;

real r1=1, r2=1+sqrt(2);
pair A=(0,0), B=(1+sqrt(2),1+sqrt(2)), C=(-1-sqrt(2),1+sqrt(2)), D=(-1-sqrt(2),-1-sqrt(2)), E=(1+sqrt(2),-1-sqrt(2));
pair A1=(1,0), B1=(2+2sqrt(2),1+sqrt(2)), C1=(0,1+sqrt(2)), D1=(0,-1-sqrt(2)), E1=(2+2sqrt(2),-1-sqrt(2));
path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r2); path circleD=Circle(D,r2); path circleE=Circle(E,r2);
draw(circleA); draw(circleB); draw(circleC); draw(circleD); draw(circleE);
draw(A--A1); draw(B--B1); draw(C--C1); draw(D--D1); draw(E--E1);

label("$1$",midpoint(A--A1),N);
label("$r$",midpoint(B--B1),N);
label("$r$",midpoint(C--C1),N);
label("$r$",midpoint(D--D1),N);
label("$r$",midpoint(E--E1),N);
</asy></center>

<math>\textbf{(A) } \sqrt{2} \qquad\textbf{(B) } 1+\sqrt{2} \qquad\textbf{(C) } \sqrt{6} \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 2+\sqrt{2}</math>

[[2007 AMC 10B Problems/Problem 18|Solution]]

==Problem 19==

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by <math>4,</math> and the second number is divided by <math>5.</math> The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?

<center><asy>
unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;

real r=2;
pair O=(0,0);
pair A=(0,2), A1=(0,-2); draw(A--A1);
pair B=(2cos(60),2sin(60)), B1=(-2cos(60),-2sin(60)); draw(B--B1);
pair C=(2cos(60),-2sin(60)), C1=(-2cos(60),2sin(60)); draw(C--C1);
path circleO=Circle(O,r);
draw(circleO);
pair[] ps={O}; dot(ps);

fill((4,0)--(4,1)--(5,1)--(5,0)--cycle,gray);
fill((6,2)--(6,1)--(5,1)--(5,2)--cycle,gray);
fill((6,0)--(6,-1)--(5,-1)--(5,0)--cycle,gray);
fill((6,0)--(6,1)--(7,1)--(7,0)--cycle,gray);
fill((4,-1)--(5,-1)--(5,-2)--(4,-2)--cycle,gray);
fill((6,-1)--(7,-1)--(7,-2)--(6,-2)--cycle,gray);
draw((4,2)--(7,2)--(7,-2)--(4,-2)--cycle);
draw((4,1)--(7,1)); draw((4,0)--(7,0)); draw((4,-1)--(7,-1));
draw((5,2)--(5,-2)); draw((6,2)--(6,-2));
label("$1$",midpoint((4,-1)--(4,-2)),W);
label("$2$",midpoint((4,0)--(4,-1)),W);
label("$3$",midpoint((4,1)--(4,0)),W);
label("$4$",midpoint((4,2)--(4,1)),W);
label("$1$",midpoint((4,-2)--(5,-2)),S);
label("$2$",midpoint((5,-2)--(6,-2)),S);
label("$3$",midpoint((7,-2)--(6,-2)),S);
</asy></center>

<math>\textbf{(A) } \frac{1}{3} \qquad\textbf{(B) } \frac{4}{9} \qquad\textbf{(C) } \frac{1}{2} \qquad\textbf{(D) } \frac{5}{9} \qquad\textbf{(E) } \frac{2}{3}</math>

[[2007 AMC 10B Problems/Problem 19|Solution]]

==Problem 20==

A set of <math>25</math> square blocks is arranged into a <math>5 \times 5</math> square. How many different combinations of <math>3</math> blocks can be selected from that set so that no two are in the same row or column?

<math>\textbf{(A) } 100 \qquad\textbf{(B) } 125 \qquad\textbf{(C) } 600 \qquad\textbf{(D) } 2300 \qquad\textbf{(E) } 3600</math>

[[2007 AMC 10B Problems/Problem 20|Solution]]

==Problem 21==

Right <math>\triangle ABC</math> has <math>AB=3, BC=4,</math> and <math>AC=5.</math> Square <math>XYZW</math> is inscribed in <math>\triangle ABC</math> with <math>X</math> and <math>Y</math> on <math>\overline{AC}, W</math> on <math>\overline{AB},</math> and <math>Z</math> on <math>\overline{BC}.</math> What is the side length of the square?

<math>\textbf{(A) } \frac{3}{2} \qquad\textbf{(B) } \frac{60}{37} \qquad\textbf{(C) } \frac{12}{7} \qquad\textbf{(D) } \frac{23}{13} \qquad\textbf{(E) 2} </math>

[[2007 AMC 10B Problems/Problem 21|Solution]]

==Problem 22==

A player chooses one of the numbers <math>1</math> through <math>4.</math> After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered <math>1</math> through <math>4.</math> If the number chosen appears on the bottom of exactly one die after it has been rolled, then the player wins <math>\</math><math>1</math>. If the number chosen appears on the bottom of both of the dice, then the player wins <math>\</math><math>2</math>. If the number chosen does not appear on the bottom of either of the dice, the player loses <math>\</math><math>1</math>. What is the expected return to the player, in dollars, for one roll of the dice?

<math>\textbf{(A) } -\frac{1}{8} \qquad\textbf{(B) } -\frac{1}{16} \qquad\textbf{(C) } 0 \qquad\textbf{(D) } \frac{1}{16} \qquad\textbf{(E) } \frac{1}{8}</math>

[[2007 AMC 10B Problems/Problem 22|Solution]]

==Problem 23==

A pyramid with a square base is cut by a plane that is parallel to its base and <math>2</math> units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?

<math>\textbf{(A) } 2 \qquad\textbf{(B) } 2+\sqrt{2} \qquad\textbf{(C) } 1+2\sqrt{2} \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 4+2\sqrt{2}</math>

[[2007 AMC 10B Problems/Problem 23|Solution]]

==Problem 24==

Let <math>n</math> denote the smallest positive integer that is divisible by both <math>4</math> and <math>9,</math> and whose base-<math>10</math> representation consists of only <math>4</math>'s and <math>9</math>'s, with at least one of each. What are the last four digits of <math>n?</math>

<math>\textbf{(A) } 4444 \qquad\textbf{(B) } 4494 \qquad\textbf{(C) } 4944 \qquad\textbf{(D) } 9444 \qquad\textbf{(E) } 9944</math>

[[2007 AMC 10B Problems/Problem 24|Solution]]

==Problem 25==

How many pairs of positive integers <math>(a,b)</math> are there such that <math>a</math> and <math>b</math> have no common factors greater than <math>1</math> and
<cmath>\frac{a}{b} + \frac{14b}{9a}</cmath>
is an integer?

<math>\textbf{(A) } 4 \qquad\textbf{(B) } 6 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } \text{infinitely many}</math>

[[2007 AMC 10B Problems/Problem 25|Solution]]