View source for 2004 AMC 10B Problems

[size=1000]<math>\Huge{Problem 1}</math>[/size]

Each row of the Misty Moon Amphitheater has <math>33</math> seats. Rows <math>12</math> through <math>22</math> are reserved for a youth club. How many seats are reserved for this club?

<math> \mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm{(D) \ } 396\qquad \mathrm{(E) \ } 726 </math>

[[2004 AMC 10B Problems/Problem 1|Solution]]

==Problem 1==

Each row of the Misty Moon Amphitheater has <math>33</math> seats. Rows <math>12</math> through <math>22</math> are reserved for a youth club. How many seats are reserved for this club?

<math> \mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm{(D) \ } 396\qquad \mathrm{(E) \ } 726 </math>

[[2004 AMC 10B Problems/Problem 1|Solution]]

== Problem 3 ==

At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made <math>48</math> free throws. How many free throws did she make at the first practice?

<math> \mathrm{(A) \ } 3 \qquad \mathrm{(B) \ } 6 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ } 15 </math>

[[2004 AMC 10B Problems/Problem 3|Solution]]

== Problem 4 ==

A standard six-sided die is rolled, and <math>P</math> is the product of the five numbers that are visible. What is the largest number that is certain to divide <math>P</math>?

<math> \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 144\qquad \mathrm{(E) \ } 720 </math>

[[2004 AMC 10B Problems/Problem 4|Solution]]

== Problem 5 ==

In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are <math>0</math>, <math>1</math>, <math>2</math>, and <math>3</math>, although not necessarily in that order. What is the maximum possible value of the result?

<math> \mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 8 \qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 10 </math>

[[2004 AMC 10B Problems/Problem 5|Solution]]

== Problem 6 ==

Which of the following numbers is a perfect square?

<math> \mathrm{(A) \ } 98! \cdot 99! \qquad \mathrm{(B) \ } 98! \cdot 100! \qquad \mathrm{(C) \ } 99! \cdot 100! \qquad \mathrm{(D) \ } 99! \cdot 101! \qquad \mathrm{(E) \ } 100! \cdot 101! </math>

[[2004 AMC 10B Problems/Problem 6|Solution]]

== Problem 7 ==

On a trip from the United States to Canada, Isabella took <math>d</math> U.S. dollars. At the border she exchanged them all, receiving <math>10</math> Canadian dollars for every <math>7</math> U.S. dollars. After spending <math>60</math> Canadian dollars, she had <math>d</math> Canadian dollars left. What is the sum of the digits of <math>d</math>?

<math> \mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 9 </math>

[[2004 AMC 10B Problems/Problem 7|Solution]]

== Problem 8 ==

Minneapolis-St. Paul International Airport is <math>8</math> miles southwest of downtown St. Paul and <math>10</math> miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?

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

[[2004 AMC 10B Problems/Problem 8|Solution]]

== Problem 9 ==

A square has sides of length <math>10</math>, and a circle centered at one of its vertices has radius <math>10</math>. What is the area of the union of the regions enclosed by the square and the circle?

<math> \mathrm{(A) \ } 200+25\pi \qquad \mathrm{(B) \ } 100+75\pi \qquad \mathrm{(C) \ } 75+100\pi \qquad \mathrm{(D) \ } 100+100\pi \qquad \mathrm{(E) \ } 100+125\pi </math>

[[2004 AMC 10B Problems/Problem 9|Solution]]

== Problem 10 ==

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains <math>100</math> cans, how many rows does it contain?

<math> \mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 411 </math>

[[2004 AMC 10B Problems/Problem 10|Solution]]

== Problem 11 ==

Two eight-sided dice each have faces numbered <math>1</math> through <math>8</math>. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum?

<math> \mathrm{(A) \ } \frac{1}{2} \qquad \mathrm{(B) \ } \frac{47}{64} \qquad \mathrm{(C) \ } \frac{3}{4} \qquad \mathrm{(D) \ } \frac{55}{64} \qquad \mathrm{(E) \ } \frac{7}{8} </math>

[[2004 AMC 10B Problems/Problem 11|Solution]]

== Problem 12 ==

An annulus is the region between two concentric circles. The concentric circles in the ﬁgure have radii <math>b</math> and <math>c</math>, with <math>b>c</math>. Let <math>OX</math> be a radius of the larger circle, let <math>XZ</math> be tangent to the smaller circle at <math>Z</math>, and let <math>OY</math> be the radius of the larger circle that contains <math>Z</math>. Let <math>a=XZ</math>, <math>d=YZ</math>, and <math>e=XY</math>. What is the area of the annulus? 

<asy>
unitsize(1.5cm);
defaultpen(0.8);
real r1=1.5, r2=2.5;
pair O=(0,0);
path inner=Circle(O,r1), outer=Circle(O,r2);
pair Y=(0,r2), Z=(0,r1), X=intersectionpoint( Z--(Z+(10,0)), outer );
filldraw(outer,lightgray,black);
filldraw(inner,white,black);
draw(X--O--Y); draw(Y--X--Z);
label("$O$",O,SW);
label("$X$",X,E);
label("$Y$",Y,N);
label("$Z$",Z,SW);
label("$a$",X--Z,N);
label("$b$",0.25*X,SE);
label("$c$",O--Z,E);
label("$d$",Y--Z,W);
label("$e$",Y*0.65 + X*0.35,SW);
defaultpen(0.5);
dot(O); dot(X); dot(Z); dot(Y);
</asy>

<math> \mathrm{(A) \ } \pi a^2 \qquad \mathrm{(B) \ } \pi b^2 \qquad \mathrm{(C) \ } \pi c^2 \qquad \mathrm{(D) \ } \pi d^2 \qquad \mathrm{(E) \ } \pi e^2 </math>

[[2004 AMC 10B Problems/Problem 12|Solution]]

== Problem 13 ==

In the United States, coins have the following thicknesses: penny, <math>1.55</math> mm; nickel, <math>1.95</math> mm; dime, <math>1.35</math> mm; quarter, <math>1.75</math> mm. If a stack of these coins is exactly <math>14</math> mm high, how many coins are in the stack?

<math> \mathrm{(A) \ } 7 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } 11 </math>

[[2004 AMC 10B Problems/Problem 13|Solution]]

== Problem 14 ==

A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only <math>\frac{1}{3}</math> of the marbles in the bag are blue. Then yellow marbles are added to the bag until only <math>\frac{1}{5}</math> of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?

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

[[2004 AMC 10B Problems/Problem 14|Solution]]

== Problem 15 ==

Patty has <math>20</math> coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have <math>70</math> cents more. How much are her coins worth?

<math> \textbf{(A)}\ \textdollar 1.15\qquad\textbf{(B)}\ \textdollar 1.20\qquad\textbf{(C)}\ \textdollar 1.25\qquad\textbf{(D)}\ \textdollar 1.30\qquad\textbf{(E)}\ \textdollar 1.35 </math>

[[2004 AMC 10B Problems/Problem 15|Solution]]

== Problem 16 ==

Three circles of radius <math>1</math> are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?

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

[[2004 AMC 10B Problems/Problem 16|Solution]]

== Problem 17 ==

The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?

<math> \mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ } 45 </math>

[[2004 AMC 10B Problems/Problem 17|Solution]]

== Problem 18 ==

In the right triangle <math>\triangle ACE</math>, we have <math>AC=12</math>, <math>CE=16</math>, and <math>EA=20</math>. Points <math>B</math>, <math>D</math>, and <math>F</math> are located on <math>AC</math>, <math>CE</math>, and <math>EA</math>, respectively, so that <math>AB=3</math>, <math>CD=4</math>, and <math>EF=5</math>. What is the ratio of the area of <math>\triangle DBF</math> to that of <math>\triangle ACE</math>?

<math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{9}{25} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{11}{25} \qquad \mathrm{(E) \ } \frac{7}{16} </math>

<asy>
unitsize(0.5cm);
defaultpen(0.8);
pair C=(0,0), A=(0,12), E=(20,0);
draw(A--C--E--cycle);
pair B=A + 3*(C-A)/length(C-A);
pair D=C + 4*(E-C)/length(E-C);
pair F=E + 5*(A-E)/length(A-E);
draw(B--D--F--cycle);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,SW);
label("$D$",D,S);
label("$E$",E,SE);
label("$F$",F,NE);
label("$3$",A--B,W);
label("$9$",C--B,W);
label("$4$",C--D,S);
label("$12$",D--E,S);
label("$5$",E--F,NE);
label("$15$",F--A,NE);
</asy>

[[2004 AMC 10B Problems/Problem 18|Solution]]

== Problem 19 ==

In the sequence <math>2001</math>, <math>2002</math>, <math>2003</math>, <math>\ldots</math> , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is <math>2001 + 2002 - 2003 = 2000</math>. What is the
<math>2004^\textrm{th}</math> term in this sequence?

<math> \mathrm{(A) \ } -2004 \qquad \mathrm{(B) \ } -2 \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } 4003 \qquad \mathrm{(E) \ } 6007 </math>

[[2004 AMC 10B Problems/Problem 19|Solution]]

== Problem 20 ==

In <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>BC</math> and <math>AC</math>, respectively. If <math>AD</math> and <math>BE</math> intersect at <math>T</math> so that <math>AT/DT=3</math> and <math>BT/ET=4</math>, what is <math>CD/BD</math>?


<math> \mathrm{(A) \ } \frac{1}{8} \qquad \mathrm{(B) \ } \frac{2}{9} \qquad \mathrm{(C) \ } \frac{3}{10} \qquad \mathrm{(D) \ } \frac{4}{11} \qquad \mathrm{(E) \ } \frac{5}{12} </math>


<asy>
unitsize(1cm);
defaultpen(0.8);
pair A=(0,0), B=5*dir(60), C=5*(1,0), D=B + (11/15)*(C-B), E = A + (11/16)*(C-A);
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
pair T=intersectionpoint(A--D,B--E);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,SE);
label("$D$",D,NE);
label("$E$",E,S);
label("$T$",T,2*WNW);
</asy>


[[2004 AMC 10B Problems/Problem 20|Solution]]

== Problem 21 ==

Let <math>1</math>; <math>4</math>; <math>\ldots</math> and <math>9</math>; <math>16</math>; <math>\ldots</math> be two arithmetic progressions. The set <math>S</math> is the union of the first <math>2004</math> terms of each sequence. How many distinct numbers are in <math>S</math>?

<math> \mathrm{(A) \ } 3722 \qquad \mathrm{(B) \ } 3732 \qquad \mathrm{(C) \ } 3914 \qquad \mathrm{(D) \ } 3924 \qquad \mathrm{(E) \ } 4007 </math>

[[2004 AMC 10B Problems/Problem 21|Solution]]

== Problem 22 ==

A triangle with sides of <math>5, 12,</math> and <math>13</math> has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?

<math> \mathrm{(A) \ } \frac{3\sqrt{5}}{2} \qquad \mathrm{(B) \ } \frac{7}{2} \qquad \mathrm{(C) \ } \sqrt{15} \qquad \mathrm{(D) \ } \frac{\sqrt{65}}{2} \qquad \mathrm{(E) \ } \frac{9}{2} </math>

[[2004 AMC 10B Problems/Problem 22|Solution]]

== Problem 23 ==

Each face of a cube is painted either red or blue, each with probability <math>1/2</math>. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?

<math> \mathrm{(A) \ } \frac{1}{4} \qquad \mathrm{(B) \ } \frac{5}{16} \qquad \mathrm{(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{7}{16} \qquad \mathrm{(E) \ } \frac{1}{2} </math>

[[2004 AMC 10B Problems/Problem 23|Solution]]

== Problem 24 ==

In <math>\bigtriangleup ABC</math> we have <math>AB = 7</math>, <math>AC = 8</math>, and <math>BC = 9</math>. Point <math>D</math> is on the circumscribed circle of the triangle so that <math>AD</math> bisects <math>\angle BAC</math>. What is the value of <math>\frac{AD}{CD}</math>?

<math> \mathrm{(A) \ } \frac{9}{8} \qquad \mathrm{(B) \ } \frac{5}{3} \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } \frac{17}{7} \qquad \mathrm{(E) \ } \frac{5}{2} </math>

[[2004 AMC 10B Problems/Problem 24|Solution]]

== Problem 25 ==

A circle of radius <math>1</math> is internally tangent to two circles of radius <math>2</math> at points <math>A</math> and <math>B</math>, where <math>AB</math> is a diameter of the smaller circle. What is the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles? 

<math> 
\mathrm{(A) \ } \frac{5}{3} \pi - 3\sqrt 2
\qquad 
\mathrm{(B) \ } \frac{5}{3} \pi - 2\sqrt 3
\qquad 
\mathrm{(C) \ } \frac{8}{3} \pi - 3\sqrt 3
\qquad 
\mathrm{(D) \ } \frac{8}{3} \pi - 3\sqrt 2
\qquad 
\mathrm{(E) \ } \frac{8}{3} \pi - 2\sqrt 3
</math>


<asy>
unitsize(1cm);
defaultpen(0.8);

pair O=(0,0), A=(0,1), B=(0,-1);
path bigc1 = Circle(A,2), bigc2 = Circle(B,2), smallc = Circle(O,1);

pair[] P = intersectionpoints(bigc1, bigc2);
filldraw( arc(A,P[0],P[1])--arc(B,P[1],P[0])--cycle, lightgray, black );
draw(bigc1);
draw(bigc2);
unfill(smallc);
draw(smallc);

dot(O); dot(A); dot(B); label("$A$",A,N); label("$B$",B,S);
draw( O--dir(30) );
draw( A--(A+2*dir(30)) );
draw( B--(B+2*dir(210)) );

label("$1$", O--dir(30), N );
label("$2$", A--(A+2*dir(30)), N );
label("$2$", B--(B+2*dir(210)), S );

</asy>

[[2004 AMC 10B Problems/Problem 25|Solution]]

== See also ==
{{AMC10 box|year=2004|ab=B|before=[[2004 AMC 10A Problems]]|after=[[2005 AMC 10A Problems]]}}
* [[AMC 10]]
* [[AMC 10 Problems and Solutions]]
* [[2004 AMC 10B]]
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=28 2004 AMC B Math Jam Transcript]
* [[Mathematics competition resources]]
{{MAA Notice}}