Difference between revisions of "2004 AMC 10B Problems"

Revision as of 15:25, 4 June 2011

Problem 1

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

$\mathrm{(A) \ } 297 \qquad \mathrm{(B) \ } 330\qquad \mathrm{(C) \ } 363\qquad \mathrm{(D) \ } 396\qquad \mathrm{(E) \ } 726$

Solution

Problem 2

How many two-digit positive integers have at least one $7$ as a digit?

$\mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 19 \qquad \mathrm{(D) \ } 20\qquad \mathrm{(E) \ } 30$

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 $48$ free throws. How many free throws did she make at the first practice?

$\mathrm{(A) \ } 3 \qquad \mathrm{(B) \ } 6 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ } 15$

Solution

Problem 4

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

$\mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 144\qquad \mathrm{(E) \ } 720$

Solution

Problem 5

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

$\mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 8 \qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 10$

Solution

Problem 6

Which of the following numbers is a perfect square?

$\mathrm{(A) \ } 98! \cdot 99! \qquad \mathrm{(B) \ } 98! \cdot 100! \qquad \mathrm{(C) \ } 99! \cdot 100! \qquad \mathrm{(D) \ } 99! \cdot 101! \qquad \mathrm{(E) \ } 100! \cdot 101!$

Solution

Problem 7

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

$\mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 9$

Solution

Problem 8

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

$\mathrm{(A) \ } 13 \qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ } 16\qquad \mathrm{(E) \ } 17$

Solution

Problem 9

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

$\mathrm{(A) \ } 200+25\pi \qquad \mathrm{(B) \ } 100+75\pi \qquad \mathrm{(C) \ } 75+100\pi \qquad \mathrm{(D) \ } 100+100\pi \qquad \mathrm{(E) \ } 100+125\pi$

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 $100$ cans, how many rows does it contain?

$\mathrm{(A) \ } 5 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 411$

Solution

Problem 11

Two eight-sided dice each have faces numbered $1$ through $8$ . 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?

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

Solution

Problem 12

An annulus is the region between two concentric circles. The concentric circles in the ﬁgure have radii $b$ and $c$ , with $b>c$ . Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$ , and let $OY$ be the radius of the larger circle that contains $Z$ . Let $a=XZ$ , $d=YZ$ , and $e=XY$ . 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$",Y*0.65 + X*0.35,SW); defaultpen(0.5); dot(O); dot(X); dot(Z); dot(Y); [/asy]$

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

Solution

Problem 13

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

$\mathrm{(A) \ } 7 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 9 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } 11$

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 $\frac{1}{3}$ of the marbles in the bag are blue. Then yellow marbles are added to the bag until only $\frac{1}{5}$ 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?

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

Solution

Problem 15

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

$\mathrm{(A) \ } $$ 1.15 \qquad \mathrm{(B) \ } $ $1.20 \qquad \mathrm{(C) \ } $$ 1.25 \qquad \mathrm{(D) \ } $ $1.30 \qquad \mathrm{(E) \ } $$ 1.35 $[[2004 AMC 10B Problems/Problem 15|Solution]]

== Problem 16 ==

Three circles of radius$ (Error compiling LaTeX. Unknown error_msg)1 $are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?$ \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{2}}{3} \qquad \mathrm{(E) \ } \frac{3 + \sqrt{3}}{2} $[[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?$ (Error compiling LaTeX. Unknown error_msg) \mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ } 45 $[[2004 AMC 10B Problems/Problem 17|Solution]]

== Problem 18 ==

In the right triangle$ (Error compiling LaTeX. Unknown error_msg)\triangle ACE $, we have$ AC=12 $,$ CE=16 $, and$ EA=20 $. Points$ B $,$ D $, and$ F $are located on$ AC $,$ CE $, and$ EA $, respectively, so that$ AB=3 $,$ CD=4 $, and$ EF=5 $. What is the ratio of the area of$ \triangle DBF $to that of$ \triangle ACE $?$ \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} $

$[asy] unitsize(0.5cm); defaultpen(0.8); pair C=(0,0), A=(0,12), E=(20,0); draw(A--C--E--cycle); pair B=A + 3*(C-A)/length(C-A); pair D=C + 4*(E-C)/length(E-C); pair F=E + 5*(A-E)/length(A-E); draw(B--D--F--cycle); label("$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]$

Solution

Problem 19

In the sequence $2001$ , $2002$ , $2003$ , $\ldots$ , 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 $2001 + 2002 - 2003 = 2000$ . What is the $2004^\textrm{th}$ term in this sequence?

$\mathrm{(A) \ } -2004 \qquad \mathrm{(B) \ } -2 \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } 4003 \qquad \mathrm{(E) \ } 6007$

Solution

Problem 20

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

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

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

Solution

Problem 21

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

$\mathrm{(A) \ } 3722 \qquad \mathrm{(B) \ } 3732 \qquad \mathrm{(C) \ } 3914 \qquad \mathrm{(D) \ } 3924 \qquad \mathrm{(E) \ } 4007$

Solution

Problem 22

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

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

Solution

Problem 23

Each face of a cube is painted either red or blue, each with probability $1/2$ . 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?

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

Solution

Problem 24

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

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

Solution

Problem 25

A circle of radius $1$ is internally tangent to two circles of radius $2$ at points $A$ and $B$ , where $AB$ 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?

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

$[asy] unitsize(1cm); defaultpen(0.8); pair O=(0,0), A=(0,1), B=(0,-1); path bigc1 = Circle(A,2), bigc2 = Circle(B,2), smallc = Circle(O,1); pair[] P = intersectionpoints(bigc1, bigc2); filldraw( arc(A,P[0],P[1])--arc(B,P[1],P[0])--cycle, lightgray, black ); draw(bigc1); draw(bigc2); unfill(smallc); draw(smallc); dot(O); dot(A); dot(B); label("$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]$

Solution

@@ Line 1: / Line 1: @@
 == Problem 1 ==
-Each row of the Misty Moon Amphitheater has 33 seats. Rows 12 through 22 are reserved for a youth club. How many seats are reserved for this club?
+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>
@@ Line 9: / Line 9: @@
 == Problem 2 ==
-How many two-digit positive integers have at least one 7 as a digit?
+How many two-digit positive integers have at least one <math>7 </math>as a digit?
 <math> \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 19 \qquad \mathrm{(D) \ } 20\qquad \mathrm{(E) \ } 30 </math>
@@ Line 17: / Line 17: @@
 == 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 48 free throws. How many free throws did she make at the first practice?
+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>
@@ Line 25: / Line 25: @@
 == Problem 4 ==
-A standard six-sided die is rolled, and P is the product of the five numbers that are visible. What is the largest number that is certain to divide P?
+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>
@@ Line 57: / Line 57: @@
 == Problem 8 ==
-Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
+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>
@@ Line 65: / Line 65: @@
 == Problem 9 ==
-A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?
+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>
@@ Line 73: / Line 73: @@
 == 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 100 cans, how many rows does it contain?
+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>
@@ Line 81: / Line 81: @@
 == Problem 11 ==
-Two eight-sided dice each have faces numbered 1 through 8. 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?
+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>
@@ Line 138: / Line 138: @@
 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> \mathrm{(A) \ } \</math> <math>1.15 \qquad \mathrm{(B) \ } \</math> <math>1.20 \qquad \mathrm{(C) \ } \</math> <math>1.25 \qquad \mathrm{(D) \ } \</math> <math>1.30 \qquad \mathrm{(E) \ } \</math> <math>1.35 </math>
+<math> \mathrm{(A) \ } &#36; </math>1.15 \qquad \mathrm{(B) \ } &#36; <math>1.20 \qquad \mathrm{(C) \ } &#36; </math>1.25 \qquad \mathrm{(D) \ } &#36; <math>1.30 \qquad \mathrm{(E) \ } &#36; </math>1.35 <math>
 [[2004 AMC 10B Problems/Problem 15|Solution]]
@@ Line 144: / Line 144: @@
 == 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?
+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{2}}{3} \qquad \mathrm{(E) \ } \frac{3 + \sqrt{3}}{2} </math>
+</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{2}}{3} \qquad \mathrm{(E) \ } \frac{3 + \sqrt{3}}{2} <math>
 [[2004 AMC 10B Problems/Problem 16|Solution]]
@@ Line 154: / Line 154: @@
 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>
+</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]]
@@ Line 160: / Line 160: @@
 == 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>?
+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>
+</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} $
 <asy>
@@ Line 235: / Line 235: @@
 == Problem 22 ==
-A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
+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>
@@ Line 243: / Line 243: @@
 == Problem 23 ==
-Each face of a cube is painted either red or blue, each with probability 1/2. 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?
+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>

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Difference between revisions of "2004 AMC 10B Problems"

Revision as of 15:25, 4 June 2011

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also