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Difference between revisions of "2004 AMC 12B Problems"

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m (→‎Problem 24)
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== Problem 24 ==
 
== Problem 24 ==
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 +
In <math>\triangle ABC</math>, <math>AB = BC</math>, and <math>\overline{BD}</math> is an [[altitude]]. Point <math>E</math> is on the extension of <math>\overline{AC}</math> such that <math>BE = 10</math>. The values of <math>\tan \angle CBE</math>, <math>\tan \angle DBE</math>, and <math>\tan \angle ABE</math> form a [[geometric progression]], and the values of <math>\cot \angle DBE,</math> <math>\cot \angle CBE,</math> <math>\cot \angle DBC</math> form an [[arithmetic progression]]. What is the area of <math>\triangle ABC</math>?
 +
 +
<center><asy>
 +
size(120);
 +
defaultpen(0.7);
 +
pair A = (0,0), D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (13*2^.5/3,0);
 +
draw(A--D--C--E--B--C--D--B--cycle);
 +
label("\(A\)",A,S);
 +
label("\(B\)",B,N);
 +
label("\(C\)",C,S);
 +
label("\(D\)",D,S);
 +
label("\(E\)",E,S);
 +
</asy></center>
 +
 +
<math>\mathrm{(A)}\ 16
 +
\qquad\mathrm{(B)}\ \frac {50}3
 +
\qquad\mathrm{(C)}\ 10\sqrt{3}
 +
\qquad\mathrm{(D)}\ 8\sqrt{5}
 +
\qquad\mathrm{(E)}\ 18</math>
 +
  
 
[[2004 AMC 12B Problems/Problem 24|Solution]]
 
[[2004 AMC 12B Problems/Problem 24|Solution]]

Revision as of 20:15, 16 September 2008

Problem 1

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 2

In the expression $c\cdot a^b-d$, 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 3

If $x$ and $y$ are positive integers for which $2^x3^y=1296$, what is the value of $x+y$?

$(\mathrm {A}) 8\qquad (\mathrm {B}) 9 \qquad (\mathrm {C}) 10 \qquad (\mathrm {D}) 11 \qquad (\mathrm {E}) 12$

Solution

Problem 4

An integer $x$, with $10\leq x\leq 99$, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of $x$ is a 7?

$(\mathrm {A}) \dfrac{1}{9} \qquad (\mathrm {B}) \dfrac{1}{5} \qquad (\mathrm {C}) dfrac{19}{90} \qquad (\mathrm {D}) \dfrac{2}{9} \qquad (\mathrm {E}) \dfrac{1}{3}$

Solution

Problem 5

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 6

Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the follow- ing 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 7

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 8

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?

$(\mathrm {A}) 5 \qquad (\mathrm {B}) 8 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 10 \qquad (\mathrm {E}) 11$

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, what is the value of $a+b$?

$\mathrm{(A)}\ -2 \qquad\mathrm{(B)}\ -1 \qquad\mathrm{(C)}\ 0 \qquad\mathrm{(D)}\ 1 \qquad\mathrm{(E)}\ 2$


Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

A function $f$ is defined by $f(z) = i\overline{z}$, where $i=\sqrt{-1}$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 2 \qquad\mathrm{(D)}\ 4 \qquad\mathrm{(E)}\ 8$


Solution

Problem 17

For some real numbers $a$ and $b$, the equation \[8x^3 + 4ax^2 + 2bx + a = 0\] has three distinct positive roots. If the sum of the base-$2$ logarithms of the roots is $5$, what is the value of $a$?

$\mathrm{(A)}\ -256 \qquad\mathrm{(B)}\ -64 \qquad\mathrm{(C)}\ -8 \qquad\mathrm{(D)}\ 64 \qquad\mathrm{(E)}\ 256$


Solution

Problem 18

Solution

Problem 19

A truncated cone has horizontal bases with radii $18$ and $2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?

$\mathrm{(A)}\ 6 \qquad\mathrm{(B)}\ 4\sqrt{5} \qquad\mathrm{(C)}\ 9 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 6\sqrt{3}$


Solution

Problem 20

Solution

Problem 21

The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\frac yx$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?

$\mathrm{(A)}\ 3 \qquad\mathrm{(B)}\ \sqrt{10} \qquad\mathrm{(C)}\ \frac 72 \qquad\mathrm{(D)}\ \frac 92 \qquad\mathrm{(E)}\ 2\sqrt{14}$


Solution

Problem 22

Solution

Problem 23

The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible?

$\mathrm{(A)}\ 250,000 \qquad\mathrm{(B)}\ 250,250 \qquad\mathrm{(C)}\ 250,500 \qquad\mathrm{(D)}\ 250,750 \qquad\mathrm{(E)}\ 251,000$

Solution

Problem 24

In $\triangle ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE = 10$. The values of $\tan \angle CBE$, $\tan \angle DBE$, and $\tan \angle ABE$ form a geometric progression, and the values of $\cot \angle DBE,$ $\cot \angle CBE,$ $\cot \angle DBC$ form an arithmetic progression. What is the area of $\triangle ABC$?

[asy] size(120); defaultpen(0.7); pair A = (0,0), D = (5*2^.5/3,0), C = (10*2^.5/3,0), B = (5*2^.5/3,5*2^.5), E = (13*2^.5/3,0); draw(A--D--C--E--B--C--D--B--cycle); label("\(A\)",A,S); label("\(B\)",B,N); label("\(C\)",C,S); label("\(D\)",D,S); label("\(E\)",E,S); [/asy]

$\mathrm{(A)}\ 16 \qquad\mathrm{(B)}\ \frac {50}3 \qquad\mathrm{(C)}\ 10\sqrt{3} \qquad\mathrm{(D)}\ 8\sqrt{5} \qquad\mathrm{(E)}\ 18$


Solution

Problem 25

Solution

See also

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