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Difference between revisions of "2009 AMC 12B Problems"

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== Problem 20 ==
 
== Problem 20 ==
 +
A convex polyhedron <math>Q</math> has vertices <math>V_1,V_2,\ldots,V_n</math>, and <math>100</math> edges. The polyhedron is cut by planes <math>P_1,P_2,\ldots,P_n</math> in such a way that plane <math>P_k</math> cuts only those edges that meet at vertex <math>V_k</math>. In addition, no two planes intersect inside or on <math>Q</math>. The cuts produce <math>n</math> pyramids and a new polyhedron <math>R</math>. How many edges does <math>R</math> have?
 +
 +
<math>\textbf{(A)}\ 200\qquad \textbf{(B)}\ 2n\qquad \textbf{(C)}\ 300\qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 4n</math>
  
 
[[2009 AMC 12B Problems/Problem 20|Solution]]
 
[[2009 AMC 12B Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
 +
Ten women sit in <math>10</math> seats in a line. All of the <math>10</math> get up and then reseat themselves using all <math>10</math> seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
 +
 +
<math>\textbf{(A)}\ 89\qquad \textbf{(B)}\ 90\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 2^{10}\qquad \textbf{(E)}\ 2^2 3^8</math>
  
 
[[2009 AMC 12B Problems/Problem 21|Solution]]
 
[[2009 AMC 12B Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
 +
Parallelogram <math>ABCD</math> has area <math>1,\!000,\!000</math>. Vertex <math>A</math> is at <math>(0,0)</math> and all other vertices are in the first quadrant. Vertices <math>B</math> and <math>D</math> are lattice points on the lines <math>y = x</math> and <math>y = kx</math> for some integer <math>k > 1</math>, respectively. How many such parallelograms are there?
 +
 +
<math>\textbf{(A)}\ 49\qquad \textbf{(B)}\ 720\qquad \textbf{(C)}\ 784\qquad \textbf{(D)}\ 2009\qquad \textbf{(E)}\ 2048</math>
  
 
[[2009 AMC 12B Problems/Problem 22|Solution]]
 
[[2009 AMC 12B Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
A region <math>S</math> in the complex plane is defined by
 +
<cmath>
 +
S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.
 +
</cmath>
 +
A complex number <math>z = x + iy</math> is chosen uniformly at random form <math>S</math>. What is the probability that <math>\left(\frac34 + \frac34i\right)z</math> is also in <math>S</math>?
 +
 +
<math>\textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac23\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac79\qquad \textbf{(E)}\ \frac78</math>
  
 
[[2009 AMC 12B Problems/Problem 23|Solution]]
 
[[2009 AMC 12B Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
For how many values of <math>x</math> in <math>[0,\pi]</math> is <math>\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)</math>?
 +
Note: The functions <math>\sin^{ - 1} = \arcsin</math> and <math>\cos^{ - 1} = \arccos</math> denote inverse trigonometric functions.
 +
 +
<math>\textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7</math>
  
 
[[2009 AMC 12B Problems/Problem 24|Solution]]
 
[[2009 AMC 12B Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== Problem 25 ==
 +
The set <math>G</math> is defined by the points <math>(x,y)</math> with integer coordinates, <math>3\le|x|\le7</math>, <math>3\le|y|\le7</math>. How many squares of side at least <math>6</math> have their four vertices in <math>G</math>?
 +
<asy>
 +
defaultpen(black+0.75bp+fontsize(8pt));
 +
size(5cm);
 +
path p = scale(.15)*unitcircle;
 +
draw((-8,0)--(8.5,0),Arrow(HookHead,1mm));
 +
draw((0,-8)--(0,8.5),Arrow(HookHead,1mm));
 +
int i,j;
 +
for (i=-7;i<8;++i) {
 +
for (j=-7;j<8;++j) {
 +
if (((-7 <= i) && (i <= -3)) || ((3 <= i) &&  (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) &&  (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp);
 +
draw((-3,-.2)--(-3,.2),black+0.5bp);
 +
draw((3,-.2)--(3,.2),black+0.5bp);
 +
draw((7,-.2)--(7,.2),black+0.5bp);
 +
draw((-.2,-7)--(.2,-7),black+0.5bp);
 +
draw((-.2,-3)--(.2,-3),black+0.5bp);
 +
draw((-.2,3)--(.2,3),black+0.5bp);
 +
draw((-.2,7)--(.2,7),black+0.5bp);
 +
label("$-7$",(-7,0),S);
 +
label("$-3$",(-3,0),S);
 +
label("$3$",(3,0),S);
 +
label("$7$",(7,0),S);
 +
label("$-7$",(0,-7),W);
 +
label("$-3$",(0,-3),W);
 +
label("$3$",(0,3),W);
 +
label("$7$",(0,7),W);
 +
</asy><math>\textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225</math>
  
 
[[2009 AMC 12B Problems/Problem 25|Solution]]
 
[[2009 AMC 12B Problems/Problem 25|Solution]]

Revision as of 19:17, 26 February 2009

Problem 1

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have?

$\textbf{(A)}\ 200\qquad \textbf{(B)}\ 2n\qquad \textbf{(C)}\ 300\qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 4n$

Solution

Problem 21

Ten women sit in $10$ seats in a line. All of the $10$ get up and then reseat themselves using all $10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?

$\textbf{(A)}\ 89\qquad \textbf{(B)}\ 90\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 2^{10}\qquad \textbf{(E)}\ 2^2 3^8$

Solution

Problem 22

Parallelogram $ABCD$ has area $1,\!000,\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there?

$\textbf{(A)}\ 49\qquad \textbf{(B)}\ 720\qquad \textbf{(C)}\ 784\qquad \textbf{(D)}\ 2009\qquad \textbf{(E)}\ 2048$

Solution

Problem 23

A region $S$ in the complex plane is defined by \[S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.\] A complex number $z = x + iy$ is chosen uniformly at random form $S$. What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$?

$\textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac23\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac79\qquad \textbf{(E)}\ \frac78$

Solution

Problem 24

For how many values of $x$ in $[0,\pi]$ is $\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)$? Note: The functions $\sin^{ - 1} = \arcsin$ and $\cos^{ - 1} = \arccos$ denote inverse trigonometric functions.

$\textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Solution

Problem 25

The set $G$ is defined by the points $(x,y)$ with integer coordinates, $3\le|x|\le7$, $3\le|y|\le7$. How many squares of side at least $6$ have their four vertices in $G$? [asy] defaultpen(black+0.75bp+fontsize(8pt)); size(5cm); path p = scale(.15)*unitcircle; draw((-8,0)--(8.5,0),Arrow(HookHead,1mm)); draw((0,-8)--(0,8.5),Arrow(HookHead,1mm)); int i,j; for (i=-7;i<8;++i) { for (j=-7;j<8;++j) { if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp); draw((-3,-.2)--(-3,.2),black+0.5bp); draw((3,-.2)--(3,.2),black+0.5bp); draw((7,-.2)--(7,.2),black+0.5bp); draw((-.2,-7)--(.2,-7),black+0.5bp); draw((-.2,-3)--(.2,-3),black+0.5bp); draw((-.2,3)--(.2,3),black+0.5bp); draw((-.2,7)--(.2,7),black+0.5bp); label("$-7$",(-7,0),S); label("$-3$",(-3,0),S); label("$3$",(3,0),S); label("$7$",(7,0),S); label("$-7$",(0,-7),W); label("$-3$",(0,-3),W); label("$3$",(0,3),W); label("$7$",(0,7),W); [/asy]$\textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225$

Solution

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