https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Lg5293&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T14:42:10ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2007_AIME_I_Problems/Problem_6&diff=372932007 AIME I Problems/Problem 62011-03-09T04:13:25Z<p>Lg5293: </p>
<hr />
<div>== Problem ==<br />
A frog is placed at the [[origin]] on the [[number line]], and moves according to the following rule: in a given move, the frog advances to either the closest [[point]] with a greater [[integer]] [[coordinate]] that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A ''move sequence'' is a [[sequence]] of coordinates which correspond to valid moves, beginning with 0, and ending with 39. For example, <math>0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39</math> is a move sequence. How many move sequences are possible for the frog?<br />
<br />
__TOC__<br />
<br />
== Solution ==<br />
=== Solution 1 ===<br />
Let us keep a careful tree of the possible number of paths around every multiple of <math>13</math>.<br />
<br />
From <math>0 \Rightarrow 13</math>, we can end at either <math>12</math> (mult. of 3) or <math>13</math> (mult. of 13). <br />
<br />
*Only <math>1</math> path leads to <math>12</math><br />
**Continuing from <math>12</math>, there is <math>1 \cdot 1 = 1</math> way to continue to <math>24</math><br />
**There are <math>1 \cdot \left(\frac{24-15}{3} + 1\right) = 4</math> ways to reach <math>26</math>.<br />
*There are <math>\frac{12 - 0}{3} + 1 = 5</math> ways to reach <math>13</math>. <br />
** Continuing from <math>13</math>, there are <math>5 \cdot 1 = 5</math> ways to get to <math>24</math><br />
**There are <math>5 \cdot \left(\frac{24-15}{3} + 1 + 1\right) = 25</math> ways (the first 1 to make it inclusive, the second to also jump from <math>13 \Rightarrow 26</math>) to get to <math>26</math>. <br />
<br />
Regrouping, work from <math>24 | 26\Rightarrow 39</math><br />
*There are <math>1 + 5 = 6</math> ways to get to <math>24</math> <br />
** Continuing from <math>24</math>, there are <math>6 \cdot \left(\frac{39 - 27}{3}\right) = 24</math> ways to continue to <math>39</math>.<br />
*There are <math>4 + 25 = 29</math> ways to reach <math>26</math>. <br />
** Continuing from <math>26</math>, there are <math>29 \cdot \left(\frac{39-27}{3} + 1\right) = 145</math> (note that the 1 is not to inclusive, but to count <math>26 \Rightarrow 39</math>). <br />
<br />
In total, we get <math>145 + 26 = 169</math>.<br />
<br />
<br><br />
<br />
In summary, we can draw the following tree, where in <math>(x,y)</math>, <math>x</math> represents the current position on the number line, and <math>y</math> represents the number of paths to get there:<br />
<br />
{| class="wikitable"<br />
|-<br />
| width="50%" |<br />
*<math>(12,1)</math><br />
**<math>(24,1)</math><br />
***<math>(39,4)</math><br />
**<math>(26,4)</math><br />
***<math>(39,20)</math><br />
|<br />
*<math>(13,5)</math><br />
**<math>(24,5)</math><br />
***<math>(39,20)</math><br />
**<math>(26,25)</math><br />
***<math>(39,125)</math><br />
|}<br />
<br />
Again, this totals <math>4 + 20 + 20 + 125 = 169</math>.<br />
<br />
=== Solution 2 === <br />
We divide it into 3 stages. The first occurs before the frog moves past 13. The second occurs before it moves past 26, and the last is everything else.<br />
<br />
For the first stage the possible paths are <math>(0,13)</math>, <math>(0,3,13)</math>, <math>(0,3,6,13)</math>, <math>(0,3,6,9,13)</math>, <math>(0,3,6,9,12,13)</math>, and <math>(0,3,6,9,12)</math>. That is a total of 6.<br />
<br />
For the second stage the possible paths are <math>(26)</math>, <math>(15,26)</math>, <math>(15,18,26)</math>, <math>(15,18,21,26)</math>, <math>(15,18,21,24,26)</math>, and <math>(15,18,21,24)</math>. That is a total of 6.<br />
<br />
For the second stage the possible paths are <math>(39)</math>, <math>(27,39)</math>, <math>(27,30,39)</math>, <math>(27,30,33,39)</math>, and <math>(27,30,33,36,39)</math>. That is a total of 5.<br />
<br />
However, we cannot jump from <math>12 \Rightarrow 26</math> (this eliminates 5 paths) or <math>24 \Rightarrow 39</math> (this eliminates 6 paths), so we must subtract <math>6 + 5 = 11</math>.<br />
<br />
The answer is <math>6*6*5 - 11=169</math><br />
<br />
=== Solution 3 ===<br />
<br />
Another way would be to use a table representing the number of ways to reach a certain number<br />
<br />
<math>\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}<br />
0 & 3 & 6 & 9 & 12 & 13 & 15 & 18 & 21 & 24 & 26 & 27 & 30 & 33 & 36 \\ <br />
\hline<br />
1 & 1 & 1 & 1 & 1 & 5 & 6 & 6 & 6 & 6 & 29 & 35 & 35 & 35 & 35 \\<br />
\end{tabular}</math><br />
<br />
How we came with each value is to just add in the number of ways that we can reach that number from previous numbers. For example, for <math>26</math>, we can reach it from <math>13, 15, 18, 21, 24</math>, so we add all those values to get the value for <math>26</math>. For <math>27</math>, it is only reachable from <math>24</math> or <math>26</math>, so we have <math>29 + 6 = 35</math>.<br />
<br />
The answer for <math>39</math> can be computed in a similar way to get <math>35 * 4 + 29 = \boxed{169}</math>.<br />
<br />
== See also ==<br />
{{AIME box|year=2007|n=I|num-b=5|num-a=7}}<br />
<br />
[[Category:Intermediate Combinatorics Problems]]</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372702011 AMC 12B Problems2011-03-06T18:15:33Z<p>Lg5293: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The arithmetic mean of two distinct positive integers <math>x</math> and <math>y</math> is a two-digit integer. The geometric mean of <math>x</math> and <math>y</math> is obtained by reversing the digits of the arithmetic mean. What is <math>|x - y|</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math><br />
<br />
[[2011 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
Let <math>T_1</math> be a triangle with sides <math>2011, 2012</math>, and <math>2013</math>. For <math>n \geq 1</math>, if <math>T_n = \Delta ABC</math> and <math>D, E</math>, and <math>F</math> are the points of tangency of the incircle of <math>\Delta ABC</math> to the sides <math>AB, BC</math>, and <math>AC</math>, respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE</math>, and <math>CF</math>, if it exists. What is the perimeter of the last triangle in the sequence <math>\left(T_n\right)</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
A bug travels in the coordinate plane, moving only along the lines that are parallel to the <math>x</math>-axis or <math>y</math>-axis. Let <math>A = (-3, 2)</math> and <math>B = (3, -2)</math>. Consider all possible paths of the bug from <math>A</math> to <math>B</math> of length at most <math>20</math>. How many points with integer coordinates lie on at least one of these paths?<br />
<br />
<math>\textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255</math><br />
<br />
[[2011 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Let <math>P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)</math>. What is the minimum perimeter amont all the <math>8</math>-sided olygons in the complex plane whose vertices are precisely the zeros of <math>P(z)</math>?<br />
<br />
<math>\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2} + 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3} + 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
For every <math>m</math> and <math>k</math> integers with <math>k</math> odd, denote by <math>\left[\frac{m}{k}\right]</math> the integer closest to <math>\frac{m}{k}</math>. For every odd integer <math>k</math>, let <math>P(k)</math> be the probability that<br />
<br />
<cmath> \left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right] </cmath><br />
<br />
for an integer <math>n</math> randomly chosen from the interval <math>1 \leq n \leq 99!</math>. What is the minimum possible value of <math>P(k)</math> over the odd integers <math>k</math> in the interval <math>1 \leq k \leq 99</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{44}{87} \qquad \textbf{(D)}\ \frac{34}{67} \qquad \textbf{(E)}\ \frac{7}{13}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 25|Solution]]</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372692011 AMC 12B Problems2011-03-06T18:15:00Z<p>Lg5293: /* Problem 25 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The arithmetic mean of two distinct positive integers <math>x</math> and <math>y</math> is a two-digit integer. The geometric mean of <math>x</math> and <math>y</math> is obtained by reversing the digits of the arithmetic mean. What is <math>|x - y|</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math><br />
<br />
[[2011 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
Let <math>T_1</math> be a triangle with sides <math>2011, 2012</math>, and <math>2013</math>. For <math>n \geq 1</math>, if <math>T_n = \Delta ABC</math> and <math>D, E</math>, and <math>F</math> are the points of tangency of the incircle of <math>\Delta ABC</math> to the sides <math>AB, BC</math>, and <math>AC</math>, respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE</math>, and <math>CF</math>, if it exists. What is the perimeter of the last triangle in the sequence <math>\left(T_n\right)</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
A bug travels in the coordinate plane, moving only along the lines that are parallel to the <math>x</math>-axis or <math>y</math>-axis. Let <math>A = (-3, 2)</math> and <math>B = (3, -2)</math>. Consider all possible paths of the bug from <math>A</math> to <math>B</math> of length at most <math>20</math>. How many points with integer coordinates lie on at least one of these paths?<br />
<br />
<math>\textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255</math><br />
<br />
[[2011 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Let <math>P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)</math>. What is the minimum perimeter amont all the <math>8</math>-sided olygons in the complex plane whose vertices are precisely the zeros of <math>P(z)</math>?<br />
<br />
<math>\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2} + 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3} + 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==<br />
<br />
For every <math>m</math> and <math>k</math> integers with <math>k</math> odd, denote by <math>\left[\frac{m}{k}\right]</math> the integer closest to <math>\frac{m}{k}</math>. FOr every odd integer <math>k</math>, let <math>P(k)</math> be the probability that<br />
<br />
<cmath> \left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right] </cmath><br />
<br />
for an integer <math>n</math> randomly chosen from the interval <math>1 \leq n \leq 99!</math>. What is the minimum possible value of <math>P(k)</math> over the odd integers <math>k</math> in the interval <math>1 \leq k \leq 99</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{44}{87} \qquad \textbf{(D)}\ \frac{34}{67} \qquad \textbf{(E)}\ \frac{7}{13}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 25|Solution]]</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372682011 AMC 12B Problems2011-03-06T18:10:35Z<p>Lg5293: /* Problem 24 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The arithmetic mean of two distinct positive integers <math>x</math> and <math>y</math> is a two-digit integer. The geometric mean of <math>x</math> and <math>y</math> is obtained by reversing the digits of the arithmetic mean. What is <math>|x - y|</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math><br />
<br />
[[2011 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
Let <math>T_1</math> be a triangle with sides <math>2011, 2012</math>, and <math>2013</math>. For <math>n \geq 1</math>, if <math>T_n = \Delta ABC</math> and <math>D, E</math>, and <math>F</math> are the points of tangency of the incircle of <math>\Delta ABC</math> to the sides <math>AB, BC</math>, and <math>AC</math>, respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE</math>, and <math>CF</math>, if it exists. What is the perimeter of the last triangle in the sequence <math>\left(T_n\right)</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
A bug travels in the coordinate plane, moving only along the lines that are parallel to the <math>x</math>-axis or <math>y</math>-axis. Let <math>A = (-3, 2)</math> and <math>B = (3, -2)</math>. Consider all possible paths of the bug from <math>A</math> to <math>B</math> of length at most <math>20</math>. How many points with integer coordinates lie on at least one of these paths?<br />
<br />
<math>\textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255</math><br />
<br />
[[2011 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
Let <math>P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)</math>. What is the minimum perimeter amont all the <math>8</math>-sided olygons in the complex plane whose vertices are precisely the zeros of <math>P(z)</math>?<br />
<br />
<math>\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2} + 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3} + 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 24|Solution]]<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372672011 AMC 12B Problems2011-03-06T18:08:51Z<p>Lg5293: /* Problem 23 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The arithmetic mean of two distinct positive integers <math>x</math> and <math>y</math> is a two-digit integer. The geometric mean of <math>x</math> and <math>y</math> is obtained by reversing the digits of the arithmetic mean. What is <math>|x - y|</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math><br />
<br />
[[2011 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
Let <math>T_1</math> be a triangle with sides <math>2011, 2012</math>, and <math>2013</math>. For <math>n \geq 1</math>, if <math>T_n = \Delta ABC</math> and <math>D, E</math>, and <math>F</math> are the points of tangency of the incircle of <math>\Delta ABC</math> to the sides <math>AB, BC</math>, and <math>AC</math>, respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE</math>, and <math>CF</math>, if it exists. What is the perimeter of the last triangle in the sequence <math>\left(T_n\right)</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
A bug travels in the coordinate plane, moving only along the lines that are parallel to the <math>x</math>-axis or <math>y</math>-axis. Let <math>A = (-3, 2)</math> and <math>B = (3, -2)</math>. Consider all possible paths of the bug from <math>A</math> to <math>B</math> of length at most <math>20</math>. How many points with integer coordinates lie on at least one of these paths?<br />
<br />
<math>\textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\ 195 \qquad \textbf{(D)}\ 227 \qquad \textbf{(E)}\ 255</math><br />
<br />
[[2011 AMC 12B Problems/Problem 23|Solution]]<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372662011 AMC 12B Problems2011-03-06T18:06:57Z<p>Lg5293: /* Problem 22 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The arithmetic mean of two distinct positive integers <math>x</math> and <math>y</math> is a two-digit integer. The geometric mean of <math>x</math> and <math>y</math> is obtained by reversing the digits of the arithmetic mean. What is <math>|x - y|</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math><br />
<br />
[[2011 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
Let <math>T_1</math> be a triangle with sides <math>2011, 2012</math>, and <math>2013</math>. For <math>n \geq 1</math>, if <math>T_n = \Delta ABC</math> and <math>D, E</math>, and <math>F</math> are the points of tangency of the incircle of <math>\Delta ABC</math> to the sides <math>AB, BC</math>, and <math>AC</math>, respectively, then <math>T_{n+1}</math> is a triangle with side lengths <math>AD, BE</math>, and <math>CF</math>, if it exists. What is the perimeter of the last triangle in the sequence <math>\left(T_n\right)</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 22|Solution]]<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372652011 AMC 12B Problems2011-03-06T18:04:43Z<p>Lg5293: /* Problem 21 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
The arithmetic mean of two distinct positive integers <math>x</math> and <math>y</math> is a two-digit integer. The geometric mean of <math>x</math> and <math>y</math> is obtained by reversing the digits of the arithmetic mean. What is <math>|x - y|</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math><br />
<br />
[[2011 AMC 12B Problems/Problem 21|Solution]]<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372642011 AMC 12B Problems2011-03-06T18:02:43Z<p>Lg5293: /* Problem 20 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
Triangle <math>ABC</math> has <math>AB = 13, BC = 14</math>, and <math>AC = 15</math>. The points <math>D, E</math>, and <math>F</math> are the midpoints of <math>\overline{AB}, \overline{BC}</math>, and <math>\overline{AC}</math> respectively. Let <math>X \not= E</math> be the intersection of the circumcircles of <math>\Delta BDE</math> and <math>\Delta CEF</math>. What is <math>XA + XB + XC</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 20|Solution]]<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372632011 AMC 12B Problems2011-03-06T17:59:20Z<p>Lg5293: /* Problem 19 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
A lattice point in an <math>xy</math>-coordinate system is any point <math>(x, y)</math> where both <math>x</math> and <math>y</math> are integers. The graph of <math>y = mx + 2</math> passes through no lattice point with <math>0 < x \leq 100</math> for all <math>m</math> such that <math>\frac{1}{2} < m < a</math>. What is the maximum possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 19|Solution]]<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372622011 AMC 12B Problems2011-03-06T17:57:44Z<p>Lg5293: /* Problem 18 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?<br />
<br />
<math>\textbf{(A)}\ 5\sqrt{2} - 7 \qquad \textbf{(B)}\ 7 - 4\sqrt{3} \qquad \textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad \textbf{(E)}\ \frac{\sqrt{3}}{9}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 18|Solution]]<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372612011 AMC 12B Problems2011-03-06T17:56:11Z<p>Lg5293: /* Problem 17 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
Let <math>f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))</math>, and <math>h_n(x) = h_1(h_{n-1}(x))</math> for integers <math>n \geq 2</math>. What is the sum of the digits of <math>h_{2011}(1)</math>?<br />
<br />
<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math><br />
<br />
[[2011 AMC 12B Problems/Problem 17|Solution]]<br />
<br />
==Problem 18==<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372602011 AMC 12B Problems2011-03-06T17:54:39Z<p>Lg5293: /* Problem 16 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120^{\circ}</math>. Region <math>R</math> consists of all points inside of the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2</math><br />
<br />
[[2011 AMC 12B Problems/Problem 16|Solution]]<br />
<br />
==Problem 17==<br />
<br />
==Problem 18==<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372592011 AMC 12B Problems2011-03-06T17:53:09Z<p>Lg5293: /* Problem 15 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^{24}-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
==Problem 17==<br />
<br />
==Problem 18==<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372582011 AMC 12B Problems2011-03-06T17:52:57Z<p>Lg5293: /* Problem 15 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
How many positive two-digits inters are factors of <math>2^24-1</math>?<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math><br />
<br />
[[2011 AMC 12B Problems/Problem 15|Solution]]<br />
<br />
==Problem 16==<br />
<br />
==Problem 17==<br />
<br />
==Problem 18==<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372572011 AMC 12B Problems2011-03-06T17:52:18Z<p>Lg5293: /* Problem 14 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
A segment through the focus <math>F</math> of a parabola with vertex <math>V</math> is perpendicular to <math>\overline{FV}</math> and intersects the parabola in points <math>A</math> and <math>B</math>. What is <math>\cos\left(\angle AVB\right)</math>?<br />
<br />
<math>\textbf{(A)}\ -\frac{3\sqrt{5}}{7} \qquad \textbf{(B)}\ -\frac{2\sqrt{5}}{5} \qquad \textbf{(C)}\ -\frac{4}{5} \qquad \textbf{(D)}\ -\frac{3}{5} \qquad \textbf{(E)}\ -\frac{1}{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 14|Solution]]<br />
<br />
==Problem 15==<br />
<br />
==Problem 16==<br />
<br />
==Problem 17==<br />
<br />
==Problem 18==<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372562011 AMC 12B Problems2011-03-06T17:50:45Z<p>Lg5293: /* Problem 13 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
Brian writes down four integers <math>w > x > y > z</math> whose sum is <math>44</math>. The pairwise positive differences of these numbers are <math>1, 3, 4, 5, 6</math> and <math>9</math>. What is the sum of the possible values of <math>w</math>?<br />
<br />
<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math><br />
<br />
[[2011 AMC 12B Problems/Problem 13|Solution]]<br />
<br />
==Problem 14==<br />
<br />
==Problem 15==<br />
<br />
==Problem 16==<br />
<br />
==Problem 17==<br />
<br />
==Problem 18==<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12B_Problems&diff=372552011 AMC 12B Problems2011-03-06T17:49:28Z<p>Lg5293: /* Problem 12 */</p>
<hr />
<div>==Problem 1==<br />
What is <center><math> \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? </math></center><br />
<br />
<br />
<math>\textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Josanna's test scores to date are <math>90</math>, <math>80</math>, <math>70</math>, <math>60</math>, and <math>85</math>. Her goal is to raise her test average at least <math>3</math> points with her next test. What is the minimum test score she would need to accomplish this goal?<br />
<br />
<math>\textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95</math><br />
<br />
[[2011 AMC 12B Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid <math>A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A<B</math>. How many dollars must LeRoy give to Bernardo so that they share the costs equally?<br />
<br />
<math>\textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B</math><br />
<br />
[[2011 AMC 12B Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was 161. What is the correct value of the product of <math>a</math> and <math>b</math>?<br />
<br />
<math>\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224</math><br />
<br />
[[2011 AMC 12B Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>N</math> be the second smallest positive integer that is divisible by every positive integer less than <math>7</math>. What is the sum of the digits of <math>N</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math><br />
<br />
[[2011 AMC 12B Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Two tangents to a circle are drawn from a point <math>A</math>. The points of contact <math>B</math> and <math>C</math> divide the circle into arcs with lengths in the ratio <math>2 : 3</math>. What is the degree measure of <math>\angle{BAC}</math>?<br />
<br />
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 60</math><br />
<br />
[[2011 AMC 12B Problems/Problem 6|Solution]]<br />
<br />
==Problem 7==<br />
Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>?<br />
<br />
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 7|Solution]]<br />
<br />
==Problem 8==<br />
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width <math>6</math> meters, and it takes her <math>36</math> seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?<br />
<br />
<math>\textbf{(A)}\ \frac{\pi}{3} \qquad \textbf{(B)}\ \frac{2\pi}{3} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{4\pi}{3} \qquad \textbf{(E)}\ \frac{5\pi}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 8|Solution]]<br />
<br />
==Problem 9==<br />
Two real numbers are selected independently and at random from the interval <math>[-20,10]</math>. What is the probability that the product of those numbers is greater than zero?<br />
<br />
<math>\textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{4}{9} \qquad \textbf{(D)}\ \frac{5}{9} \qquad \textbf{(E)}\ \frac{2}{3}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 9|Solution]]<br />
<br />
==Problem 10==<br />
Rectangle <math>ABCD</math> has <math>AB=6</math> and <math>BC=3</math>. Point <math>M</math> is chosen on side <math>AB</math> so that <math>\angle AMD=\angle CMD</math>. What is the degree measure of <math>\angle AMD</math>?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math><br />
<br />
[[2011 AMC 12B Problems/Problem 10|Solution]]<br />
<br />
==Problem 11==<br />
A frog located at <math>(x,y)</math>, with both <math>x</math> and <math>y</math> integers, makes successive jumps of length <math>5</math> and always lands on points with integer coordinates. Suppose that the frog starts at <math>(0,0)</math> and ends at <math>(1,0)</math>. What is the smallest possible number of jumps the frog makes?<br />
<br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
<br />
[[2011 AMC 12B Problems/Problem 11|Solution]]<br />
<br />
==Problem 12==<br />
<br />
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br />
<br />
[Needs picture]<br />
<br />
<math>\textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad \textbf{(E)}\ 2 - \sqrt{2}</math><br />
<br />
[[2011 AMC 12B Problems/Problem 12|Solution]]<br />
<br />
==Problem 13==<br />
<br />
==Problem 14==<br />
<br />
==Problem 15==<br />
<br />
==Problem 16==<br />
<br />
==Problem 17==<br />
<br />
==Problem 18==<br />
<br />
==Problem 19==<br />
<br />
==Problem 20==<br />
<br />
==Problem 21==<br />
<br />
==Problem 22==<br />
<br />
==Problem 23==<br />
<br />
==Problem 24==<br />
<br />
==Problem 25==</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12A_Problems/Problem_11&diff=371092011 AMC 12A Problems/Problem 112011-02-25T02:47:53Z<p>Lg5293: </p>
<hr />
<div>== Problem ==<br />
Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math>C</math> has a point of tangency with the midpoint of <math>\overline{AB}.</math> What is the area inside circle <math>C</math> but outside circle <math>A</math> and circle <math>B?</math><br />
<br />
<math><br />
\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad<br />
\textbf{(B)}\ \frac{\pi}{2} \qquad<br />
\textbf{(C)}\ 2 \qquad<br />
\textbf{(D)}\ \frac{3\pi}{4} \qquad<br />
\textbf{(E)}\ 1+\frac{\pi}{2}} </math><br />
<br />
== Solution ==<br />
<br />
<asy><br />
unitsize(1.1cm);<br />
defaultpen(linewidth(.8pt));<br />
dotfactor=4;<br />
<br />
pair A=(0,0), B=(2,0), C=(1,-1);<br />
pair M=(1,0);<br />
pair D=(2,-1);<br />
dot (A);<br />
dot (B);<br />
dot (C);<br />
dot (D);<br />
dot (M);<br />
<br />
draw(Circle(A,1));<br />
draw(Circle(B,1));<br />
draw(Circle(C,1));<br />
<br />
draw(A--B);<br />
draw(M--D);<br />
draw(D--B);<br />
<br />
label("$A$",A,W);<br />
label("$B$",B,E);<br />
label("$C$",C,W);<br />
label("$M$",M,NE);<br />
label("$D$",D,SE);<br />
</asy><br />
<br />
The requested area is the area of <math>C</math> minus the area shared between circles <math>A</math>, <math>B</math> and <math>C</math>.<br />
<br />
Let <math>M</math> be the midpoint of <math>\overline{AB}</math> and <math>D</math> be the other intersection of circles <math>C</math> and <math>B</math>.<br />
<br />
Then area shared between <math>C</math>, <math>A</math> and <math>B</math> is <math>4</math> of the regions between arc <math>\widehat {MD}</math> and line <math>\overline{MD}</math>, which is (considering the arc on circle <math>B</math>) a quarter of the circle <math>B</math> minus <math>\triangle MDB</math>:<br />
<br />
<math>\frac{\pi r^2}{4}-\frac{bh}{2}</math><br />
<br />
<math>b = h = r = 1</math><br />
<br />
(We can assume this because <math>\angle DBM</math> is 90 degrees, since <math>CDBM</math> is a square, due the application of the tangent chord theorem at point <math>M</math>)<br />
<br />
So the area of the small region is<br />
<br />
<math>\frac{\pi}{4}-\frac{1}{2}</math><br />
<br />
The requested area is area of circle <math>C</math> minus 4 of this area:<br />
<br />
<math>\pi 1^2 - 4(\frac{\pi}{4}-\frac{1}{2})<br />
= \pi - \pi + 2<br />
= 2</math><br />
<br />
<math>\boxed{\textbf{C}}</math>.<br />
<br />
== Solution 2 ==<br />
<asy><br />
unitsize(1.1cm);<br />
defaultpen(linewidth(.8pt));<br />
dotfactor=4;<br />
<br />
pair A=(0,0), B=(2,0), C=(1,1);<br />
pair D=(2,1);<br />
pair E=(0,1);<br />
pair F = (1, 2);<br />
pair M = (1, 0);<br />
dot (A);<br />
dot (B);<br />
dot (C);<br />
dot (D);<br />
dot (E);<br />
dot (F);<br />
dot (M);<br />
<br />
draw(Circle(A,1));<br />
draw(Circle(B,1));<br />
draw(Circle(C,1));<br />
<br />
draw (D--F--E--M--D);<br />
<br />
label("$A$",A,W);<br />
label("$B$",B,E);<br />
label("$C$",C,W);<br />
label("$M$",M,NE);<br />
label("$D$",D,E);<br />
label("$E$",E,W);<br />
label("$F$",F,N);<br />
</asy><br />
<br />
We can move the area above the part of the circle above the segment <math>EF</math> down, and similarly for the other side. Then, we have a square, whose diagonal is <math>2</math>, so the area is then just <math>\left(\frac{2}{\sqrt{2}}\right)^2 = 2</math>.<br />
<br />
== See also ==<br />
{{AMC12 box|year=2011|num-b=10|num-a=12|ab=A}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12A_Problems/Problem_11&diff=371082011 AMC 12A Problems/Problem 112011-02-25T02:45:58Z<p>Lg5293: </p>
<hr />
<div>== Problem ==<br />
Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math>C</math> has a point of tangency with the midpoint of <math>\overline{AB}.</math> What is the area inside circle <math>C</math> but outside circle <math>A</math> and circle <math>B?</math><br />
<br />
<math><br />
\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad<br />
\textbf{(B)}\ \frac{\pi}{2} \qquad<br />
\textbf{(C)}\ 2 \qquad<br />
\textbf{(D)}\ \frac{3\pi}{4} \qquad<br />
\textbf{(E)}\ 1+\frac{\pi}{2}} </math><br />
<br />
== Solution ==<br />
<br />
<asy><br />
unitsize(1.1cm);<br />
defaultpen(linewidth(.8pt));<br />
dotfactor=4;<br />
<br />
pair A=(0,0), B=(2,0), C=(1,-1);<br />
pair M=(1,0);<br />
pair D=(2,-1);<br />
dot (A);<br />
dot (B);<br />
dot (C);<br />
dot (D);<br />
dot (M);<br />
<br />
draw(Circle(A,1));<br />
draw(Circle(B,1));<br />
draw(Circle(C,1));<br />
<br />
draw(A--B);<br />
draw(M--D);<br />
draw(D--B);<br />
<br />
label("$A$",A,W);<br />
label("$B$",B,E);<br />
label("$C$",C,W);<br />
label("$M$",M,NE);<br />
label("$D$",D,SE);<br />
</asy><br />
<br />
The requested area is the area of <math>C</math> minus the area shared between circles <math>A</math>, <math>B</math> and <math>C</math>.<br />
<br />
Let <math>M</math> be the midpoint of <math>\overline{AB}</math> and <math>D</math> be the other intersection of circles <math>C</math> and <math>B</math>.<br />
<br />
Then area shared between <math>C</math>, <math>A</math> and <math>B</math> is <math>4</math> of the regions between arc <math>\widehat {MD}</math> and line <math>\overline{MD}</math>, which is (considering the arc on circle <math>B</math>) a quarter of the circle <math>B</math> minus <math>\triangle MDB</math>:<br />
<br />
<math>\frac{\pi r^2}{4}-\frac{bh}{2}</math><br />
<br />
<math>b = h = r = 1</math><br />
<br />
(We can assume this because <math>\angle DBM</math> is 90 degrees, since <math>CDBM</math> is a square, due the application of the tangent chord theorem at point <math>M</math>)<br />
<br />
So the area of the small region is<br />
<br />
<math>\frac{\pi}{4}-\frac{1}{2}</math><br />
<br />
The requested area is area of circle <math>C</math> minus 4 of this area:<br />
<br />
<math>\pi 1^2 - 4(\frac{\pi}{4}-\frac{1}{2})<br />
= \pi - \pi + 2<br />
= 2</math><br />
<br />
<math>\boxed{\textbf{C}}</math>.<br />
<br />
== Solution 2 ==<br />
<asy><br />
unitsize(1.1cm);<br />
defaultpen(linewidth(.8pt));<br />
dotfactor=4;<br />
<br />
pair A=(0,0), B=(2,0), C=(1,1);<br />
pair D=(2,1);<br />
pair E=(0,1);<br />
pair F = (1, 2);<br />
pair M = (1, 0);<br />
dot (A);<br />
dot (B);<br />
dot (C);<br />
dot (D);<br />
dot (E);<br />
dot (F);<br />
dot (M);<br />
<br />
draw(Circle(A,1));<br />
draw(Circle(B,1));<br />
draw(Circle(C,1));<br />
<br />
draw (D--F--E--M--D);<br />
<br />
label("$A$",A,W);<br />
label("$B$",B,E);<br />
label("$C$",C,W);<br />
label("$M$",M,NE);<br />
label("$D$",D,E);<br />
label("$E$",E,W);<br />
label("$F$",F,N);<br />
</asy><br />
<br />
Instead, we can move the area above the region<br />
<br />
== See also ==<br />
{{AMC12 box|year=2011|num-b=10|num-a=12|ab=A}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_12A_Problems/Problem_11&diff=371072011 AMC 12A Problems/Problem 112011-02-25T02:44:13Z<p>Lg5293: </p>
<hr />
<div>== Problem ==<br />
Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math>C</math> has a point of tangency with the midpoint of <math>\overline{AB}.</math> What is the area inside circle <math>C</math> but outside circle <math>A</math> and circle <math>B?</math><br />
<br />
<math><br />
\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad<br />
\textbf{(B)}\ \frac{\pi}{2} \qquad<br />
\textbf{(C)}\ 2 \qquad<br />
\textbf{(D)}\ \frac{3\pi}{4} \qquad<br />
\textbf{(E)}\ 1+\frac{\pi}{2}} </math><br />
<br />
== Solution ==<br />
<br />
<asy><br />
unitsize(1.1cm);<br />
defaultpen(linewidth(.8pt));<br />
dotfactor=4;<br />
<br />
pair A=(0,0), B=(2,0), C=(1,-1);<br />
pair M=(1,0);<br />
pair D=(2,-1);<br />
dot (A);<br />
dot (B);<br />
dot (C);<br />
dot (D);<br />
dot (M);<br />
<br />
draw(Circle(A,1));<br />
draw(Circle(B,1));<br />
draw(Circle(C,1));<br />
<br />
draw(A--B);<br />
draw(M--D);<br />
draw(D--B);<br />
<br />
label("$A$",A,W);<br />
label("$B$",B,E);<br />
label("$C$",C,W);<br />
label("$M$",M,NE);<br />
label("$D$",D,SE);<br />
</asy><br />
<br />
The requested area is the area of <math>C</math> minus the area shared between circles <math>A</math>, <math>B</math> and <math>C</math>.<br />
<br />
Let <math>M</math> be the midpoint of <math>\overline{AB}</math> and <math>D</math> be the other intersection of circles <math>C</math> and <math>B</math>.<br />
<br />
Then area shared between <math>C</math>, <math>A</math> and <math>B</math> is <math>4</math> of the regions between arc <math>\widehat {MD}</math> and line <math>\overline{MD}</math>, which is (considering the arc on circle <math>B</math>) a quarter of the circle <math>B</math> minus <math>\triangle MDB</math>:<br />
<br />
<math>\frac{\pi r^2}{4}-\frac{bh}{2}</math><br />
<br />
<math>b = h = r = 1</math><br />
<br />
(We can assume this because <math>\angle DBM</math> is 90 degrees, since <math>CDBM</math> is a square, due the application of the tangent chord theorem at point <math>M</math>)<br />
<br />
So the area of the small region is<br />
<br />
<math>\frac{\pi}{4}-\frac{1}{2}</math><br />
<br />
The requested area is area of circle <math>C</math> minus 4 of this area:<br />
<br />
<math>\pi 1^2 - 4(\frac{\pi}{4}-\frac{1}{2})<br />
= \pi - \pi + 2<br />
= 2</math><br />
<br />
<math>\boxed{\textbf{C}}</math>.<br />
<br />
== Solution 2 ==<br />
<asy><br />
unitsize(1.1cm);<br />
defaultpen(linewidth(.8pt));<br />
dotfactor=4;<br />
<br />
pair A=(0,0), B=(2,0), C=(1,1)<br />
pair D=(2,1);<br />
pair E=(0,1);<br />
pair F = (1, 2);<br />
pair M = (1, 0);<br />
dot (A);<br />
dot (B);<br />
dot (C);<br />
dot (D);<br />
dot (E);<br />
dot (F);<br />
dot (M);<br />
<br />
draw(Circle(A,1));<br />
draw(Circle(B,1));<br />
draw(Circle(C,1));<br />
<br />
draw(A--B);<br />
draw(M--E);<br />
draw(E--B);<br />
draw (D--F--E--M);<br />
<br />
label("$A$",A,W);<br />
label("$B$",B,E);<br />
label("$C$",C,W);<br />
label("$M$",M,NE);<br />
label("$D$",D,SE);<br />
label("$E$",E,SE);<br />
label("$F$",F,SE);<br />
</asy><br />
<br />
== See also ==<br />
{{AMC12 box|year=2011|num-b=10|num-a=12|ab=A}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_24&diff=362122010 AMC 12B Problems/Problem 242010-12-25T21:36:51Z<p>Lg5293: </p>
<hr />
<div>== Problem 24 ==<br />
The set of real numbers <math>x</math> for which <br />
<br />
<cmath>\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1</cmath><br />
<br />
is the union of intervals of the form <math>a<x\le b</math>. What is the sum of the lengths of these intervals?<br />
<br />
<math>\textbf{(A)}\ \dfrac{1003}{335} \qquad \textbf{(B)}\ \dfrac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \dfrac{403}{134} \qquad \textbf{(E)}\ \dfrac{202}{67}</math></div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_24&diff=362112010 AMC 12B Problems/Problem 242010-12-25T21:36:22Z<p>Lg5293: </p>
<hr />
<div>== Problem 24 ==<br />
The set of real numbers <math>x</math> for which <br />
<br />
<cmath>\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1</cmath><br />
<br />
is the union of intervals of the form <math>a<x\le b</math>. What is the sum of the lengths of these intervals?<br />
<br />
<math>\textbf{(A)}\ \dfrac{1003}{335} \qquad \textbf{(B)}\ \dfrac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \dfrac{403}{134} \qquad \textbf{(E)}\ \dfrac{202}{67}</math><br />
<br />
== Solution ==<br />
First, we shift the graph so it will be easier to manipulate<br />
<cmath> \dfrac{1}{x - 1} + \dfrac{1}{x} + \dfrac{1}{x+1} \ge 1 </cmath><br />
We can see that this is equivalent to the original problem. We now want to solve this equation.<br />
<br />
<br />
<cmath> \frac{(x)(x + 1) + (x-1)(x+1) + (x)(x-1)}{(x-1)(x)(x+1)} \ge 1 </cmath><br />
<cmath> \frac{(x^2+x+x^2-1+x^2-x)}{x^3-x} \ge 1 </cmath><br />
<cmath> \frac{3x^2-1}{x^3-x}\ge 1 </cmath><br />
We first find where this will lead to equality. We have <math>x^3 - 3x^2 - x + 1 = 0</math>. <br />
Our answer is just the sum of the roots, and using Vieta's, we get that to be <math>\boxed{(C)}</math>.</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_25&diff=353602010 AMC 12B Problems/Problem 252010-07-12T21:32:20Z<p>Lg5293: </p>
<hr />
<div>== Problem 25 ==<br />
For every integer <math>n\ge2</math>, let <math>\text{pow}(n)</math> be the largest power of the largest prime tha divides <math>n</math>. For example <math>\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2</math>. What is the largest integer <math>m</math> such that <math>2010^m</math> divides<br />
<br />
<center><br />
<math>\prod_{n=2}^{5300}\text{pow}(n)</math>?<br />
</center><br />
<br />
<br />
<math>\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78</math></div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_25&diff=353592010 AMC 12B Problems/Problem 252010-07-12T21:31:43Z<p>Lg5293: Created page with 'For every integer <math>n\ge2</math>, let <math>\text{pow}(n)</math> be the largest power of the largest prime tha divides <math>n</math>. For example <math>\text{pow}(144)=\text…'</p>
<hr />
<div>For every integer <math>n\ge2</math>, let <math>\text{pow}(n)</math> be the largest power of the largest prime tha divides <math>n</math>. For example <math>\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2</math>. What is the largest integer <math>m</math> such that <math>2010^m</math> divides<br />
<br />
<center><br />
<math>\prod_{n=2}^{5300}\text{pow}(n)</math>?<br />
</center><br />
<br />
<br />
<math>\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 75 \qquad \textbf{(C)}\ 76 \qquad \textbf{(D)}\ 77 \qquad \textbf{(E)}\ 78</math></div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_24&diff=353582010 AMC 12B Problems/Problem 242010-07-12T21:31:28Z<p>Lg5293: Created page with '== Problem 24 == The set of real numbers <math>x</math> for which <cmath>\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1</cmath> is the union of intervals of the for…'</p>
<hr />
<div>== Problem 24 ==<br />
The set of real numbers <math>x</math> for which <br />
<br />
<cmath>\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1</cmath><br />
<br />
is the union of intervals of the form <math>a<x\le b</math>. What is the sum of the lengths of these intervals?<br />
<br />
<math>\textbf{(A)}\ \dfrac{1003}{335} \qquad \textbf{(B)}\ \dfrac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \dfrac{403}{134} \qquad \textbf{(E)}\ \dfrac{202}{67}</math></div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_23&diff=353572010 AMC 12B Problems/Problem 232010-07-12T21:30:50Z<p>Lg5293: Created page with ' == Problem 23 == Monic quadratic polynomial <math>P(x)</math> and <math>Q(x)</math> have the property that <math>P(Q(x))</math> has zeros at <math>x=-23, -21, -17,</math> and <m…'</p>
<hr />
<div><br />
== Problem 23 ==<br />
Monic quadratic polynomial <math>P(x)</math> and <math>Q(x)</math> have the property that <math>P(Q(x))</math> has zeros at <math>x=-23, -21, -17,</math> and <math>-15</math>, and <math>Q(P(x))</math> has zeros at <math>x=-59,-57,-51</math> and <math>-49</math>. What is the sum of the minimum values of <math>P(x)</math> and <math>Q(x)</math>? <br />
<br />
<math>\textbf{(A)}\ -100 \qquad \textbf{(B)}\ -82 \qquad \textbf{(C)}\ -73 \qquad \textbf{(D)}\ -64 \qquad \textbf{(E)}\ 0</math></div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_1&diff=353562010 AMC 12B Problems/Problem 12010-07-12T21:24:41Z<p>Lg5293: </p>
<hr />
<div>== Problem 1 ==<br />
Makarla attended two meetings during her <math>9</math>-hour work day. The first meeting took <math>45</math> minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?<br />
<br />
<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35</math><br />
<br />
== Solution ==<br />
The total number of minutes in here <math>9</math>-hour work day is <math>9 \times 60 = 540</math>.<br />
The total amount of time spend in meetings in minutes is <math>45 + 45 \times 2 = 135</math>.<br />
The answer is then <math>\frac{135}{540} = .25</math> <math>\LongRightArrow</math> <math>(C)</math><br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=1|num-a=2|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_16&diff=353552010 AMC 12B Problems/Problem 162010-07-12T21:24:15Z<p>Lg5293: </p>
<hr />
<div>== Problem 16 ==<br />
Positive integers <math>a</math>, <math>b</math>, and <math>c</math> are randomly and independently selected with replacement from the set <math>\{1, 2, 3,\dots, 2010\}</math>. What is the probability that <math>abc + ab + a</math> is divisible by <math>3</math>?<br />
<br />
<math>\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=15|num-a=17|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_17&diff=353542010 AMC 12B Problems/Problem 172010-07-12T21:24:02Z<p>Lg5293: </p>
<hr />
<div>== Problem 17 ==<br />
The entries in a <math>3 \times 3</math> array include all the digits from <math>1</math> through <math>9</math>, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?<br />
<br />
<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=16|num-a=18|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_18&diff=353532010 AMC 12B Problems/Problem 182010-07-12T21:23:38Z<p>Lg5293: </p>
<hr />
<div>== Problem 18 ==<br />
A frog makes <math>3</math> jumps, each exactly <math>1</math> meter long. The directions of the jumps are chosen independenly at random. What is the probability that the frog's final position is no more than <math>1</math> meter from its starting position?<br />
<br />
<math>\textbf{(A)}\ \dfrac{1}{6} \qquad \textbf{(B)}\ \dfrac{1}{5} \qquad \textbf{(C)}\ \dfrac{1}{4} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{1}{2}</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=17|num-a=19|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_19&diff=353522010 AMC 12B Problems/Problem 192010-07-12T21:23:22Z<p>Lg5293: </p>
<hr />
<div>== Problem 19 ==<br />
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than <math>100</math> points. What was the total number of points scored by the two teams in the first half?<br />
<br />
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=18|num-a=20|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_20&diff=353512010 AMC 12B Problems/Problem 202010-07-12T21:22:54Z<p>Lg5293: </p>
<hr />
<div>== Problem 20 ==<br />
A geometric sequence <math>(a_n)</math> has <math>a_1=\sin x</math>, <math>a_2=\cos x</math>, and <math>a_3= \tan x</math> for some real number <math>x</math>. For what value of <math>n</math> does <math>a_n=1+\cos x</math>?<br />
<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=19|num-a=21|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_21&diff=353502010 AMC 12B Problems/Problem 212010-07-12T21:22:36Z<p>Lg5293: </p>
<hr />
<div>== Problem 21 ==<br />
Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that<br />
<br />
<center><br />
<math>P(1) = P(3) = P(5) = P(7) = a</math>, and<br/><br />
<math>P(2) = P(4) = P(6) = P(8) = -a</math>.<br />
</center><br />
<br />
What is the smallest possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=20|num-a=22|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_22&diff=353492010 AMC 12B Problems/Problem 222010-07-12T21:22:17Z<p>Lg5293: Created page with '== Problem 22 == Let <math>ABCD</math> be a cyclic quadralateral. The side lengths of <math>ABCD</math> are distinct integers less than <math>15</math> such that <math>BC\cdot CD…'</p>
<hr />
<div>== Problem 22 ==<br />
Let <math>ABCD</math> be a cyclic quadralateral. The side lengths of <math>ABCD</math> are distinct integers less than <math>15</math> such that <math>BC\cdot CD=AB\cdot DA</math>. What is the largest possible value of <math>BD</math>?<br />
<br />
<math>\textbf{(A)}\ \sqrt{\dfrac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)}\ \sqrt{\dfrac{389}{2}} \qquad \textbf{(D)}\ \sqrt{\dfrac{425}{2}} \qquad \textbf{(E)}\ \sqrt{\dfrac{533}{2}}</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=21|num-a=23|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_21&diff=353482010 AMC 12B Problems/Problem 212010-07-12T21:21:46Z<p>Lg5293: Created page with '== Problem 21 == Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that <center> <math>P(1) = P(3) = P(5) = P(7) = a</math>, and<b…'</p>
<hr />
<div>== Problem 21 ==<br />
Let <math>a > 0</math>, and let <math>P(x)</math> be a polynomial with integer coefficients such that<br />
<br />
<center><br />
<math>P(1) = P(3) = P(5) = P(7) = a</math>, and<br/><br />
<math>P(2) = P(4) = P(6) = P(8) = -a</math>.<br />
</center><br />
<br />
What is the smallest possible value of <math>a</math>?<br />
<br />
<math>\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_20&diff=353472010 AMC 12B Problems/Problem 202010-07-12T21:21:27Z<p>Lg5293: Created page with '== Problem 20 == A geometric sequence <math>(a_n)</math> has <math>a_1=\sin x</math>, <math>a_2=\cos x</math>, and <math>a_3= \tan x</math> for some real number <math>x</math>. F…'</p>
<hr />
<div>== Problem 20 ==<br />
A geometric sequence <math>(a_n)</math> has <math>a_1=\sin x</math>, <math>a_2=\cos x</math>, and <math>a_3= \tan x</math> for some real number <math>x</math>. For what value of <math>n</math> does <math>a_n=1+\cos x</math>?<br />
<br />
<br />
<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_19&diff=353462010 AMC 12B Problems/Problem 192010-07-12T21:21:11Z<p>Lg5293: Created page with '== Problem 19 == A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of th…'</p>
<hr />
<div>== Problem 19 ==<br />
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than <math>100</math> points. What was the total number of points scored by the two teams in the first half?<br />
<br />
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_18&diff=353452010 AMC 12B Problems/Problem 182010-07-12T21:20:28Z<p>Lg5293: Created page with '== Problem 18 == A frog makes <math>3</math> jumps, each exactly <math>1</math> meter long. The directions of the jumps are chosen independenly at random. What is the probability…'</p>
<hr />
<div>== Problem 18 ==<br />
A frog makes <math>3</math> jumps, each exactly <math>1</math> meter long. The directions of the jumps are chosen independenly at random. What is the probability that the frog's final position is no more than <math>1</math> meter from its starting position?<br />
<br />
<math>\textbf{(A)}\ \dfrac{1}{6} \qquad \textbf{(B)}\ \dfrac{1}{5} \qquad \textbf{(C)}\ \dfrac{1}{4} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{1}{2}</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_17&diff=353442010 AMC 12B Problems/Problem 172010-07-12T21:19:58Z<p>Lg5293: Created page with '== Problem 17 == The entries in a <math>3 \times 3</math> array include all the digits from <math>1</math> through <math>9</math>, arranged so that the entries in every row and c…'</p>
<hr />
<div>== Problem 17 ==<br />
The entries in a <math>3 \times 3</math> array include all the digits from <math>1</math> through <math>9</math>, arranged so that the entries in every row and column are in increasing order. How many such arrays are there?<br />
<br />
<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_16&diff=353432010 AMC 12B Problems/Problem 162010-07-12T21:19:40Z<p>Lg5293: Created page with '== Problem 16 == Positive integers <math>a</math>, <math>b</math>, and <math>c</math> are randomly and independently selected with replacement from the set <math>\{1, 2, 3,\dots,…'</p>
<hr />
<div>== Problem 16 ==<br />
Positive integers <math>a</math>, <math>b</math>, and <math>c</math> are randomly and independently selected with replacement from the set <math>\{1, 2, 3,\dots, 2010\}</math>. What is the probability that <math>abc + ab + a</math> is divisible by <math>3</math>?<br />
<br />
<math>\textbf{(A)}\ \dfrac{1}{3} \qquad \textbf{(B)}\ \dfrac{29}{81} \qquad \textbf{(C)}\ \dfrac{31}{81} \qquad \textbf{(D)}\ \dfrac{11}{27} \qquad \textbf{(E)}\ \dfrac{13}{27}</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_15&diff=353422010 AMC 12B Problems/Problem 152010-07-12T21:19:22Z<p>Lg5293: Created page with '== Problem 15 == For how many ordered triples <math>(x,y,z)</math> of nonnegative integers less than <math>20</math> are there exactly two distinct elements in the set <math>\{i^…'</p>
<hr />
<div>== Problem 15 ==<br />
For how many ordered triples <math>(x,y,z)</math> of nonnegative integers less than <math>20</math> are there exactly two distinct elements in the set <math>\{i^x, (1+i)^y, z\}</math>, where <math>i=\sqrt{-1}</math>?<br />
<br />
<math>\textbf{(A)}\ 149 \qquad \textbf{(B)}\ 205 \qquad \textbf{(C)}\ 215 \qquad \textbf{(D)}\ 225 \qquad \textbf{(E)}\ 235</math><br />
<br />
== Solution ==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_14&diff=353412010 AMC 12B Problems/Problem 142010-07-12T21:18:59Z<p>Lg5293: Created page with '== Problem 14 == Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> be postive integers with <math>a+b+c+d+e=2010</math> and let <math>M</math…'</p>
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<div>== Problem 14 ==<br />
Let <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math>, and <math>e</math> be postive integers with <math>a+b+c+d+e=2010</math> and let <math>M</math> be the largest of the sum <math>a+b</math>, <math>b+c</math>, <math>c+d</math> and <math>d+e</math>. What is the smallest possible value of <math>M</math>?<br />
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<math>\textbf{(A)}\ 670 \qquad \textbf{(B)}\ 671 \qquad \textbf{(C)}\ 802 \qquad \textbf{(D)}\ 803 \qquad \textbf{(E)}\ 804</math><br />
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== Solution ==<br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_11&diff=353402010 AMC 12B Problems/Problem 112010-07-12T21:18:35Z<p>Lg5293: Created page with '== Problem 11 == A palindrome between <math>1000</math> and <math>10,000</math> is chosen at random. What is the probability that it is divisible by <math>7</math>? <math>\textb…'</p>
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<div>== Problem 11 ==<br />
A palindrome between <math>1000</math> and <math>10,000</math> is chosen at random. What is the probability that it is divisible by <math>7</math>?<br />
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<math>\textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6} \qquad \textbf{(E)}\ \dfrac{1}{5}</math><br />
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== Solution ==<br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_10&diff=353392010 AMC 12B Problems/Problem 102010-07-12T21:18:16Z<p>Lg5293: Created page with '== Problem 10 == The average of the numbers <math>1, 2, 3,\cdots, 98, 99,</math> and <math>x</math> is <math>100x</math>. What is <math>x</math>? <math>\textbf{(A)}\ \dfrac{49}{…'</p>
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<div>== Problem 10 ==<br />
The average of the numbers <math>1, 2, 3,\cdots, 98, 99,</math> and <math>x</math> is <math>100x</math>. What is <math>x</math>?<br />
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<math>\textbf{(A)}\ \dfrac{49}{101} \qquad \textbf{(B)}\ \dfrac{50}{101} \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ \dfrac{51}{101} \qquad \textbf{(E)}\ \dfrac{50}{99}</math><br />
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== Solution ==<br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_9&diff=353382010 AMC 12B Problems/Problem 92010-07-12T21:17:57Z<p>Lg5293: Created page with '== Problem 9 == Let <math>n</math> be the smallest positive integer such that <math>n</math> id divisible by <math>20</math>, <math>n^2</math> is a perfect cube, and <math>n^3</m…'</p>
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<div>== Problem 9 ==<br />
Let <math>n</math> be the smallest positive integer such that <math>n</math> id divisible by <math>20</math>, <math>n^2</math> is a perfect cube, and <math>n^3</math> is a perfect square. What is the number of digits of <math>n</math>?<br />
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<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7</math><br />
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== Solution ==<br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_8&diff=353372010 AMC 12B Problems/Problem 82010-07-12T21:17:19Z<p>Lg5293: Created page with '== Problem 8 == Every high school in the city of Euclid sent a team of <math>3</math> students to a math contest. Each participant in the contest received a different score. Andr…'</p>
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<div>== Problem 8 ==<br />
Every high school in the city of Euclid sent a team of <math>3</math> students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed <math>37</math><sup>th</sup> and <math>64</math><sup>th</sup>, respectively. How many schools are in the city?<br />
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<math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 26</math><br />
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== Solution ==<br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_7&diff=353362010 AMC 12B Problems/Problem 72010-07-12T21:16:53Z<p>Lg5293: Created page with '== Problem 7 == Shelby drives her scooter at a speed of <math>30</math> miles per hour if it is not raining, and <math>20</math> miles per hour if it is raining. Today she drove …'</p>
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<div>== Problem 7 ==<br />
Shelby drives her scooter at a speed of <math>30</math> miles per hour if it is not raining, and <math>20</math> miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of <math>16</math> miles in <math>40</math> minutes. How many minutes did she drive in the rain?<br />
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<math>\textbf{(A)}\ 18 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30</math><br />
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== Solution ==<br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_4&diff=353352010 AMC 12B Problems/Problem 42010-07-12T21:16:04Z<p>Lg5293: Created page with '== Problem 4 == A month with <math>31</math> days has the same number of Mondays and Wednesdays.How many of the seven days of the week could be the first day of this month? <mat…'</p>
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<div>== Problem 4 ==<br />
A month with <math>31</math> days has the same number of Mondays and Wednesdays.How many of the seven days of the week could be the first day of this month?<br />
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<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math><br />
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== Solution ==<br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_12&diff=351862010 AMC 12B Problems/Problem 122010-07-09T22:53:25Z<p>Lg5293: </p>
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<div>== Problem 12 ==<br />
For what value of <math>x</math> does<br />
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<cmath>\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?</cmath><br />
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<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024</math><br />
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== Solution ==<br />
<cmath> \log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) = 40 </cmath><br />
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<cmath> \frac{1}{2} \frac{\log_2x}{\log_2\sqrt{2}} + \log_2x + \frac{2\log_2x}{\log_24} + \frac{3\log_2x}{\log_28} + \frac{4\log_2x}{\log_216} = 40 </cmath><br />
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<cmath> \log_2x + \log_2x + \log_2x + \log_2x + \log_2x = 40 </cmath><br />
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<cmath> 5\log_2x = 40 </cmath><br />
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<cmath> \log_2x = 8 </cmath><br />
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<cmath> x = 256 \;\; (D) </cmath><br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293https://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12B_Problems/Problem_12&diff=351852010 AMC 12B Problems/Problem 122010-07-09T22:53:05Z<p>Lg5293: </p>
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<div>== Problem 12 ==<br />
For what value of <math>x</math> does<br />
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<cmath>\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?</cmath><br />
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<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024</math><br />
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== Solution ==<br />
<cmath> \log_{\sqrt{2}}\sqrt{x} + \log_2x + \log_4(x^2) + \log_8(x^3) + \log_{16}(x^4) = 40 </cmath>\\<br />
<cmath> \frac{1}{2} \frac{\log_2x}{\log_2\sqrt{2}} + \log_2x + \frac{2\log_2x}{\log_24} + \frac{3\log_2x}{\log_28} + \frac{4\log_2x}{\log_216} = 40 </cmath><br />
<cmath> \log_2x + \log_2x + \log_2x + \log_2x + \log_2x = 40 </cmath><br />
<cmath> 5\log_2x = 40 </cmath><br />
<cmath> \log_2x = 8 </cmath><br />
<cmath> x = 256 \;\; (D) </cmath><br />
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== See also ==<br />
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}}</div>Lg5293