https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Zverevab&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T17:46:35ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_18&diff=514742013 AMC 12B Problems/Problem 182013-02-22T22:29:03Z<p>Zverevab: Undo revision 51473 by Zverevab (talk)</p>
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<div>==Problem==<br />
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove <math>2</math> or <math>4</math> coins, unless only one coin remains, in which case she loses her turn. What it is Jenna’s turn, she must remove <math>1</math> or <math>3</math> coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with <math>2013</math> coins and when the game starts with <math>2014</math> coins?<br />
<br />
<math> \textbf{(A)}</math> Barbara will win with <math>2013</math> coins and Jenna will win with <math>2014</math> coins. <br />
<br />
<math>\textbf{(B)}</math> Jenna will win with <math>2013</math> coins, and whoever goes first will win with <math>2014</math> coins. <br />
<br />
<math>\textbf{(C)}</math> Barbara will win with <math>2013</math> coins, and whoever goes second will win with <math>2014</math> coins.<br />
<br />
<math>\textbf{(D)}</math> Jenna will win with <math>2013</math> coins, and Barbara will win with <math>2014</math> coins.<br />
<br />
<math>\textbf{(E)}</math> Whoever goes first will win with <math>2013</math> coins, and whoever goes second will win with <math>2014</math> coins.<br />
<br />
<br />
==Solution==<br />
<br />
== See also ==<br />
{{AMC12 box|year=2013|ab=B|num-b=17|num-a=19}}</div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_18&diff=514732013 AMC 12B Problems/Problem 182013-02-22T22:27:34Z<p>Zverevab: </p>
<hr />
<div>==Problem==<br />
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove <math>2</math> or <math>4</math> coins, unless only one coin remains, in which case she loses her turn. What it is Jenna’s turn, she must remove <math>1</math> or <math>3</math> coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with <math>2013</math> coins and when the game starts with <math>2014</math> coins?<br />
<br />
<math> \textbf{(A)}</math> Barbara will win with <math>2013</math> coins and Jenna will win with <math>2014</math> coins. <br />
<br />
<math>\textbf{(B)}</math> Jenna will win with <math>2013</math> coins, and whoever goes first will win with <math>2014</math> coins. <br />
<br />
<math>\textbf{(C)}</math> Barbara will win with <math>2013</math> coins, and whoever goes second will win with <math>2014</math> coins.<br />
<br />
<math>\textbf{(D)}</math> Jenna will win with <math>2013</math> coins, and Barbara will win with <math>2014</math> coins.<br />
<br />
<math>\textbf{(E)}</math> Whoever goes first will win with <math>2013</math> coins, and whoever goes second will win with <math>2014</math> coins.<br />
<br />
<br />
==Solution==</div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_2&diff=514252013 AMC 12B Problems/Problem 22013-02-22T21:49:08Z<p>Zverevab: </p>
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<div>==Problem==<br />
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is <math>15</math> steps by <math>20</math> steps. Each of Mr. Green's steps is <math>2</math> feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?<br />
<br />
<math>\textbf{(A)}\ 600 \qquad \textbf{(B)}\ 800 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1200 \qquad \textbf{(E)}\ 1400</math><br />
<br />
==Solution==<br />
<br />
Since each step is <math>2</math> feet, his garden is <math>30</math> by <math>40</math> feet. Thus, the area of <math>30(40) = 1200</math> square feet. Since he is expecting <math>\frac{1}{2}</math> of a pound per square foot, the total amount of potatoes expected is <math>1200 \times \frac{1}{2} = \boxed{\textbf{(A) }600}</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_24&diff=514202013 AMC 12B Problems/Problem 242013-02-22T21:44:43Z<p>Zverevab: Blanked the page</p>
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<div></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_18&diff=514162013 AMC 12B Problems/Problem 182013-02-22T21:43:03Z<p>Zverevab: /* Problem */</p>
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<div></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_25&diff=513822013 AMC 12B Problems/Problem 252013-02-22T20:02:50Z<p>Zverevab: Created page with "==Problem== Let <math>G</math> be the set of polynomials of the form <cmath> P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, </cmath> where <math> c_1,c_2,\cdots, c_{n-1} </math>..."</p>
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<div>==Problem==<br />
<br />
Let <math>G</math> be the set of polynomials of the form<br />
<cmath> P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, </cmath><br />
where <math> c_1,c_2,\cdots, c_{n-1} </math> are integers and <math>P(z)</math> has distinct roots of the form <math>a+ib</math> with <math>a</math> and <math>b</math> integers. How many polynomials are in <math>G</math>?<br />
<br />
<math> \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 </math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_24&diff=513812013 AMC 12B Problems/Problem 242013-02-22T20:02:28Z<p>Zverevab: Created page with "==Problem== 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 among all the <math>8</math>-sided polygons i..."</p>
<hr />
<div>==Problem==<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 among all the <math>8</math>-sided polygons 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></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_23&diff=513802013 AMC 12B Problems/Problem 232013-02-22T20:02:05Z<p>Zverevab: Created page with "==Problem== Bernardo chooses a three-digit positive integer <math>N</math> and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers..."</p>
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<div>==Problem==<br />
Bernardo chooses a three-digit positive integer <math>N</math> and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer <math>S</math>. For example, if <math>N=749</math>, Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum <math>S=13,689</math>. For how many choices of <math>N</math> are the two rightmost digits of <math>S</math>, in order, the same as those of <math>2N</math>?<br />
<br />
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25 </math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_22&diff=513782013 AMC 12B Problems/Problem 222013-02-22T20:01:17Z<p>Zverevab: Created page with "==Problem== Let <math>m>1</math> and <math>n>1</math> be integers. Suppose that the product of the solutions for <math>x</math> of the equation <cmath> 8(\log_n x)(\log_m x)-7\lo..."</p>
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<div>==Problem==<br />
Let <math>m>1</math> and <math>n>1</math> be integers. Suppose that the product of the solutions for <math>x</math> of the equation<br />
<cmath> 8(\log_n x)(\log_m x)-7\log_n x-6 log_m x-2013 = 0 </cmath><br />
is the smallest possible integer. What is <math>m+n</math>?<br />
<br />
<math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 48\qquad\textbf{(E)}\ 272 </math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_21&diff=513772013 AMC 12B Problems/Problem 212013-02-22T20:00:54Z<p>Zverevab: Created page with "==Problem== Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form <math>y=ax+b</math> with a and ..."</p>
<hr />
<div>==Problem==<br />
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form <math>y=ax+b</math> with a and b integers such that <math>a\in \{-2,-1,0,1,2\}</math> and <math>b\in \{-3,-2,-1,1,2,3\}</math>. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?<br />
<br />
<math> \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D}}\ 840\qquad\textbf{(E)}\ 870 </math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_20&diff=513762013 AMC 12B Problems/Problem 202013-02-22T20:00:23Z<p>Zverevab: Created page with "==Problem== For <math>135^\circ < x < 180^\circ</math>, points <math>P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)</math> and <math>S =(\tan x, \tan^2 x)</math..."</p>
<hr />
<div>==Problem==<br />
For <math>135^\circ < x < 180^\circ</math>, points <math>P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)</math> and <math>S =(\tan x, \tan^2 x)</math> are the vertices of a trapezoid. What is <math>\sin(2x)</math>?<br />
<br />
<math> \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D}}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3}</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_19&diff=513742013 AMC 12B Problems/Problem 192013-02-22T19:59:25Z<p>Zverevab: Created page with "==Problem== 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 <mat..."</p>
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<div>==Problem==<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></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_18&diff=513722013 AMC 12B Problems/Problem 182013-02-22T19:59:05Z<p>Zverevab: Created page with "==Problem== 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 th..."</p>
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<div>==Problem==<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></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_17&diff=513712013 AMC 12B Problems/Problem 172013-02-22T19:58:33Z<p>Zverevab: Created page with "==Problem== Let <math>a,b,</math> and <math>c</math> be real numbers such that <math>a+b+c=2,</math> and <math> a^2+b^2+c^2=12 </math> What is the difference between the max..."</p>
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<div>==Problem==<br />
<br />
<br />
Let <math>a,b,</math> and <math>c</math> be real numbers such that <br />
<math>a+b+c=2,</math> and <br />
<math> a^2+b^2+c^2=12 </math><br />
<br />
What is the difference between the maximum and minimum possible values of <math>c</math>?<br />
<br />
<math> \text{(A) }2\qquad \text{ (B) }\frac{10}{3}\qquad \text{ (C) }4 \qquad \text{ (D) }\frac{16}{3}\qquad \text{ (E) }\frac{20}{3} </math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_16&diff=513702013 AMC 12B Problems/Problem 162013-02-22T19:58:04Z<p>Zverevab: Created page with "==Problem== 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 ..."</p>
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<div>==Problem==<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></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_15&diff=513682013 AMC 12B Problems/Problem 152013-02-22T19:57:36Z<p>Zverevab: Created page with "==Problem== the number <math>2013</math> is expressed in the form <br \> <center> <math>2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}</math>,</center><br />where <math>a_1 \ge..."</p>
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<div>==Problem==<br />
<br />
the number <math>2013</math> is expressed in the form <br \> <center> <math>2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}</math>,</center><br />where <math>a_1 \ge a_2 \ge ... \ge a_m</math> and <math>b_1 \ge b_2 \ge ... \ge b_n</math> are positive integers and <math>a_1 + b_1</math> is as small as possible. What is <math>|a_1 - b_1|</math>?<br />
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_14&diff=513672013 AMC 12B Problems/Problem 142013-02-22T19:57:08Z<p>Zverevab: Created page with "==Problem== Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of..."</p>
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<div>==Problem==<br />
<br />
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is <math>N</math>. What is the smallest possible value of <math>N</math> ?<br />
<br />
<math>\textbf{(A)}\ 55 \qquad \textbf{(B)}\ 89 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 273</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_13&diff=513662013 AMC 12B Problems/Problem 132013-02-22T19:56:44Z<p>Zverevab: Created page with "==Problem== The internal angles of quadrilateral <math>ABCD</math> form an arithmetic progression. Triangles <math>ABD</math> and <math>DCB</math> are similar with <math>\angle ..."</p>
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<div>==Problem==<br />
<br />
The internal angles of quadrilateral <math>ABCD</math> form an arithmetic progression. Triangles <math>ABD</math> and <math>DCB</math> are similar with <math>\angle DBA = \angle DCB</math> and <math>\angle ADB = \angle CBD</math>. Moreover, the angles in each of these two triangles also form an arithemetic progression. In degrees, what is the largest possible sum of the two largest angles of <math>ABCD</math>?<br />
<br />
<math>\textbf{(A)}\ 210 \qquad \textbf{(B)}\ 220 \qquad \textbf{(C)}\ 230 \qquad \textbf{(D)}\ 240 \qquad \textbf{(E)}\ 250</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_12&diff=513642013 AMC 12B Problems/Problem 122013-02-22T19:56:05Z<p>Zverevab: Created page with "==Problem== Cities <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are connected by roads <math>AB</math>, <math>AD</math>, <math>AE</math>, <..."</p>
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<div>==Problem==<br />
<br />
Cities <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are connected by roads <math>AB</math>, <math>AD</math>, <math>AE</math>, <math>BC</math>, <math>BD</math>, <math>CD</math>, and <math>DE</math>. How many different routes are there from <math>A</math> to <math>B</math> that use each road exactly once? (Such a route will necessarily visit some cities more than once.)<br />
<asy><br />
unitsize(10mm);<br />
defaultpen(linewidth(1.2pt)+fontsize(10pt));<br />
dotfactor=4;<br />
pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08);<br />
dot (A);<br />
dot (B);<br />
dot (C);<br />
dot (D);<br />
dot (E);<br />
label("$A$",A,S);<br />
label("$B$",B,SE);<br />
label("$C$",C,E);<br />
label("$D$",D,N);<br />
label("$E$",E,W);<br />
draw(A--B--C--D--E--cycle);<br />
draw(A--D);<br />
draw(B--D);</asy><br />
<br />
<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_11&diff=513632013 AMC 12B Problems/Problem 112013-02-22T19:55:33Z<p>Zverevab: Created page with "==Problem== Two bees start at the same spot and fly at the same rate in the following directions. Bee <math>A</math> travels <math>1</math> foot north, then <math>1</math> foot e..."</p>
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<div>==Problem==<br />
Two bees start at the same spot and fly at the same rate in the following directions. Bee <math>A</math> travels <math>1</math> foot north, then <math>1</math> foot east, then <math>1</math> foot upwards, and then continues to repeat this pattern. Bee <math>B</math> travels <math>1</math> foot south, then <math>1</math> foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly <math>10</math> feet away from each other?<br />
<br />
<math>\textbf{(A)}\ A</math> east, <math>B</math> west<br \><math>\qquad \textbf{(B)}\ A</math> north, <math>B</math> south<br \><math> \qquad \textbf{(C)}\ A</math> north, <math>B</math> west<br \><math> \qquad \textbf{(D)}\ A</math> up, <math>B</math> south<br \><math> \qquad \textbf{(E)}\ A</math> up, <math>B</math> west<br \></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_10&diff=513622013 AMC 12B Problems/Problem 102013-02-22T19:54:56Z<p>Zverevab: Created page with "==Problem== Alex has <math>75</math> red tokens and <math>75</math> blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a bl..."</p>
<hr />
<div>==Problem==<br />
Alex has <math>75</math> red tokens and <math>75</math> blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?<br />
<br />
<math>\textbf{(A)}\ 62 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 83 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 103</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_9&diff=513612013 AMC 12B Problems/Problem 92013-02-22T19:54:28Z<p>Zverevab: Created page with "==Problem== What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides <math>12!</math> ? <math>\textbf{(A)}\ 5 \qquad \..."</p>
<hr />
<div>==Problem==<br />
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides <math>12!</math> ?<br />
<br />
<math>\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12 </math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_8&diff=513602013 AMC 12B Problems/Problem 82013-02-22T19:54:08Z<p>Zverevab: Created page with "==Problem== Line <math>l_1</math> has equation <math>3x - 2y = 1</math> and goes through <math>A = (-1, -2)</math>. Line <math>l_2</math> has equation <math>y = 1</math> and meet..."</p>
<hr />
<div>==Problem==<br />
Line <math>l_1</math> has equation <math>3x - 2y = 1</math> and goes through <math>A = (-1, -2)</math>. Line <math>l_2</math> has equation <math>y = 1</math> and meets line <math>l_1</math> at point <math>B</math>. Line <math>l_3</math> has positive slope, goes through point <math>A</math>, and meets <math>l_2</math> at point <math>C</math>. The area of <math>\triangle ABC</math> is 3. What is the slope of <math>l_3</math>?<br />
<br />
<math>\textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{3}{4} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \frac{4}{3} \qquad \textbf{(E)}\ \frac{3}{2}</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_7&diff=513592013 AMC 12B Problems/Problem 72013-02-22T19:53:46Z<p>Zverevab: Created page with "==Problem== Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saing "1", so Blair follows by saying "1, 2". Jo then ..."</p>
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<div>==Problem==<br />
Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saing "1", so Blair follows by saying "1, 2". Jo then says "1, 2, 3", and so on. What is the <math>53^{rd}</math> number said?<br \><br />
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_6&diff=513572013 AMC 12B Problems/Problem 62013-02-22T19:52:56Z<p>Zverevab: Created page with "==Problem== Real numbers <math>x</math> and <math>y</math> satisfy the equation <math>x^2 + y^2 = 10x - 6y - 34</math>. What is <math>x + y</math>? <math>\textbf{(A)}\ 1 \qquad ..."</p>
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<div>==Problem==<br />
Real numbers <math>x</math> and <math>y</math> satisfy the equation <math>x^2 + y^2 = 10x - 6y - 34</math>. What is <math>x + y</math>?<br />
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<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_5&diff=513562013 AMC 12B Problems/Problem 52013-02-22T19:52:30Z<p>Zverevab: Created page with "==Problem== The average age of <math>33</math> fifth-graders is <math>11</math>. The average age of <math>55</math> of their parents is <math>33</math>. What is the average age o..."</p>
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<div>==Problem==<br />
The average age of <math>33</math> fifth-graders is <math>11</math>. The average age of <math>55</math> of their parents is <math>33</math>. What is the average age of all of these parents and fifth-graders?<br />
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<math>\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23.25 \qquad \textbf{(C)}\ 24.75 \qquad \textbf{(D)}\ 26.25 \qquad \textbf{(E)}\ 28</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_4&diff=513552013 AMC 12B Problems/Problem 42013-02-22T19:52:04Z<p>Zverevab: Created page with "==Problem== Ray's car averages <math>40</math> miles per gallon of gasoline, and Tom's car averages <math>10</math> miles per gallon of gasoline. Ray and Tom each drive the same ..."</p>
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<div>==Problem==<br />
Ray's car averages <math>40</math> miles per gallon of gasoline, and Tom's car averages <math>10</math> miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?<br \><br />
<math>\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 40</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_3&diff=513542013 AMC 12B Problems/Problem 32013-02-22T19:51:20Z<p>Zverevab: Created page with "==Problem== When counting from <math>3</math> to <math>201</math>, <math>53</math> is the <math>51^{st}</math> number counted. When counting backwards from <math>201</math> to <m..."</p>
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<div>==Problem==<br />
When counting from <math>3</math> to <math>201</math>, <math>53</math> is the <math>51^{st}</math> number counted. When counting backwards from <math>201</math> to <math>3</math>, <math>53</math> is the <math>n^{th}</math> number counted. What is <math>n</math>?<br />
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<math>\textbf{(A)}\ 146 \qquad \textbf{(B)}\ 147 \qquad \textbf{(C)}\ 148 \qquad \textbf{(D)}\ 149 \qquad \textbf{(E)}\ 150</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_2&diff=513522013 AMC 12B Problems/Problem 22013-02-22T19:50:38Z<p>Zverevab: Created page with "==Problem== Mr. Green measures his rectangular garden by walking two of the sides and finds that it is <math>15</math> steps by <math>20</math> steps. Each of Mr. Green's steps i..."</p>
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<div>==Problem==<br />
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is <math>15</math> steps by <math>20</math> steps. Each of Mr. Green's steps is <math>2</math> feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?<br />
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<math>\textbf{(A)}\ 600 \qquad \textbf{(B)}\ 800 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1200 \qquad \textbf{(E)}\ 1400</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_1&diff=513502013 AMC 12B Problems/Problem 12013-02-22T19:49:24Z<p>Zverevab: Created page with "==Problem== On a particular January day, the high temperature in Lincoln, Nebraska, was <math>16</math> degrees higher than the low temperature, and the average of the high and l..."</p>
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<div>==Problem==<br />
On a particular January day, the high temperature in Lincoln, Nebraska, was <math>16</math> degrees higher than the low temperature, and the average of the high and low temperatures was <math>3\textdegree</math>. In degrees, what was the low temperature in Lincoln that day?<br />
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<math>\textbf{(A)}\ -13 \qquad \textbf{(B)}\ -8 \qquad \textbf{(C)}\ -5 \qquad \textbf{(D)}\ -3 \qquad \textbf{(E)}\ 11</math></div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Answer_Key&diff=513472013 AMC 12B Answer Key2013-02-22T19:39:13Z<p>Zverevab: Created page with "1. C 2. A 3. D 4. B 5. C 6. B 7. E 8. B 9. C 10. E 11. A 12. D 13. D 14. C 15. B 16. A 17. D 18. B 19. B 20. A 21. C 22. A 23. E 24. A 25. B"</p>
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<div>1. C<br />
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2. A<br />
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3. D<br />
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4. B<br />
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5. C<br />
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6. B<br />
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7. E<br />
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8. B<br />
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9. C<br />
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10. E<br />
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11. A<br />
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12. D<br />
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13. D<br />
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14. C<br />
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15. B<br />
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16. A<br />
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17. D<br />
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18. B<br />
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19. B<br />
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20. A<br />
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21. C<br />
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22. A<br />
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23. E<br />
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24. A<br />
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25. B</div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B&diff=513462013 AMC 12B2013-02-22T19:34:59Z<p>Zverevab: Created page with "*2013 AMC 12B Problems *2013 AMC 12B Answer Key **Problem 1 **Problem 2 **[[2013 AMC 12B Problems/..."</p>
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<div>*[[2013 AMC 12B Problems]]<br />
*[[2013 AMC 12B Answer Key]]<br />
**[[2013 AMC 12B Problems/Problem 1|Problem 1]]<br />
**[[2013 AMC 12B Problems/Problem 2|Problem 2]]<br />
**[[2013 AMC 12B Problems/Problem 3|Problem 3]]<br />
**[[2013 AMC 12B Problems/Problem 4|Problem 4]]<br />
**[[2013 AMC 12B Problems/Problem 5|Problem 5]]<br />
**[[2013 AMC 12B Problems/Problem 6|Problem 6]]<br />
**[[2013 AMC 12B Problems/Problem 7|Problem 7]]<br />
**[[2013 AMC 12B Problems/Problem 8|Problem 8]]<br />
**[[2013 AMC 12B Problems/Problem 9|Problem 9]]<br />
**[[2013 AMC 12B Problems/Problem 10|Problem 10]]<br />
**[[2013 AMC 12B Problems/Problem 11|Problem 11]]<br />
**[[2013 AMC 12B Problems/Problem 12|Problem 12]]<br />
**[[2013 AMC 12B Problems/Problem 13|Problem 13]]<br />
**[[2013 AMC 12B Problems/Problem 14|Problem 14]]<br />
**[[2013 AMC 12B Problems/Problem 15|Problem 15]]<br />
**[[2013 AMC 12B Problems/Problem 16|Problem 16]]<br />
**[[2013 AMC 12B Problems/Problem 17|Problem 17]]<br />
**[[2013 AMC 12B Problems/Problem 18|Problem 18]]<br />
**[[2013 AMC 12B Problems/Problem 19|Problem 19]]<br />
**[[2013 AMC 12B Problems/Problem 20|Problem 20]]<br />
**[[2013 AMC 12B Problems/Problem 21|Problem 21]]<br />
**[[2013 AMC 12B Problems/Problem 22|Problem 22]]<br />
**[[2013 AMC 12B Problems/Problem 23|Problem 23]]<br />
**[[2013 AMC 12B Problems/Problem 24|Problem 24]]<br />
**[[2013 AMC 12B Problems/Problem 25|Problem 25]]</div>Zverevabhttps://artofproblemsolving.com/wiki/index.php?title=AMC_12_Problems_and_Solutions&diff=51345AMC 12 Problems and Solutions2013-02-22T19:32:54Z<p>Zverevab: </p>
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<div>[[AMC 12]] problems and solutions by test:<br />
<br />
* [[2000 AMC 12]]<br />
* [[2001 AMC 12]]<br />
* [[2002 AMC 12A]]<br />
* [[2002 AMC 12B]]<br />
* [[2003 AMC 12A]]<br />
* [[2003 AMC 12B]]<br />
* [[2004 AMC 12A]]<br />
* [[2004 AMC 12B]]<br />
* [[2005 AMC 12A]]<br />
* [[2005 AMC 12B]]<br />
* [[2006 AMC 12A]]<br />
* [[2006 AMC 12B]]<br />
* [[2007 AMC 12A]]<br />
* [[2007 AMC 12B]]<br />
* [[2008 AMC 12A]]<br />
* [[2008 AMC 12B]]<br />
* [[2009 AMC 12A]]<br />
* [[2009 AMC 12B]]<br />
* [[2010 AMC 12A]]<br />
* [[2010 AMC 12B]]<br />
* [[2011 AMC 12A]]<br />
* [[2011 AMC 12B]]<br />
* [[2012 AMC 12A]]<br />
* [[2012 AMC 12B]]<br />
* [[2013 AMC 12A]]<br />
* [[2013 AMC 12B]]<br />
== Resources ==<br />
* [[American Mathematics Competitions]]<br />
* [[AMC Problems and Solutions]]<br />
* [[Mathematics competition resources]]<br />
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[[Category:Math Contest Problems]]</div>Zverevab