Difference between revisions of "2013 AMC 12B Problems"

Latest revision as of 22:05, 4 July 2024

2013 AMC 12B (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

On a particular January day, the high temperature in Lincoln, Nebraska, was $16$ degrees higher than the low temperature, and the average of the high and low temperatures was $3$ . In degrees, what was the low temperature in Lincoln that day?

$\textbf{(A)}\ -13 \qquad \textbf{(B)}\ -8 \qquad \textbf{(C)}\ -5 \qquad \textbf{(D)}\ -3 \qquad \textbf{(E)}\ 11$

Solution

Problem 2

Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green’s steps is $2$ 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?

$\textbf{(A)}\ 600 \qquad \textbf{(B)}\ 800 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1200 \qquad \textbf{(E)}\ 1400$

Solution

Problem 3

When counting from $3$ to $201$ , $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$ , $53$ is the $n^{\text{th}}$ number counted. What is $n$ ?

$\textbf{(A)}\ 146 \qquad \textbf{(B)}\ 147 \qquad \textbf{(C)}\ 148 \qquad \textbf{(D)}\ 149 \qquad \textbf{(E)}\ 150$

Solution

Problem 4

Ray's car averages $40$ miles per gallon of gasoline, and Tom's car averages $10$ 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?

$\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 40$

Solution

Problem 5

The average age of $33$ fifth-graders is $11$ . The average age of $55$ of their parents is $33$ . What is the average age of all of these parents and fifth-graders?

$\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23.25 \qquad \textbf{(C)}\ 24.75 \qquad \textbf{(D)}\ 26.25 \qquad \textbf{(E)}\ 28$

Solution

Problem 6

Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$ . What is $x + y$ ?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Solution

Problem 7

Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying $``1"$ , so Blair follows by saying $``1, 2"$ . Jo then says $``1, 2, 3"$ , and so on. What is the $53^{\text{rd}}$ number said?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Solution

Problem 8

Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$ . Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$ . Line $l_3$ has positive slope, goes through point $A$ , and meets $l_2$ at point $C$ . The area of $\triangle ABC$ is $3$ . What is the slope of $l_3$ ?

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

Solution

Problem 9

What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$ ?

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

Solution

Problem 10

Alex has $75$ red tokens and $75$ 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?

$\textbf{(A)}\ 62 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 83 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 103$

Solution

Problem 11

Two bees start at the same spot and fly at the same rate in the following directions. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly $10$ feet away from each other?

$\textbf{(A)}\ A$ east, $B$ west
$\textbf{(B)}\ A$ north, $B$ south
$\textbf{(C)}\ A$ north, $B$ west
$\textbf{(D)}\ A$ up, $B$ south
$\textbf{(E)}\ A$ up, $B$ west

Solution

Problem 12

Cities $A$ , $B$ , $C$ , $D$ , and $E$ are connected by roads $\widetilde{AB}$ , $\widetilde{AD}$ , $\widetilde{AE}$ , $\widetilde{BC}$ , $\widetilde{BD}$ , $\widetilde{CD}$ , and $\widetilde{DE}$ . How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some cities more than once.) $[asy] unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (C); dot (D); dot (E); label("$A$",A,S); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$E$",E,W); guide squiggly(path g, real stepsize, real slope=45) { real len = arclength(g); real step = len / round(len / stepsize); guide squig; for (real u = 0; u < len; u += step){ real a = arctime(g, u); real b = arctime(g, u + step / 2); pair p = point(g, a); pair q = point(g, b); pair np = unit( rotate(slope) * dir(g,a)); pair nq = unit( rotate(0 - slope) * dir(g,b)); squig = squig .. p{np} .. q{nq}; } squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))}; return squig; } pen pp = defaultpen + 2.718; draw(squiggly(A--B, 4.04, 30), pp); draw(squiggly(A--D, 7.777, 20), pp); draw(squiggly(A--E, 5.050, 15), pp); draw(squiggly(B--C, 5.050, 15), pp); draw(squiggly(B--D, 4.04, 20), pp); draw(squiggly(C--D, 2.718, 20), pp); draw(squiggly(D--E, 2.718, -60), pp);[/asy]$

$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18$

Solution

Problem 13

The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$ . Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$ ?

$\textbf{(A)}\ 210 \qquad \textbf{(B)}\ 220 \qquad \textbf{(C)}\ 230 \qquad \textbf{(D)}\ 240 \qquad \textbf{(E)}\ 250$

Solution

Problem 14

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 $N$ . What is the smallest possible value of $N$ ?

$\textbf{(A)}\ 55 \qquad \textbf{(B)}\ 89 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 273$

Solution

Problem 15

The number $2013$ is expressed in the form

$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$ ,

where $a_1 \ge a_2 \ge ... \ge a_m$ and $b_1 \ge b_2 \ge ... \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$ ?

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

Solution

Problem 16

Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$ . The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$ ?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)}\ \frac{\sqrt{5}+1}{2} \qquad \textbf{(E)}\ \sqrt{5}$

Solution

Problem 17

Let $a,b,$ and $c$ be real numbers such that

$a+b+c=2, \text{ and}$ $a^2+b^2+c^2=12$

What is the difference between the maximum and minimum possible values of $c$ ?

$\textbf{(A) }2\qquad \textbf{ (B) }\frac{10}{3}\qquad \textbf{ (C) }4 \qquad \textbf{ (D) }\frac{16}{3}\qquad \textbf{ (E) }\frac{20}{3}$

Solution

Problem 18

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 $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove $1$ or $3$ 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 $2013$ coins and when the game starts with $2014$ coins?

$\textbf{(A)}$ Barbara will win with $2013$ coins and Jenna will win with $2014$ coins.

$\textbf{(B)}$ Jenna will win with $2013$ coins, and whoever goes first will win with $2014$ coins.

$\textbf{(C)}$ Barbara will win with $2013$ coins, and whoever goes second will win with $2014$ coins.

$\textbf{(D)}$ Jenna will win with $2013$ coins, and Barbara will win with $2014$ coins.

$\textbf{(E)}$ Whoever goes first will win with $2013$ coins, and whoever goes second will win with $2014$ coins.

Solution

Problem 19

In triangle $ABC$ , $AB=13$ , $BC=14$ , and $CA=15$ . Distinct points $D$ , $E$ , and $F$ lie on segments $\overline{BC}$ , $\overline{CA}$ , and $\overline{DE}$ , respectively, such that $\overline{AD}\perp\overline{BC}$ , $\overline{DE}\perp\overline{AC}$ , and $\overline{AF}\perp\overline{BF}$ . The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?

$\textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30$

Solution

Problem 20

For $135^\circ < x < 180^\circ$ , points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$ ?

$\textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)} 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3}$

Solution

Problem 21

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 $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$ . No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?

$\textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 870$

Solution

Problem 22

Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation $8(\log_n x)(\log_m x)-7\log_n x-6 \log_m x-2013 = 0$ is the smallest possible integer. What is $m+n$ ?

$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 272$

Solution

Problem 23

Bernardo chooses a three-digit positive integer $N$ 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 $S$ . For example, if $N=749$ , Bernardo writes the numbers $10444$ and $3245$ , and LeRoy obtains the sum $S=13,689$ . For how many choices of $N$ are the two rightmost digits of $S$ , in order, the same as those of $2N$ ?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25$

Solution

Problem 24

Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$ , and $\overline{CN}$ is the angle bisector of $\angle{ACB}$ with $N$ on $\overline{AB}$ . Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$ . In addition $\triangle BXN$ is equilateral with $AC=2$ . What is $BX^2$ ?

$\textbf{(A)}\ \frac{10-6\sqrt{2}}{7} \qquad \textbf{(B)}\ \frac{2}{9} \qquad \textbf{(C)}\ \frac{5\sqrt{2}-3\sqrt{3}}{8} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad \textbf{(E)}\ \frac{3\sqrt{3}-4}{5}$

Solution

Problem 25

Let $G$ be the set of polynomials of the form $P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,$ where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$ ?

$\textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D)}\ 992\qquad\textbf{(E)}\ 1056$

Solution

2013 AMC 12B (Problems • Answer Key • Resources)
Preceded by 2013 AMC 12A Problems	Followed by 2014 AMC 12A Problems
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All AMC 12 Problems and Solutions

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Difference between revisions of "2013 AMC 12B Problems"

Latest revision as of 22:05, 4 July 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 2: / Line 2: @@
 ==Problem 1==
-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?
+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</math>. In degrees, what was the low temperature in Lincoln that day?
 <math>\textbf{(A)}\ -13 \qquad \textbf{(B)}\ -8 \qquad \textbf{(C)}\ -5 \qquad \textbf{(D)}\ -3 \qquad \textbf{(E)}\ 11</math>
@@ Line 9: / Line 9: @@
 ==Problem 2==
-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?
+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?
 <math>\textbf{(A)}\ 600 \qquad \textbf{(B)}\ 800 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1200 \qquad \textbf{(E)}\ 1400</math>
@@ Line 16: / Line 16: @@
 ==Problem 3==
-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>?
+When counting from <math>3</math> to <math>201</math>, <math>53</math> is the <math>51^{\text{st}}</math> number counted. When counting backwards from <math>201</math> to <math>3</math>, <math>53</math> is the <math>n^{\text{th}}</math> number counted. What is <math>n</math>?
 <math>\textbf{(A)}\ 146 \qquad \textbf{(B)}\ 147 \qquad \textbf{(C)}\ 148 \qquad \textbf{(D)}\ 149 \qquad \textbf{(E)}\ 150</math>
@@ Line 24: / Line 24: @@
 ==Problem 4==
 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 \>
 <math>\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 40</math>
@@ Line 43: / Line 44: @@
 ==Problem 7==
-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 \>
+Jo and Blair take turns counting from <math>1</math> to one more than the last number said by the other person. Jo starts by saying <math>``1"</math>, so Blair follows by saying <math>``1, 2"</math>. Jo then says <math>``1, 2, 3"</math>, and so on. What is the <math>53^{\text{rd}}</math> number said?
 <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8</math>
@@ Line 49: / Line 51: @@
 ==Problem 8==
-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?
+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 <math>3</math>. What is the slope of <math>l_3</math>?
-<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>
+<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>
 [[2013 AMC 12B Problems/Problem 8|Solution]]
 ==Problem 9==
-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?
+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)}\ \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>
+<math>\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12 </math>
 [[2013 AMC 12B Problems/Problem 9|Solution]]
 ==Problem 10==
-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>?
+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?
-<math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75</math>
+<math>\textbf{(A)}\ 62 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 83 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 103</math>
 [[2013 AMC 12B Problems/Problem 10|Solution]]
 ==Problem 11==
-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?
+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?
-<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math>
+<math>\textbf{(A)}\ A</math> east, <math>B</math> west<br \><math>\textbf{(B)}\ A</math> north, <math>B</math> south<br \><math>\textbf{(C)}\ A</math> north, <math>B</math> west<br \><math>\textbf{(D)}\ A</math> up, <math>B</math> south<br \><math>\textbf{(E)}\ A</math> up, <math>B</math> west<br \>
 [[2013 AMC 12B Problems/Problem 11|Solution]]
@@ Line 78: / Line 80: @@
 ==Problem 12==
-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?
+Cities <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are connected by roads <math>\widetilde{AB}</math>, <math>\widetilde{AD}</math>, <math>\widetilde{AE}</math>, <math>\widetilde{BC}</math>, <math>\widetilde{BD}</math>, <math>\widetilde{CD}</math>, and <math>\widetilde{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.)
 <asy>
 unitsize(10mm);
-defaultpen(linewidth(.8pt)+fontsize(10pt));
+defaultpen(linewidth(1.2pt)+fontsize(10pt));
 dotfactor=4;
-pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2));
+pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08);
-draw(A--B--C--D--E--F--G--H--cycle);
+dot (A);
-draw(A--D);
+dot (B);
-draw(B--G);
+dot (C);
-draw(C--F);
+dot (D);
-draw(E--H);</asy>
+dot (E);
+label("$A$",A,S);
+label("$B$",B,SE);
+label("$C$",C,E);
+label("$D$",D,N);
+label("$E$",E,W);
+guide squiggly(path g, real stepsize, real slope=45)
+{
+ real len = arclength(g);
+ real step = len / round(len / stepsize);
+ guide squig;
+ for (real u = 0; u < len; u += step){
+ real a = arctime(g, u);
+ real b = arctime(g, u + step / 2);
+ pair p = point(g, a);
+ pair q = point(g, b);
+ pair np = unit( rotate(slope) * dir(g,a));
+ pair nq = unit( rotate(0 - slope) * dir(g,b));
+ squig = squig .. p{np} .. q{nq};
+ }
+ squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))};
+ return squig;
+}
+pen pp = defaultpen + 2.718;
+draw(squiggly(A--B, 4.04, 30), pp);
+draw(squiggly(A--D, 7.777, 20), pp);
+draw(squiggly(A--E, 5.050, 15), pp);
+draw(squiggly(B--C, 5.050, 15), pp);
+draw(squiggly(B--D, 4.04, 20), pp);
+draw(squiggly(C--D, 2.718, 20), pp);
+draw(squiggly(D--E, 2.718, -60), pp);</asy>
-<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>
+<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math>
 [[2013 AMC 12B Problems/Problem 12|Solution]]
@@ Line 97: / Line 128: @@
 ==Problem 13==
-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>?
+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 arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of <math>ABCD</math>?
-<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93</math>
+<math>\textbf{(A)}\ 210 \qquad \textbf{(B)}\ 220 \qquad \textbf{(C)}\ 230 \qquad \textbf{(D)}\ 240 \qquad \textbf{(E)}\ 250</math>
 [[2013 AMC 12B Problems/Problem 13|Solution]]
@@ Line 105: / Line 136: @@
 ==Problem 14==
-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>?
+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> ?
-<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>
+<math>\textbf{(A)}\ 55 \qquad \textbf{(B)}\ 89 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 273</math>
 [[2013 AMC 12B Problems/Problem 14|Solution]]
@@ Line 113: / Line 144: @@
 ==Problem 15==
-How many positive two-digit integers are factors of <math>2^{24}-1</math>?
+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>?
+<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math>
-<math>\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 14</math>
 [[2013 AMC 12B Problems/Problem 15|Solution]]
@@ Line 121: / Line 151: @@
 ==Problem 16==
-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>?
+Let <math>ABCDE</math> be an equiangular convex pentagon of perimeter <math>1</math>. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let <math>s</math> be the perimeter of this star. What is the difference between the maximum and the minimum possible values of <math>s</math>?
-<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>
+<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2} \qquad \textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)}\  \frac{\sqrt{5}+1}{2} \qquad \textbf{(E)}\ \sqrt{5}</math>
 [[2013 AMC 12B Problems/Problem 16|Solution]]
@@ Line 129: / Line 159: @@
 ==Problem 17==
-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>?
-<math>\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099</math>
+Let <math>a,b,</math> and <math>c</math> be real numbers such that
+<cmath>a+b+c=2, \text{ and} </cmath>
+<cmath> a^2+b^2+c^2=12 </cmath>
+What is the difference between the maximum and minimum possible values of <math>c</math>?
+<math> \textbf{(A) }2\qquad \textbf{ (B) }\frac{10}{3}\qquad \textbf{ (C) }4 \qquad \textbf{ (D) }\frac{16}{3}\qquad \textbf{ (E) }\frac{20}{3} </math>
 [[2013 AMC 12B Problems/Problem 17|Solution]]
@@ Line 137: / Line 173: @@
 ==Problem 18==
-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?
+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. When 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?
+<math> \textbf{(A)}</math> Barbara will win with <math>2013</math> coins and Jenna will win with <math>2014</math> coins.
+<math>\textbf{(B)}</math> Jenna will win with <math>2013</math> coins, and whoever goes first will win with <math>2014</math> coins.
+<math>\textbf{(C)}</math> Barbara will win with <math>2013</math> coins, and whoever goes second will win with <math>2014</math> coins.
-<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>
+<math>\textbf{(D)}</math> Jenna will win with <math>2013</math> coins, and Barbara will win with <math>2014</math> coins.
+<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.
 [[2013 AMC 12B Problems/Problem 18|Solution]]
 ==Problem 19==
+In triangle <math>ABC</math>, <math>AB=13</math>, <math>BC=14</math>, and <math>CA=15</math>. Distinct points <math>D</math>, <math>E</math>, and <math>F</math> lie on segments <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{DE}</math>, respectively, such that <math>\overline{AD}\perp\overline{BC}</math>, <math>\overline{DE}\perp\overline{AC}</math>, and <math>\overline{AF}\perp\overline{BF}</math>. The length of segment <math>\overline{DF}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>?
-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>?
+<math> \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30 </math>
-<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>
 [[2013 AMC 12B Problems/Problem 19|Solution]]
 ==Problem 20==
+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>?
-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>?
+<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>
-<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>
 [[2013 AMC 12B Problems/Problem 20|Solution]]
 ==Problem 21==
+Consider the set of <math>30</math> parabolas defined as follows: all parabolas have as focus the point <math>(0,0)</math> and the directrix lines have the form <math>y=ax+b</math> with <math>a</math> and <math>b</math> 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?
-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>?
+<math> \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 870 </math>
-<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70</math>
 [[2013 AMC 12B Problems/Problem 21|Solution]]
 ==Problem 22==
+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\log_n x-6 \log_m x-2013 = 0 </cmath>
+is the smallest possible integer. What is <math>m+n</math>?
-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>?
+<math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 272 </math>
-<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>
 [[2013 AMC 12B Problems/Problem 22|Solution]]
 ==Problem 23==
+Bernardo chooses a three-digit positive integer <math>N</math> and writes both its base-<math>5</math> and base-<math>6</math> representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-<math>10</math> integers, he adds them to obtain an integer <math>S</math>. For example, if <math>N=749</math>, Bernardo writes the numbers <math>10444</math> and <math>3245</math>, 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>?
-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?
+<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25 </math>
-<math>\textbf{(A)}\ 161 \qquad \textbf{(B)}\ 185 \qquad \textbf{(C)}\  195 \qquad \textbf{(D)}\  227 \qquad \textbf{(E)}\  255</math>
-[[2013 AMC 12B Problems/Problem 23|Solution]]
+[[2013 AMC 10B Problems/Problem 25|Solution]]
 ==Problem 24==
-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>?
+Let <math>ABC</math> be a triangle where <math>M</math> is the midpoint of <math>\overline{AC}</math>, and <math>\overline{CN}</math> is the angle bisector of <math>\angle{ACB}</math> with <math>N</math> on <math>\overline{AB}</math>. Let <math>X</math> be the intersection of the median <math>\overline{BM}</math> and the bisector <math>\overline{CN}</math>. In addition <math>\triangle BXN</math> is equilateral with <math>AC=2</math>. What is <math>BX^2</math>?
-<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>
+<math>\textbf{(A)}\  \frac{10-6\sqrt{2}}{7} \qquad \textbf{(B)}\ \frac{2}{9} \qquad \textbf{(C)}\ \frac{5\sqrt{2}-3\sqrt{3}}{8} \qquad \textbf{(D)}\ \frac{\sqrt{2}}{6} \qquad \textbf{(E)}\ \frac{3\sqrt{3}-4}{5}</math>
 [[2013 AMC 12B Problems/Problem 24|Solution]]
@@ Line 193: / Line 234: @@
 ==Problem 25==
-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
+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>
-<cmath> \left[\frac{n}{k}\right] + \left[\frac{100 - n}{k}\right] = \left[\frac{100}{k}\right] </cmath>
+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>?
-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>?
-<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>
+<math> \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D)}\ 992\qquad\textbf{(E)}\ 1056 </math>
 [[2013 AMC 12B Problems/Problem 25|Solution]]
@@ Line 205: / Line 244: @@
 == See also ==
 {{AMC12 box|year=2013|ab=B|before=[[2013 AMC 12A Problems]]|after=[[2014 AMC 12A Problems]]}}
+{{MAA Notice}}