Difference between revisions of "2013 AMC 12B Problems"
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==Problem 4== | ==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 \> | 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> | <math>\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 40</math> | ||
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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? | 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)}\ A</math> east, <math>B</math> west<br \><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]] | [[2013 AMC 12B Problems/Problem 11|Solution]] | ||
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==Problem 12== | ==Problem 12== | ||
− | 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.) | + | 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> | <asy> | ||
unitsize(10mm); | unitsize(10mm); | ||
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label("$D$",D,N); | label("$D$",D,N); | ||
label("$E$",E,W); | label("$E$",E,W); | ||
− | draw(A--B | + | guide squiggly(path g, real stepsize, real slope=45) |
− | draw( | + | { |
− | draw( | + | 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)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math> | <math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math> | ||
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==Problem 13== | ==Problem 13== | ||
− | 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 | + | 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)}\ 210 \qquad \textbf{(B)}\ 220 \qquad \textbf{(C)}\ 230 \qquad \textbf{(D)}\ 240 \qquad \textbf{(E)}\ 250</math> | <math>\textbf{(A)}\ 210 \qquad \textbf{(B)}\ 220 \qquad \textbf{(C)}\ 230 \qquad \textbf{(D)}\ 240 \qquad \textbf{(E)}\ 250</math> | ||
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==Problem 15== | ==Problem 15== | ||
− | + | 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)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5</math> | ||
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==Problem 16== | ==Problem 16== | ||
− | 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> | + | 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)}\ 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> | <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> | ||
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Let <math>a,b,</math> and <math>c</math> be real numbers such that | 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>? | What is the difference between the maximum and minimum possible values of <math>c</math>? | ||
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==Problem 18== | ==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 <math>2</math> or <math>4</math> coins, unless only one coin remains, in which case she loses her turn. | + | 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{(A)}</math> Barbara will win with <math>2013</math> coins and Jenna will win with <math>2014</math> coins. | ||
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==Problem 21== | ==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 <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? | + | 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? |
<math> \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 870 </math> | <math> \textbf{(A)}\ 720\qquad\textbf{(B)}\ 760\qquad\textbf{(C)}\ 810\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 870 </math> | ||
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==Problem 23== | ==Problem 23== | ||
− | 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 | + | 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>? |
<math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25 </math> | <math> \textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25 </math> | ||
− | [[2013 AMC | + | [[2013 AMC 10B Problems/Problem 25|Solution]] |
==Problem 24== | ==Problem 24== | ||
− | 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> | + | 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)}\ \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> | <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> | ||
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== See also == | == See also == | ||
{{AMC12 box|year=2013|ab=B|before=[[2013 AMC 12A Problems]]|after=[[2014 AMC 12A Problems]]}} | {{AMC12 box|year=2013|ab=B|before=[[2013 AMC 12A Problems]]|after=[[2014 AMC 12A Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 02:46, 30 January 2021
2013 AMC 12B (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
On a particular January day, the high temperature in Lincoln, Nebraska, was degrees higher than the low temperature, and the average of the high and low temperatures was . In degrees, what was the low temperature in Lincoln that day?
Problem 2
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is steps by steps. Each of Mr. Green’s steps is 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?
Problem 3
When counting from to , is the number counted. When counting backwards from to , is the number counted. What is ?
Problem 4
Ray's car averages miles per gallon of gasoline, and Tom's car averages 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?
Problem 5
The average age of fifth-graders is . The average age of of their parents is . What is the average age of all of these parents and fifth-graders?
Problem 6
Real numbers and satisfy the equation . What is ?
Problem 7
Jo and Blair take turns counting from to one more than the last number said by the other person. Jo starts by saying , so Blair follows by saying . Jo then says , and so on. What is the number said?
Problem 8
Line has equation and goes through . Line has equation and meets line at point . Line has positive slope, goes through point , and meets at point . The area of is . What is the slope of ?
Problem 9
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides ?
Problem 10
Alex has red tokens and 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?
Problem 11
Two bees start at the same spot and fly at the same rate in the following directions. Bee travels foot north, then foot east, then foot upwards, and then continues to repeat this pattern. Bee travels foot south, then foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly feet away from each other?
east, west
north, south
north, west
up, south
up, west
Problem 12
Cities , , , , and are connected by roads , , , , , , and . How many different routes are there from to that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
Problem 13
The internal angles of quadrilateral form an arithmetic progression. Triangles and are similar with and . 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 ?
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 . What is the smallest possible value of ?
Problem 15
The number is expressed in the form
where and are positive integers and is as small as possible. What is ?
Problem 16
Let be an equiangular convex pentagon of perimeter . The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let be the perimeter of this star. What is the difference between the maximum and the minimum possible values of ?
Problem 17
Let and be real numbers such that
What is the difference between the maximum and minimum possible values of ?
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 or coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove or 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 coins and when the game starts with coins?
Barbara will win with coins and Jenna will win with coins.
Jenna will win with coins, and whoever goes first will win with coins.
Barbara will win with coins, and whoever goes second will win with coins.
Jenna will win with coins, and Barbara will win with coins.
Whoever goes first will win with coins, and whoever goes second will win with coins.
Problem 19
In triangle , , , and . Distinct points , , and lie on segments , , and , respectively, such that , , and . The length of segment can be written as , where and are relatively prime positive integers. What is ?
Problem 20
For , points and are the vertices of a trapezoid. What is ?
Problem 21
Consider the set of parabolas defined as follows: all parabolas have as focus the point and the directrix lines have the form with and integers such that and . No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Problem 22
Let and be integers. Suppose that the product of the solutions for of the equation is the smallest possible integer. What is ?
Problem 23
Bernardo chooses a three-digit positive integer and writes both its base- and base- representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base- integers, he adds them to obtain an integer . For example, if , Bernardo writes the numbers and , and LeRoy obtains the sum . For how many choices of are the two rightmost digits of , in order, the same as those of ?
Problem 24
Let be a triangle where is the midpoint of , and is the angle bisector of with on . Let be the intersection of the median and the bisector . In addition is equilateral with . What is ?
Problem 25
Let be the set of polynomials of the form where are integers and has distinct roots of the form with and integers. How many polynomials are in ?
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2013 AMC 12A Problems |
Followed by 2014 AMC 12A Problems |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.