¶ 2013 AMC 12B Problems and Solutions
Problem Set Workbook
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Discussion Forum
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Individual Problems and Solutions
For problems and detailed solutions to each of the 2013 AMC 12B problems, please refer below:
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?
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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?
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Problem 3: When counting from to is the number counted. When counting backwards from to is the number counted. What is ?
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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?
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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?
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Problem 6: Real numbers and satisfy the equation . What is ?
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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?
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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 3 . What is the slope of ?
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Problem 9: What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides ?
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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?
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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?
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A. east, west
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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.)
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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 ?
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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 ?
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Problem 15: The number is expressed in the form
where and are positive integers and is as small as possible. What is ?
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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 fivepointed star polygon. Let be the perimeter of this star. What is the difference between the maximum and the minimum possible values of ?
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Problem 17: Let , and be real numbers such that
What is the difference between the maximum and minimum possible values of ?
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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?
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A. Barbara will win with coins, and Jenna will win with coins.
B. Jenna will win with coins, and whoever goes first will win with coins.
C. Barbara will win with coins, and whoever goes second will win with coins.
D. Jenna will win with coins, and Barbara will win with coins.
E. Whoever goes first will win with coins, and whoever goes second will win with coins.
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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 ?
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Problem 20: For , points , and are the vertices of a trapezoid. What is
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Problem 21: Consider the set of 30 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?
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Problem 22: Let and be integers. Suppose that the product of the solutions for of the equation
is the smallest possible integer. What is ?
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Problem 23: Bernardo chooses a three-digit positive integer 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- 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 ?
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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 and . What is ?
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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 ?
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The problems on this page are the property of the MAA's American Mathematics Competitions