¶ 2014 AMC 12A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2014 AMC 12A problems here.
Discussion Forum
Engage in discussion about the 2014 AMC 12A math contest by visiting Random Math AMC 12A 2014 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2014 AMC 12A problems, please refer below:
Problem 1: What is ?
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Problem 2: At the theater children get in for half price. The price for adult tickets and child tickets is . How much would adult tickets and child tickets cost?
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Problem 3: Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
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Problem 4: Suppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days?
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Problem 5: On an algebra quiz, of the students scored points, scored 80 points, scored points, and the rest scored points. What is the difference between the mean and the median of the students' scores on this quiz?
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Problem 6: The difference between a two-digit number and the number obtained by reversing its digits is times the sum of the digits of either number. What is the sum of the two-digit number and its reverse?
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Problem 7: The first three terms of a geometric progression are , and . What is the fourth term?
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Problem 8: A customer who intends to purchase an appliance has three coupons, only one of which may be used:
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Coupon : off the listed price if the listed price is at least
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Coupon : off the listed price if the listed price is at least
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Coupon : off the amount by which the listed price exceeds
For which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?
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Problem 9: Five positive consecutive integers starting with have average . What is the average of consecutive integers that start with ?
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Problem 10: Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
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Problem 11: David drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if he continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?
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Problem 12: Two circles intersect at points and . The minor arcs measure on one circle and on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
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Problem 13: A fancy bed and breakfast inn has rooms, each with a distinctive color-coded decor. One day friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than friends per room. In how many ways can the innkeeper assign the guests to the rooms?
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Problem 14: Let be three integers such that is an arithmetic progression and is a geometric progression. What is the smallest possible value for ?
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Problem 15: A five-digit palindrome is a positive integer with respective digits , where is not zero. Let be the sum of all five-digit palindromes. What is the sum of the digits of ?
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Problem 16: The product , where the second factor has digits, is an integer whose digits have a sum of 1000 . What is ?
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Problem 17: A rectangular box contains a sphere of radius and eight smaller spheres of radius . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is ?
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Problem 18: The domain of the function is an interval of length , where and are relatively prime positive integers. What is
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Problem 19: There are exactly distinct rational numbers such that and
has at least one integer solution for . What is ?
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Problem 20: In , and . Points and lie on and , respectively. What is the minimum possible value of ?
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Problem 21: For every real number , let denote the greatest integer not exceeding , and let
The set of all numbers such that and is a union of disjoint intervals. What is the sum of the lengths of those intervals?
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Problem 22: The number is between and . How many pairs of integers are there such that and
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Problem 23: The fraction
where is the length of the period of the repeating decimal expansion. What is the sum ?
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Problem 24: Let , and for , let . For how many values of is ?
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Problem 25: The parabola has focus and goes through the points and . For how many points with integer coordinates is it true that
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The problems on this page are the property of the MAA's American Mathematics Competitions