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2014 AMC 12A problems

All 25 problems from the 2014 AMC 12A, with answer choices, worked solutions and hints. Problems are roughly ordered by difficulty: 1–10 are the most approachable, 19–25 the hardest.

Problems

  1. 1Problem 1What is 10 · (1/2 + 1/5 + 1/10)^-1?Algebra
  2. 2Problem 2At the theater children get in for half price. The price for 5 adult tickets and 4 child tickets is $24.50. How much would 8 adult tickets and 6…Algebra
  3. 3Problem 3Walking 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…Counting & Probability
  4. 4Problem 4Suppose that a cows give b gallons of milk in c days. At this rate, how many gallons of milk will d cows give in e days?Algebra
  5. 5Problem 5On an algebra quiz, 10% of the students scored 70 points, 35% scored 80 points, 30% scored 90 points, and the rest scored 100 points. What is the…Algebra
  6. 6Problem 6The difference between a two-digit number and the number obtained by reversing its digits is 5 times the sum of the digits of either number. What is…Number Theory
  7. 7Problem 7The first three terms of a geometric progression are √(3), √(3), and √(3). What is the fourth term?Algebra
  8. 8Problem 8A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: 10% off the listed price if the listed…Algebra
  9. 9Problem 9Five positive consecutive integers starting with a have average b. What is the average of 5 consecutive integers that start with b?Algebra
  10. 10Problem 10Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length 1. The sum of the areas…Geometry
  11. 11Problem 11David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he…Algebra
  12. 12Problem 12Two circles intersect at points A and B. The minor arcs AB measure 30° on one circle and 60° on the other circle. What is the ratio of the area of…Geometry
  13. 13Problem 13A fancy bed and breakfast inn has 5 rooms, each with a distinctive color-coded decor. One day 5 friends arrive to spend the night. There are no other…Counting & Probability
  14. 14Problem 14Let a< b< c be three integers such that a, b, c is an arithmetic progression and a, c, b is a geometric progression. What is the smallest possible…Algebra
  15. 15Problem 15A five-digit palindrome is a positive integer with respective digits abcba, where a is not zero. Let S be the sum of all five-digit palindromes. What…Number Theory
  16. 16Problem 16The product (8)(888ldots 8), where the second factor has k digits, is an integer whose digits have a sum of 1000. What is k?Number Theory
  17. 17Problem 17A 4times 4× h rectangular box contains a sphere of radius 2 and eight smaller spheres of radius 1. The smaller spheres are each tangent to three…Geometry
  18. 18Problem 18The domain of the function tiny f(x)=log _1/2 (log _4 (log _1/4 (log _16 (log _1/16x)))) is an interval of length m/n, where m and n are relatively…Algebra
  19. 19Problem 19There are exactly N distinct rational numbers k such that |k|lt 200 and 5x^2+kx+12=0 has at least one integer solution for x. What is N?Algebra
  20. 20Problem 20In △ BAC, ∠ BAC=40°, AB=10, and AC=6. Points D and E lie on AB and AC, respectively. What is the minimum possible value of BE+DE+CD?Geometry
  21. 21Problem 21For every real number x, let ⌊ x⌋ denote the greatest integer not exceeding x, and let f(x)=⌊ x⌋(2014^ x-⌊ x⌋-1). The set of all numbers x such that…Algebra
  22. 22Problem 22The number 5^867 is between 2^2013 and 2^2014. How many pairs of integers (m,n) are there such that 1≤ mle 2012 and 5^nlt 2^mlt 2^m+2lt 5^n+1?Algebra
  23. 23Problem 23The fraction 1/99^2=0.overline b_n-1b_n-2… b_2b_1b_0, where n is the length of the period of the repeating decimal expansion. What is the sum…Number Theory
  24. 24Problem 24Let f_0(x)=x+|x-100| -|x+100|, and for nge 1, let f_n(x)=|f_n-1(x)|-1. For how many values of x is f_100(x)=0?Algebra
  25. 25Problem 25The parabola P has focus (0,0) and goes through the points (4,3) and (-4,-3). For how many points (x,y)∈ P with integer coordinates is it true that…Geometry

Practise the same ideas across every year on the topic pages, or browse the full AMC 12 archive.