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2014 AMC 10A problems

All 25 problems from the 2014 AMC 10A, 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 the value of the following expression? 10·(1/2+1/5+1/10)^-1Algebra
  2. 2Problem 2Roy’s cat eats 1/3 of a can of cat food every morning and 1/4 of a can of cat food every evening. Before feeding his cat on Monday morning, Roy…Algebra
  3. 3Problem 3Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for $ 2.50 each. In the afternoon she sells two thirds of what…Algebra
  4. 4Problem 4Walking 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
  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 6Suppose 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
  7. 7Problem 7Nonzero real numbers x, y, a, and b satisfy x < a and y < b. How many of the following inequalities must be true? (I) x + y < a + b (II) x - y < a -…Algebra
  8. 8Problem 8Which of the following numbers is a perfect square?Number Theory
  9. 9Problem 9The two legs of a right triangle, which are altitudes, have lengths 2sqrt 3 and 6. How long is the third altitude of the triangle?Geometry
  10. 10Problem 10Five positive consecutive integers starting with a have average b. What is the average of 5 consecutive integers that start with b?Algebra
  11. 11Problem 11A 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
  12. 12Problem 12A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as…Geometry
  13. 13Problem 13Equilateral △ ABC has side length 1, and squares ABDE, BCHI, CAFG lie outside the triangle. What is the area of hexagon DEFGHI?Geometry
  14. 14Problem 14The y-intercepts, P and Q, of two perpendicular lines intersecting at the point A(6,8) have a sum of zero. What is the area of △ APQ?Geometry
  15. 15Problem 15David 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
  16. 16Problem 16In rectangle ABCD, AB=1, BC=2, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of GE. What is the area…Geometry
  17. 17Problem 17Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?Counting & Probability
  18. 18Problem 18A square in the coordinate plane has vertices whose y-coordinates are 0, 1, 4, and 5. What is the area of the square?Geometry
  19. 19Problem 19Four cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portion of XY contained in the cube with edge length 3?Geometry
  20. 20Problem 20The product (8)(888dots 8), where the second factor has k digits, is an integer whose digits have a sum of 1000. What is k?Number Theory
  21. 21Problem 21Positive integers a and b are such that the graphs of y=ax+5 and y=3x+b intersect the x-axis at the same point. What is the sum of all possible…Algebra
  22. 22Problem 22In rectangle ABCD, AB=20 and BC=10. Let E be a point on CD such that ∠ CBE=15°. What is AE?Geometry
  23. 23Problem 23A rectangular piece of paper whose length is sqrt 3 times the width has area A. The paper is divided into three equal sections along the opposite…Geometry
  24. 24Problem 24A sequence of natural numbers is constructed by listing the first 4, then skipping one, listing the next 5, skipping 2, listing 6, skipping 3, and on…Algebra
  25. 25Problem 25The number 5^867 is between 2^2013 and 2^2014. How many pairs of integers (m,n) are there such that 1≤ m≤ 2012 and 5^n < 2^m < 2^m+2 < 5^n+1?Algebra

Practise the same ideas across every year on the topic pages, or browse the full AMC 10 archive.