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2015 AMC 10A problems

All 25 problems from the 2015 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? (2^0-1+5^2+0)^-1 × 5Algebra
  2. 2Problem 2A box contains a collection of triangular and square tiles. There are 25 tiles in the box, containing 84 edges total. How many square tiles are there…Algebra
  3. 3Problem 3Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase?Algebra
  4. 4Problem 4Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo…Algebra
  5. 5Problem 5Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone’s test except Payton’s, the average grade for…Algebra
  6. 6Problem 6The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller number?Algebra
  7. 7Problem 7How many terms are in the arithmetic sequence 13, 16, 19, dotsc, 70, 73?Algebra
  8. 8Problem 8Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will…Algebra
  9. 9Problem 9Two right circular cylinders have the same volume. The radius of the second cylinder is 10% more than the radius of the first. What is the…Geometry
  10. 10Problem 10How many rearrangements of abcd are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such…Counting & Probability
  11. 11Problem 11The ratio of the length to the width of a rectangle is 4:3. If the rectangle has diagonal of length d, then the area may be expressed as kd^2 for…Geometry
  12. 12Problem 12Points (√(π), a) and (√(π), b) are distinct points on the graph of y^2 + x^4 = 2x^2 y + 1. What is |a-b|?Algebra
  13. 13Problem 13Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of…Algebra
  14. 14Problem 14The diagram below shows the circular face of a clock with radius 20 cm and a circular disk with radius 10 cm externally tangent to the clock face at…Geometry
  15. 15Problem 15Consider the set of all fractions x/y, where x and y are relatively prime positive integers. How many of these fractions have the property that if…Algebra
  16. 16Problem 16If y+4 = (x-2)^2, x+4 = (y-2)^2, and x ≠ y, what is the value of x^2+y^2Algebra
  17. 17Problem 17A line that passes through the origin intersects both the line x = 1 and the line y=1+ √(3)/3 x. The three lines create an equilateral triangle. What…Geometry
  18. 18Problem 18Hexadecimal (base-16) numbers are written using numeric digits 0 through 9 as well as the letters A through F to represent 10 through 15. Among the…Number Theory
  19. 19Problem 19The isosceles right triangle ABC has right angle at C and area 12.5. The rays trisecting ∠ ACB intersect AB at D and E. What is the area of △ CDE?Geometry
  20. 20Problem 20A rectangle with positive integer side lengths in cm has area A cm^2 and perimeter P cm. Which of the following numbers cannot equal A+P?Geometry
  21. 21Problem 21Tetrahedron ABCD has AB=5, AC=3, BC=4, BD=4, AD=3, and CD=tfrac 125sqrt 2. What is the volume of the tetrahedron?Geometry
  22. 22Problem 22Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while…Counting & Probability
  23. 23Problem 23The zeros of the function f(x)=x^2-ax+2a are integers. What is the sum of the possible values of a?Algebra
  24. 24Problem 24For some positive integers p, there is a quadrilateral ABCD with positive integer side lengths, perimeter p, right angles at B and C, AB=2, and…Geometry
  25. 25Problem 25Let S be a square of side length 1. Two points are chosen independently at random on the sides of S. The probability that the straight-line distance…Counting & Probability

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