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2018 AMC 10A problems

All 25 problems from the 2018 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 (((2+1)^-1+1)^-1+1)^-1+1?Algebra
  2. 2Problem 2Liliane has 50% more soda than Jacqueline, and Alice has 25% more soda than Jacqueline. What is the relationship between the amounts of soda that…Algebra
  3. 3Problem 3A unit of blood expires after 10! = 10 · 9 · 8 … 1 seconds. Yasin donates a unit of blood at noon on January 1. On what day does his unit of blood…Algebra
  4. 4Problem 4How many ways can a student schedule 3 mathematics courses — algebra, geometry, and number theory — in a 6-period day if no two mathematics courses…Counting & Probability
  5. 5Problem 5Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, “We are at least 6 miles away,” Bob…Algebra
  6. 6Problem 6Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score…Algebra
  7. 7Problem 7For how many (not necessarily positive) integer values of n is the following value an integer? 4000 · (2/5)^nAlgebra
  8. 8Problem 8Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and…Algebra
  9. 9Problem 9All of the triangles in the diagram below are similar to isosceles triangle ABC, in which AB=AC. Each of the 7 smallest triangles has area 1, and △…Geometry
  10. 10Problem 10Suppose that real number x satisfies √(49-x^2)-√(25-x^2)=3. What is the value of √(49-x^2)+√(25-x^2)?Algebra
  11. 11Problem 11When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as dfrac n6^7, where…Counting & Probability
  12. 12Problem 12How many ordered pairs of real numbers (x,y) satisfy the following system of equations? ~x+3y =3 ~big ||x|-|y|big | =1Algebra
  13. 13Problem 13A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point A falls on point B. What is the length in inches of the…Geometry
  14. 14Problem 14What is the greatest integer less than or equal to dfrac 3^100+2^1003^96+2^96?Algebra
  15. 15Problem 15Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points A and B, as shown in the…Geometry
  16. 16Problem 16Right triangle ABC has leg lengths AB=20 and BC=21. Including AB and BC, how many line segments with integer length can be drawn from vertex B to a…Geometry
  17. 17Problem 17Let S be a set of 6 integers taken from {1,2,…,12} with the property that if a and b are elements of S with a < b, then b is not a multiple of a…Algebra
  18. 18Problem 18How many nonnegative integers can be written in the form a_7cdot 3^7+a_6cdot 3^6+a_5cdot 3^5+a_4cdot 3^4+a_3cdot 3^3+a_2cdot 3^2+a_1cdot 3^1+a_0cdot…Number Theory
  19. 19Problem 19A number m is randomly selected from the set {11,13,15,17,19}, and a number n is randomly selected from {1999,2000,2001,…,2018}. What is the…Number Theory
  20. 20Problem 20A scanning code consists of a 7 × 7 grid of squares, with some of its squares colored black and the rest colored white. There must be at least one…Counting & Probability
  21. 21Problem 21Which of the following describes the set of values of a for which the curves x^2+y^2=a^2 and y=x^2-a in the real xy-plane intersect at exactly 3…Geometry
  22. 22Problem 22Let a, b, c, and d be positive integers such that gcd (a, b)=24, gcd (b, c)=36, gcd (c, d)=54, and 70 < gcd (d, a) < 100. Which of the following must…Number Theory
  23. 23Problem 23Farmer Pythagoras has a field in the shape of a right triangle. The right triangle’s legs have lengths 3 and 4 units. In the corner where those sides…Geometry
  24. 24Problem 24Triangle ABC with AB=50 and AC=10 has area 120. Let D be the midpoint of AB, and let E be the midpoint of AC. The angle bisector of ∠ BAC intersects…Geometry
  25. 25Problem 25For a positive integer n and nonzero digits a, b, and c, let A_n be the n-digit integer each of whose digits is equal to a; let B_n be the n-digit…Algebra

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