Skip to main content

2019 AMC 12B problems

All 25 problems from the 2019 AMC 12B, 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 1Alicia had two containers. The first was 5/6 full of water and the second was empty. She poured all the water from the first container into the…Algebra
  2. 2Problem 2Consider the statement, “If n is not prime, then n-2 is prime.” Which of the following values of n is a counterexample to this statement?Number Theory
  3. 3Problem 3Which one of the following rigid transformations (isometries) maps the line segment AB onto the line segment A'B' so that the image of A(-2,1) is…Geometry
  4. 4Problem 4A positive integer n satisfies the equation (n+1)!+(n+2)!=440· n!. What is the sum of the digits of n?Algebra
  5. 5Problem 5Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of…Number Theory
  6. 6Problem 6In a given plane, points A and B are 10 units apart. How many points C are there in the plane such that the perimeter of △ ABC is 50 units and the…Algebra
  7. 7Problem 7What is the sum of all real numbers x for which the median of the numbers 4, 6, 8, 17, and x is equal to the mean of those five numbers?Algebra
  8. 8Problem 8Let f(x)=x^2(1-x)^2. What is the value of the sum f (1/2019)-f (2/2019) +f (3/2019)-f (4/2019) +…+f (2017/2019) -f (2018/2019)?Counting & Probability
  9. 9Problem 9For how many integral values of x can a triangle of positive area be formed having side lengths log _2 x, log _4 x, and 3?Algebra
  10. 10Problem 10The figure below is a map showing 12 cities and 17 roads connecting certain pairs of cities. Paula wishes to travel along exactly 13 of those roads…Number Theory
  11. 11Problem 11How many unordered pairs of edges of a given cube determine a plane?Geometry
  12. 12Problem 12Right triangle ACD with right angle at C is constructed outwards on the hypotenuse AC of isosceles right triangle ABC with leg length 1, as shown, so…Geometry
  13. 13Problem 13A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the…Counting & Probability
  14. 14Problem 14Let S be the set of all positive integer divisors of 100,000. How many numbers are the product of two distinct elements of S?Number Theory
  15. 15Problem 15As shown in the figure, line segment AD is trisected by points B and C so that AB=BC=CD=2. Three semicircles of radius 1, AEB, BFC, and CGD, have…Geometry
  16. 16Problem 16There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona…Counting & Probability
  17. 17Problem 17How many nonzero complex numbers z have the property that 0, z, and z^3, when represented by points in the complex plane, are the three distinct…Algebra
  18. 18Problem 18Square pyramid ABCDE has base ABCD, which measures 3 cm on a side, and altitude AE perpendicular to the base, which measures 6 cm. Point P lies on…Geometry
  19. 19Problem 19Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every 15 seconds, at which time each of the players who currently…Counting & Probability
  20. 20Problem 20Points A(6,13) and B(12,11) lie on circle ω in the plane. Suppose that the tangent lines to ω at A and B intersect at a point on the x-axis. What is…Geometry
  21. 21Problem 21How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the…Algebra
  22. 22Problem 22Define a sequence recursively by x_0=5 and x_n+1=x_n^2+5x_n+4/x_n+6 for all nonnegative integers n. Let m be the least positive integer such that…Algebra
  23. 23Problem 23How many sequences of 0s and 1s of length 19 are there that begin with a 0, end with a 0, contain no two consecutive 0s, and contain no three…Counting & Probability
  24. 24Problem 24Let ω=-dfrac 12+dfrac 12 isqrt 3. Let S denote all points in the complex plane of the form a+bω+cω^2, where 0≤ ale 1, 0≤ ble 1, and 0≤ cle 1. What is…Geometry
  25. 25Problem 25Let ABCD be a convex quadrilateral with BC=2 and CD=6. Suppose that the centroids of △ ABC, △ BCD, and △ ACD form the vertices of an equilateral…Geometry

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