Skip to main content

2016 AMC 12B problems

All 25 problems from the 2016 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 1What is the value of frac 2a^-1+frac a^-12a when a=tfrac 12?Algebra
  2. 2Problem 2The harmonic mean of two numbers can be computed as twice their product divided by their sum. The harmonic mean of 1 and 2016 is closest to which…Algebra
  3. 3Problem 3Let x=-2016. What is the value of Big | big | |x|-x big |-|x| Big |-x?Algebra
  4. 4Problem 4The ratio of the measures of two acute angles is 5:4, and the complement of one of these two angles is twice as large as the complement of the other…Algebra
  5. 5Problem 5The War of 1812 started with a declaration of war on Thursday, June 18, 1812. The peace treaty to end the war was signed 919 days later, on December…Algebra
  6. 6Problem 6All three vertices of △ ABC lie on the parabola defined by y=x^2, with A at the origin and BC parallel to the x-axis. The area of the triangle is 64…Geometry
  7. 7Problem 7Josh writes the numbers 1, 2, 3, …, 99, 100. He marks out 1, skips the next number (2), marks out 3, and continues skipping and marking out the next…Number Theory
  8. 8Problem 8A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second piece of the…Algebra
  9. 9Problem 9Carl decided to fence in his rectangular garden. He bought 20 fence posts, placed one on each of the four corners, and spaced out the rest evenly…Geometry
  10. 10Problem 10A quadrilateral has vertices P(a,b), Q(b,a), R(-a,-b), and S(-b,-a), where a and b are integers with a> bgt 0. The area of PQRS is 16. What is a+b?Geometry
  11. 11Problem 11How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by…Geometry
  12. 12Problem 12All the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a 3times 3 array of squares, one number in each square, in such a way that if two numbers…Number Theory
  13. 13Problem 13Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his…Geometry
  14. 14Problem 14The sum of an infinite geometric series is a positive number S, and the second term in the series is 1. What is the smallest possible value of S?Algebra
  15. 15Problem 15All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a…Algebra
  16. 16Problem 16In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?Algebra
  17. 17Problem 17In △ ABC shown in the figure, AB=7, BC=8, CA=9, and AH is an altitude. Points D and E lie on sides AC and AB, respectively, so that BD and CE are…Geometry
  18. 18Problem 18What is the area of the region enclosed by the graph of the equation x^2+y^2=|x|+|y|?Algebra
  19. 19Problem 19Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which…Counting & Probability
  20. 20Problem 20A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won 10 games and lost 10 games…Counting & Probability
  21. 21Problem 21Let ABCD be a unit square. Let Q_1 be the midpoint of CD. For i=1, 2, …, let P_i be the intersection of AQ_i and BD, and let Q_i+1 be the foot of the…Algebra
  22. 22Problem 22For a certain positive integer n less than 1000, the decimal equivalent of dfrac 1n is 0.abcdef, a repeating decimal of period 6, and the decimal…Number Theory
  23. 23Problem 23What is the volume of the region in three-dimensional space defined by the inequalities |x|+|y|+|z|le 1 and |x|+|y|+|z-1|le 1?Geometry
  24. 24Problem 24There are exactly 77,000 ordered quadruples (a,b,c,d) such that gcd (a,b,c,d)=77 and lcm(a,b,c,d)=n. What is the smallest possible value of n?Number Theory
  25. 25Problem 25The sequence (a_n) is defined recursively by a_0=1, a_1=√(2), and a_n=a_n-1a_n-2^2 for nge 2. What is the smallest positive integer k such that the…Number Theory

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