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2016 AMC 10B problems

All 25 problems from the 2016 AMC 10B, 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 dfrac 2a^-1+frac a^-12a when a= 1/2?Algebra
  2. 2Problem 2If nheartsuit m=n^3m^2, what is 2heartsuit 4/4heartsuit 2?Algebra
  3. 3Problem 3Let x=-2016. What is the value of Bigg vert Big vert |x|-xBig vert -|x|Bigg vert -x?Algebra
  4. 4Problem 4Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to…Algebra
  5. 5Problem 5The mean age of Amanda’s 4 cousins is 8, and their median age is 5. What is the sum of the ages of Amanda’s youngest and oldest cousins?Algebra
  6. 6Problem 6Laura added two three-digit positive integers. All six digits in these numbers are different. Laura’s sum is a three-digit number S. What is the…Algebra
  7. 7Problem 7The 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
  8. 8Problem 8What is the tens digit of 2015^2016-2017?Number Theory
  9. 9Problem 9All three vertices of bigtriangleup 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…Geometry
  10. 10Problem 10A 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…Geometry
  11. 11Problem 11Carl 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
  12. 12Problem 12Two different numbers are selected at random from {1, 2, 3, 4, 5} and multiplied together. What is the probability that the product is even?Counting & Probability
  13. 13Problem 13At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for 1000 of the babies…Algebra
  14. 14Problem 14How 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
  15. 15Problem 15All 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
  16. 16Problem 16The 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
  17. 17Problem 17All 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 product…Algebra
  18. 18Problem 18In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?Algebra
  19. 19Problem 19Rectangle ABCD has AB=5 and BC=4. Point E lies on AB so that EB=1, point G lies on BC so that CG=1, and point F lies on CD so that DF=2. Segments AG…Geometry
  20. 20Problem 20A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius 2 centered at A(2,2) to the circle of…Geometry
  21. 21Problem 21What is the area of the region enclosed by the graph of the equation x^2+y^2=|x|+|y|?Geometry
  22. 22Problem 22A 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
  23. 23Problem 23In regular hexagon ABCDEF, points W, X, Y, and Z are chosen on sides BC, CD, EF, and FA respectively, so lines AB, ZW, YX, and ED are parallel and…Geometry
  24. 24Problem 24How many four-digit positive integers abcd, with a ≠ 0, have the property that the three two-digit integers ab < bc < cd form an increasing…Algebra
  25. 25Problem 25Let f(x)=sum _k=2^10(⌊ kx ⌋ -k ⌊ x ⌋), where ⌊ r ⌋ denotes the greatest integer less than or equal to r. How many distinct values does f(x) assume…Algebra

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