Skip to main content

2016 AMC 10A problems

All 25 problems from the 2016 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 11!-10!/9!?Algebra
  2. 2Problem 2For what value of x does 10^x · 100^2x=1000^5?Algebra
  3. 3Problem 3For every dollar Ben spent on bagels, David spent 25 cents less. Ben paid $12.50 more than David. How much did they spend in the bagel store together?Algebra
  4. 4Problem 4The remainder function can be defined for all real numbers x and y with y ≠ 0 by rem(x,y)=x-ylfloorx /y⌋, where ⌊ x/y ⌋ denotes the greatest integer…Algebra
  5. 5Problem 5A rectangular box has integer side lengths in the ratio 1:3:4. Which of the following could be the volume of the box?Algebra
  6. 6Problem 6Ximena lists the whole numbers 1 through 30 once. Emilio copies Ximena’s numbers, replacing each occurrence of the digit 2 by the digit 1. Ximena…Number Theory
  7. 7Problem 7The mean, median, and mode of the 7 data values 60, 100, x, 40, 50, 200, 90 are all equal to x. What is the value of x?Algebra
  8. 8Problem 8Trickster Rabbit agrees with Foolish Fox to double Fox’s money every time Fox crosses the bridge by Rabbit’s house, as long as Fox pays 40 coins in…
  9. 9Problem 9A triangular array of 2016 coins has 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to N coins in the Nth…Algebra
  10. 10Problem 10A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner…Algebra
  11. 11Problem 11What is the area of the shaded region of the given 8times 5 rectangle?Geometry
  12. 12Problem 12Three distinct integers are selected at random between 1 and 2016, inclusive. Which of the following is a correct statement about the probability p…Counting & Probability
  13. 13Problem 13Five friends sat in a movie theater in a row containing 5 seats, numbered 1 to 5 from left to right. (The directions “left” and “right” are from the…
  14. 14Problem 14How many ways are there to write 2016 as the sum of twos and threes, ignoring order? (For example, 1008· 2 + 0· 3 and 402· 2 + 404· 3 are two such…Number Theory
  15. 15Problem 15Seven cookies of radius 1 inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are…Geometry
  16. 16Problem 16A triangle with vertices A(0, 2), B(-3, 2), and C(-3, 0) is reflected about the x-axis, then the image △ A'B'C' is rotated counterclockwise about the…Geometry
  17. 17Problem 17Let N be a positive multiple of 5. One red ball and N green balls are arranged in a line in random order. Let P(N) be the probability that at least…Algebra
  18. 18Problem 18Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers…Geometry
  19. 19Problem 19In rectangle ABCD, AB=6 and BC=3. Point E between B and C, and point F between E and C are such that BE=EF=FC. Segments AE and AF intersect BD at P…Geometry
  20. 20Problem 20For some particular value of N, when (a+b+c+d+1)^N is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that…Counting & Probability
  21. 21Problem 21Circles with centers P, Q and R, having radii 1, 2 and 3, respectively, lie on the same side of line l and are tangent to l at P', Q' and R'…Geometry
  22. 22Problem 22For some positive integer n, the number 110n^3 has 110 positive integer divisors, including 1 and the number 110n^3. How many positive integer…Number Theory
  23. 23Problem 23A binary operation ♦ has the properties that a ♦ (b ♦ c) = (a ♦ b)· c and that a ♦ a=1 for all nonzero real numbers a, b, and c. (Here the dot ·…Algebra
  24. 24Problem 24A quadrilateral is inscribed in a circle of radius 200√(2). Three of the sides of this quadrilateral have length 200. What is the length of the…Geometry
  25. 25Problem 25How many ordered triples (x,y,z) of positive integers satisfy lcm(x,y) = 72,lcm(x,z) = 600, and lcm(y,z)=900?Number Theory

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