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2016 AMC 12A problems

All 25 problems from the 2016 AMC 12A, 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 3The remainder function can be defined for all real numbers x and y with y≠ 0 by rem(x,y)=x-y⌊ x/y⌋, where ⌊ x/y⌋ denotes the greatest integer less…Algebra
  4. 4Problem 4The 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
  5. 5Problem 5Goldbach’s conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, 2016=13+2003). So…
  6. 6Problem 6A 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
  7. 7Problem 7Which of these describes the graph of x^2(x+y+1)=y^2(x+y+1)?Algebra
  8. 8Problem 8What is the area of the shaded region of the given 8× 5 rectangle?Geometry
  9. 9Problem 9The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square…Algebra
  10. 10Problem 10Five 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…
  11. 11Problem 11Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all…Counting & Probability
  12. 12Problem 12In △ ABC, AB=6, BC=7, and CA=8. Point D lies on BC, and AD bisects ∠ BAC. Point E lies on AC, and BE bisects ∠ ABC. The bisectors intersect at F…Algebra
  13. 13Problem 13Let 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
  14. 14Problem 14Each vertex of a cube is to be labeled with an integer from 1 through 8, with each integer being used once, in such a way that the sum of the four…Geometry
  15. 15Problem 15Circles 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
  16. 16Problem 16The graphs of y=log _3 x, y=log _x 3, y=log _1/3 x, and y=log _x1/3 are plotted on the same set of axes. How many points in the plane with positive…Algebra
  17. 17Problem 17Let ABCD be a square. Let E, F, G, and H be the centers, respectively, of equilateral triangles with bases AB, BC, CD, and DA, each exterior to the…Geometry
  18. 18Problem 18For 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
  19. 19Problem 19Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets…Counting & Probability
  20. 20Problem 20A 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 · represents…Algebra
  21. 21Problem 21A 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 its…Geometry
  22. 22Problem 22How 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
  23. 23Problem 23Three numbers in the interval [0,1] are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a…Geometry
  24. 24Problem 24There is a smallest positive real number a such that there exists a positive real number b such that all the roots of the polynomial x^3-ax^2+bx-a…Algebra
  25. 25Problem 25Let k be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the…Number Theory

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