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2019 AMC 10B problems

All 25 problems from the 2019 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 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 3In a high school with 500 students, 40% of the seniors play a musical instrument, while 30% of the non-seniors do not play a musical instrument. In…Algebra
  4. 4Problem 4All lines with equation ax+by=c such that a, b, c form an arithmetic progression pass through a common point. What are the coordinates of that point?Algebra
  5. 5Problem 5Triangle ABC lies in the first quadrant. Points A, B, and C are reflected across the line y=x to points A', B', and C', respectively. Assume that…Geometry
  6. 6Problem 6A positive integer n satisfies the equation (n+1)! + (n+2)! = n! · 440. What is the sum of the digits of n?Algebra
  7. 7Problem 7Each piece of candy in a shop costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green…Number Theory
  8. 8Problem 8The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each…Geometry
  9. 9Problem 9The function f is defined by f(x) = ⌊|x|⌋ - |⌊ x ⌋| for all real numbers x, where ⌊ r ⌋ denotes the greatest integer less than or equal to the real…Algebra
  10. 10Problem 10In 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…Geometry
  11. 11Problem 11Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 9:1, and…Algebra
  12. 12Problem 12What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than 2019?Number Theory
  13. 13Problem 13What 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
  14. 14Problem 14The base-ten representation for 19! is 121, 6T5, 100, 40M, 832, H00, where T, M, and H denote digits that are not given. What is T+M+H?Number Theory
  15. 15Problem 15Right triangles T_1 and T_2 have areas 1 and 2, respectively. A side of T_1 is congruent to a side of T_2, and a different side of T_1 is congruent…Geometry
  16. 16Problem 16In △ ABC with a right angle at C, point D lies in the interior of AB and point E lies in the interior of BC so that AC=CD, DE=EB, and the ratio…Geometry
  17. 17Problem 17A 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
  18. 18Problem 18Henry decides one morning to do a workout, and he walks 3/4 of the way from his home to his gym. The gym is 2 kilometers away from Henry’s home. At…Algebra
  19. 19Problem 19Let 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
  20. 20Problem 20As 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, widehat AEB, widehat BFC…Geometry
  21. 21Problem 21Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a…Algebra
  22. 22Problem 22Raashan, 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
  23. 23Problem 23Points A=(6,13) and B=(12,11) lie on a 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…Geometry
  24. 24Problem 24Define 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
  25. 25Problem 25How 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

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