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2010 AMC 12B problems

All 25 problems from the 2010 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 1Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent…Algebra
  2. 2Problem 2A big L is formed as shown. What is its area?Geometry
  3. 3Problem 3A ticket to a school play costs x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th…Number Theory
  4. 4Problem 4A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?Algebra
  5. 5Problem 5Lucky Larry’s teacher asked him to substitute numbers for a, b, c, d, and e in the expression a-(b-(c-(d+e))) and evaluate the result. Larry ignored…Algebra
  6. 6Problem 6At the beginning of the school year, 50% of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and 50% answered…Algebra
  7. 7Problem 7Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in…Algebra
  8. 8Problem 8Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score…Algebra
  9. 9Problem 9Let n be the smallest positive integer such that n is divisible by 20, n^2 is a perfect cube, and n^3 is a perfect square. What is the number of…Number Theory
  10. 10Problem 10The average of the numbers 1, 2, 3, …, 98, 99, and x is 100x. What is x?Algebra
  11. 11Problem 11A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?Number Theory
  12. 12Problem 12For what value of x does log _sqrt 2√(x)+log _2 x +log _4(x^2)+log _8(x^3) +log _16(x^4)=40?Algebra
  13. 13Problem 13In △ ABC, cos (2A-B)+sin (A+B)=2 and AB=4. What is BC?Geometry
  14. 14Problem 14Let a, b, c, d, and e be positive integers with a+b+c+d+e=2010, and let M be the largest of the sums a+b, b+c, c+d, and d+e. What is the smallest…Algebra
  15. 15Problem 15For how many ordered triples (x, y, z) of nonnegative integers less than 20 are there exactly two distinct elements in the set {i^x, (1+i)^y, z}…Algebra
  16. 16Problem 16Positive integers a, b, and c are randomly and independently selected with replacement from the set {1, 2, 3, …, 2010}. What is the probability that…Number Theory
  17. 17Problem 17The entries in a 3times 3 array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing…Counting & Probability
  18. 18Problem 18A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently and at random. What is the probability that the…Counting & Probability
  19. 19Problem 19A high school basketball game between the Raiders and the Wildcats was tied at the end of the first quarter. The number of points scored by the…Algebra
  20. 20Problem 20A geometric sequence (a_n) has a_1=sin x, a_2=cos x, and a_3=tan x for some real number x. For what value of n does a_n=1+cos x?Algebra
  21. 21Problem 21Let agt 0, and let P(x) be a polynomial with integer coefficients such that P(1)=P(3)=P(5) =P(7)=a, and P(2)=P(4)=P(6) =P(8)=-a. What is the smallest…Number Theory
  22. 22Problem 22Let ABCD be a cyclic quadrilateral. The side lengths of ABCD are distinct integers less than 15 such that BC· CD=AB· DA. What is the largest possible…Geometry
  23. 23Problem 23Monic quadratic polynomials P(x) and Q(x) have the property that P(Q(x)) has zeros at x=-23, -21, -17, and -15, and Q(P(x)) has zeros at x=-59, -57…Algebra
  24. 24Problem 24The set of real numbers x for which 1/x-2009+1/x-2010 +1/x-2011ge 1 is the union of intervals of the form a< x≤ b. What is the sum of the lengths of…Algebra
  25. 25Problem 25For every integer nge 2, let pow(n) be the largest power of the largest prime that divides n. For example, pow(144)=pow(2^4cdot 3^2)=3^2. What is the…Number Theory

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