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2010 AMC 10B problems

All 25 problems from the 2010 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 the following expression? 100(100-3)-(100 · 100-3)Algebra
  2. 2Problem 2Makayla 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
  3. 3Problem 3A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the…Counting & Probability
  4. 4Problem 4For a real number x, define heartsuit (x) to be the average of x and x^2. What is the value of the following expression? heartsuit (1)+heartsuit…Algebra
  5. 5Problem 5A 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
  6. 6Problem 6A circle is centered at O, AB is a diameter and C is a point on the circle with ∠ COB = 50°. What is the degree measure of ∠ CAB?Geometry
  7. 7Problem 7A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this…Geometry
  8. 8Problem 8A 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
  9. 9Problem 9Lucky 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
  10. 10Problem 10Shelby 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
  11. 11Problem 11A shopper plans to purchase an item that has a listed price greater than $100 and can use any one of the three coupons. Coupon A gives 15% off the…Algebra
  12. 12Problem 12At 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
  13. 13Problem 13What is the sum of all the solutions of the equation below? x = |2x-|60-2x||Algebra
  14. 14Problem 14The average of the numbers 1, 2, 3, …, 98, 99, and x is 100x. What is x?Algebra
  15. 15Problem 15On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an…Algebra
  16. 16Problem 16A square of side length 1 and a circle of radius √(3)/3 share the same center. What is the area inside the circle, but outside the square?Geometry
  17. 17Problem 17Every 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
  18. 18Problem 18Positive 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
  19. 19Problem 19A circle with center O has area 156π. Triangle ABC is equilateral, BC is a chord on the circle, OA = 4√(3), and point O is outside △ ABC. What is the…Geometry
  20. 20Problem 20Two circles lie outside regular hexagon ABCDEF. The first is tangent to AB, and the second is tangent to DE. Both are tangent to lines BC and FA…Geometry
  21. 21Problem 21A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?Number Theory
  22. 22Problem 22Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy…Counting & Probability
  23. 23Problem 23The entries in a 3 × 3 array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order…Counting & Probability
  24. 24Problem 24A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders…Algebra
  25. 25Problem 25Let a > 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…Number Theory

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