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2013 AMC 10B problems

All 25 problems from the 2013 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? 2+4+6/1+3+5 - 1+3+5/2+4+6Algebra
  2. 2Problem 2Mr. Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr. Green’s steps is 2 feet…Algebra
  3. 3Problem 3On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high…Algebra
  4. 4Problem 4When counting from 3 to 201, the number 53 is in position 51. When counting backward from 201 to 3, the number 53 is in position n. What is n?Counting & Probability
  5. 5Problem 5Positive integers a and b are each less than 6. What is the smallest possible value of the following expression? 2 · a - a · bAlgebra
  6. 6Problem 6The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and…Algebra
  7. 7Problem 7Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor…Geometry
  8. 8Problem 8Ray’s car averages 40 miles per gallon of gasoline, and Tom’s car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of…Algebra
  9. 9Problem 9Three positive integers are each greater than 1, have a product of 27000, and are pairwise relatively prime. What is their sum?Number Theory
  10. 10Problem 10A basketball team’s players were successful on 50% of their two-point shots and 40% of their three-point shots, which resulted in 54 points. They…Algebra
  11. 11Problem 11Real numbers x and y satisfy the equation x^2+y^2=10x-6y-34. What is x+y?Algebra
  12. 12Problem 12Let S be the set of sides and diagonals of a regular pentagon. A pair of elements of S are selected at random without replacement. What is the…Counting & Probability
  13. 13Problem 13Jo and Blair take turns counting from 1 to one more than the last number said by the other person. Jo starts by saying “1”, so Blair follows by…Algebra
  14. 14Problem 14Define aclubsuit b=a^2b-ab^2. Which of the following describes the set of points (x, y) for which xclubsuit y=yclubsuit x?Algebra
  15. 15Problem 15A wire is cut into two pieces, one of length a and the other of length b. The piece of length a is bent to form an equilateral triangle, and the…Geometry
  16. 16Problem 16In triangle △ ABC, medians AD and CE intersect at P, PE=1.5, PD=2, and DE=2.5. What is the area of AEDC?Geometry
  17. 17Problem 17Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token…Number Theory
  18. 18Problem 18The number 2013 has the property that its units digit is the sum of its other digits, that is 2+0+1=3. How many integers less than 2013 but greater…Algebra
  19. 19Problem 19The real numbers c, b, a form an arithmetic sequence with a ≥ b ≥ c ≥ 0. The quadratic ax^2+bx+c has exactly one root. What is this root?Algebra
  20. 20Problem 20The number 2013 is expressed in the form 2013 = a_1!a_2!… a_m!/b_1!b_2!… b_n!, where a_1 ≥ a_2 ≥ … ≥ a_m and b_1 ≥ b_2 ≥ … ≥ b_n are positive…Number Theory
  21. 21Problem 21Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the…Algebra
  22. 22Problem 22The regular octagon ABCDEFGH has its center at J. Each of the vertices and the center are to be associated with one of the digits 1 through 9, with…Counting & Probability
  23. 23Problem 23In triangle △ ABC, AB=13, BC=14, and CA=15. Distinct points D, E, and F lie on segments BC, CA, and DE, respectively, such that ADperpBC, DEperpAC…Geometry
  24. 24Problem 24A positive integer n is “nice” if there is a positive integer m with exactly four positive divisors (including 1 and m) such that the sum of the four…Number Theory
  25. 25Problem 25Bernardo chooses a three-digit positive integer N and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two…Number Theory

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