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2010 AMC 10A problems

All 25 problems from the 2010 AMC 10A, 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 1Mary’s top book shelf holds five books with the following widths, in centimeters: 6, 1/2, 1, 2.5, and 10. What is the average book width, in…Algebra
  2. 2Problem 2Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as…Geometry
  3. 3Problem 3Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric…Algebra
  4. 4Problem 4A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of reading. Assume that the…Number Theory
  5. 5Problem 5The area of a circle whose circumference is 24π is kπ. What is the value of k?Geometry
  6. 6Problem 6For positive numbers x and y the operation spadesuit (x,y) is defined as spadesuit (x,y) = x-1/y What is spadesuit (2,spadesuit (2,2))?Algebra
  7. 7Problem 7Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one…Geometry
  8. 8Problem 8Tony works 2 hours a day and is paid $0.50 per hour for each full year of his age. During a six month period Tony worked 50 days and earned $630. How…Algebra
  9. 9Problem 9A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x + 32 are three-digit and four-digit…Number Theory
  10. 10Problem 10Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will his birthday next fall on a Saturday?Algebra
  11. 11Problem 11The length of the interval of solutions of the inequality a ≤ 2x + 3 ≤ b is 10. What is b - a?Algebra
  12. 12Problem 12Logan is constructing a scaled model of his town. The city’s water tower stands 40 meters high, and the top portion is a sphere that holds 100,000…Algebra
  13. 13Problem 13Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether…Algebra
  14. 14Problem 14Triangle ABC has AB=2 · AC. Let D and E be on AB and BC, respectively, such that ∠ BAE = ∠ ACD. Let F be the intersection of segments AE and CD, and…Geometry
  15. 15Problem 15In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always…Counting & Probability
  16. 16Problem 16Nondegenerate △ ABC has integer side lengths, BD is an angle bisector, AD = 3, and DC = 8. What is the smallest possible value of the perimeter?Geometry
  17. 17Problem 17A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the…Geometry
  18. 18Problem 18Bernardo randomly picks 3 distinct numbers from the set {1,2,3,4,5,6,7,8,9} and arranges them in descending order to form a 3-digit number. Silvia…Counting & Probability
  19. 19Problem 19Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r. The area of △ ACE is 70% of the area of the hexagon. What is the sum of all…Geometry
  20. 20Problem 20A fly trapped inside a cubical box with side length 1 meter decides to relieve its boredom by visiting each corner of the box. It will begin and end…Algebra
  21. 21Problem 21The polynomial x^3-ax^2+bx-2010 has three positive integer zeros. What is the smallest possible value of a?Algebra
  22. 22Problem 22Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the…Counting & Probability
  23. 23Problem 23Each of 2010 boxes in a line contains a single red marble, and for 1 ≤ k ≤ 2010, the box in the kth position also contains k white marbles. Isabella…Algebra
  24. 24Problem 24The number obtained from the last two nonzero digits of 90! is equal to n. What is n?Number Theory
  25. 25Problem 25Jim starts with a positive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible…Algebra

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