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2009 AMC 10B problems

All 25 problems from the 2009 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 1Each morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the week was a whole number of…Algebra
  2. 2Problem 2Which of the following is equal to frac 13-frac 14 / frac 12-frac 13?Algebra
  3. 3Problem 3Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so…Algebra
  4. 4Problem 4A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape…Geometry
  5. 5Problem 5Twenty percent less than 60 is one-third more than what number?Algebra
  6. 6Problem 6Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?Number Theory
  7. 7Problem 7By inserting parentheses, it is possible to give the expression 2times 3+4times 5 several values. How many different values can be obtained?Algebra
  8. 8Problem 8In a certain year the price of gasoline rose by 20% during January, fell by 20% during February, rose by 25% during March, and fell by x% during…Algebra
  9. 9Problem 9Segment BD and AE intersect at C, as shown, AB=BC=CD=CE, and ∠ A=dfrac 52∠ B. What is the degree measure of ∠ D?Geometry
  10. 10Problem 10A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point x meters above the ground so that the upper part, still attached to…Geometry
  11. 11Problem 11How many 7-digit palindromes (numbers that read the same backward as forward) can be formed using the digits 2, 2, 3, 3, 5, 5, 5?Number Theory
  12. 12Problem 12Distinct points A, B, C, and D lie on a line, with AB=BC=CD=1. Points E and F lie on a second line, parallel to the first, with EF=1. A triangle with…Geometry
  13. 13Problem 13As shown below, convex pentagon ABCDE has sides AB=3, BC=4, CD=6, DE=3, and EA=7. The pentagon is originally positioned in the plane with vertex A at…Geometry
  14. 14Problem 14On Monday, Millie puts a quart of seeds, 25% of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix…Algebra
  15. 15Problem 15When a bucket is two-thirds full of water, the bucket and water weigh a kilograms. When the bucket is one-half full of water the total weight is b…Algebra
  16. 16Problem 16Points A and C lie on a circle centered at O, each of BA and BC are tangent to the circle, and △ ABC is equilateral. The circle intersects BO at D…Geometry
  17. 17Problem 17Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from (a,0) to…Geometry
  18. 18Problem 18Rectangle ABCD has AB=8 and BC=6. Point M is the midpoint of diagonal AC, and E is on AB with MEperpAC. What is the area of △ AME?Geometry
  19. 19Problem 19A particular 12-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 1, it mistakenly…Algebra
  20. 20Problem 20Triangle ABC has a right angle at B, AB=1, and BC=2. The bisector of ∠ BAC meets BC at D. What is BD?Geometry
  21. 21Problem 21What is the remainder when 3^0+3^1+3^2+…+3^2009 is divided by 8?Algebra
  22. 22Problem 22A cubical cake with edge length 2 inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where M…Geometry
  23. 23Problem 23Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes…Algebra
  24. 24Problem 24The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides…Geometry
  25. 25Problem 25Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge…Counting & Probability

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