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2006 AMC 10B problems

All 25 problems from the 2006 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 (-1)^1 + (-1)^2 + … + (-1)^2006?Algebra
  2. 2Problem 2For real numbers x and y, define x spadesuit y = (x+y)(x-y). What is 3 spadesuit (4 spadesuit 5)?Algebra
  3. 3Problem 3A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a…Algebra
  4. 4Problem 4Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and…Geometry
  5. 5Problem 5A 2 × 3 rectangle and a 3 × 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are…Geometry
  6. 6Problem 6A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure 2/π, as shown. What is the perimeter of this region?Geometry
  7. 7Problem 7Which of the following is equivalent to √(x/1-x-1/x) when x < 0?Algebra
  8. 8Problem 8A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?Geometry
  9. 9Problem 9Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon…Algebra
  10. 10Problem 10In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the…Geometry
  11. 11Problem 11What is the tens digit in the sum 7! + 8! + 9! + … + 2006!?Number Theory
  12. 12Problem 12The lines x=tfrac 14 y+a and y=tfrac 14 x+b intersect at the point (1,2). What is a+b?Algebra
  13. 13Problem 13Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2…Algebra
  14. 14Problem 14Let a and b be the roots of the equation x^2-mx+2=0. Suppose that a+tfrac 1b and b+tfrac 1a are the roots of the equation x^2-px+q=0. What is q?Algebra
  15. 15Problem 15Rhombus ABCD is similar to rhombus BFDE. The area of rhombus ABCD is 24, and ∠ BAD=60°. What is the area of rhombus BFDE?Geometry
  16. 16Problem 16Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?Algebra
  17. 17Problem 17Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from…Counting & Probability
  18. 18Problem 18Let a_1, a_2, … be a sequence for which a_1=2, a_2=3, and a_n=dfrac a_n-1a_n-2 for each positive integer n≥ 3. What is a_2006?Algebra
  19. 19Problem 19A circle of radius 2 is centered at O. Square OABC has side length 1. Sides AB and CB are extended past B to meet the circle at D and E…Geometry
  20. 20Problem 20In rectangle ABCD, we have A=(6,-22), B=(2006,178), and D=(8,y) for some integer y. What is the area of rectangle ABCD?Geometry
  21. 21Problem 21For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5, and 6 on each die are in the ratio 1:2:3:4:5:6. What is the…Counting & Probability
  22. 22Problem 22Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4¢ per glob and J blobs of jam at 5¢ per blob. The…Number Theory
  23. 23Problem 23A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three…Algebra
  24. 24Problem 24Circles with centers at O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B on the circle with center O and points C…Geometry
  25. 25Problem 25Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which…Number Theory

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