ΒΆ 2006 AMC 10B Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2006 AMC 10B problems here.
Discussion Forum
Engage in discussion about the 2006 AMC 10B math contest by visiting Random Math AMC 10B 2006 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2006 AMC 10B problems, please refer below:
Problem 1: What is ?
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Problem 2: For real numbers and , define . What is ?
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Problem 3: A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of points, and the Cougars won by a margin of points. How many points did the Panthers score?
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Problem 4: Circles of diameter inch and inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
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Problem 5: A rectangle and a rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
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Problem 6: A region is bounded by semicircular arcs constructed on the side of a square whose sides measure , as shown. What is the perimeter of this region?
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Problem 7: Which of the following is equivalent to
when ?
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Problem 8: A square of area is inscribed in a semicircle as shown. What is the area of the semicircle?
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Problem 9: Francesca uses grams of lemon juice, grams of sugar, and grams of water to make lemonade. There are calories in grams of lemon juice and calories in grams of sugar. Water contains no calories. How many calories are in grams of her lemonade?
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Problem 10: In 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 . What is the greatest possible perimeter of the triangle?
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Problem 11: What is the tens digit in the sum
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Problem 12: The lines and intersect at the point . What is ?
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Problem 13: Joe and JoAnn each bought ounces of coffee in a -ounce cup. Joe drank ounces of his coffee and then added ounces of cream. JoAnn added ounces of cream, stirred the coffee well, and then drank ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
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Problem 14: Let and be the roots of the equation . Suppose that and are the roots of the equation
What is ?
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Problem 15: Rhombus is similar to rhombus . The area of rhombus is , and . What is the area of rhombus ?
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Problem 16: Leap Day, February , , occurred on a Sunday. On what day of the week will Leap Day, February , , occur?
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A. Tuesday
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Problem 17: Bob 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 her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?
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Problem 18: Let be a sequence for which
What is ?
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Problem 19: A circle of radius is centered at . Square has side length . Sides and are extended past to meet the circle at and , respectively. What is the area of the shaded region in the figure, which is bounded by , , and the minor arc connecting and ?
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Problem 20: In rectangle , we have , and , for some integer . What is the area of rectangle ?
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Problem 21: For a particular peculiar pair of dice, the probabilities of rolling , and on each die are in the ratio . What is the probability of rolling a total of on the two dice?
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Problem 22: Elmo makes sandwiches for a fundraiser. For each sandwich he uses globs of peanut butter at cents per glob and blobs of jam at cents per blob. The cost of the peanut butter and jam to make all the sandwiches is . Assume that , , and are positive integers with . What is the cost of the jam Elmo uses to make the sandwiches?
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Problem 23: A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are , and , as shown. What is the area of the shaded quadrilateral?
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Problem 24: Circles with centers at and have radii and , respectively, and are externally tangent. Points and on the circle with center and points and on the circle with center are such that and are common external tangents to the circles. What is the area of the concave hexagon ?
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Problem 25: Mr. Jones has eight children of different ages. On a family trip his oldest child, who is , spots a license plate with a -digit number in which each of two digits appears two times. "Look, daddy!!!!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
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The problems on this page are the property of the MAA's American Mathematics Competitions