2001 AMC 10A ws-25.pdf - The median of the list n n 3 n 4 n 5 n 6 n 8 n 10 n 12 n 15 is 10 What is the mean(A 4(B 6(C 7(D 10(E 11 2001 AMC 10

2001 AMC 10A ws-25.pdf - The median of the list n n 3 n 4 n...

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The median of the list n, n +3 , n +4 , n +5 , n +6 , n +8 , n +10 , n +12 , n +15 is 10. What is the mean? (A) 4 (B) 6 (C) 7 (D) 10 (E) 11 2001 AMC 10, Problem #1 “The middle number in the 9-number list is n + 6 .” Solution (E) The middle number in the 9-number list is n +6 , which is given as 10. Thus n = 4 . Add the terms together to get 9 n + 63 = 9 · 4 + 63 = 99 . Thus the mean is 99 / 9 = 11 . Difficulty: Medium NCTM Standard: Number and Operations Standard for Grades 9–12: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Mathworld.com Classification: Probability and Statistics > Rank Statistics > Median Calculus and Analysis > Special Functions > Means
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A number x is 2 more than the product of its reciprocal and its additive inverse. In which interval does the number lie? (A) - 4 x ≤ - 2 (B) - 2 < x 0 (C) 0 < x 2 (D) 2 < x 4 (E) 4 < x 6 2001 AMC 10, Problem #2 “The reciprocal of x is 1 x , and the additive inverse of x is - x .” Solution (C) The reciprocal of x is 1 x , and the additive inverse of x is - x . The product of these is ( 1 x ) · ( - x ) = - 1 . So x = - 1 + 2 = 1 , which is in the interval 0 < x 2 . Difficulty: Medium-hard NCTM Standard: Number and Operations Standard for Grades 9–12: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Mathworld.com Classification: Number Theory > Arithmetic > Multiplication and Division > Reciprocal Algebra > Group Theory > Group Properties > Additive Inverse
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The sum of two numbers is S . Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? (A) 2 S + 3 (B) 3 S + 2 (C) 3 S + 6 (D) 2 S + 6 (E) 2 S + 12 2001 AMC 10, Problem #3 2001 AMC 12, Problem #1 “The order of arithmetic operations is add 3 then double the sum.” Solution (E) Suppose the two numbers are a and b . Then the desired sum is 2( a + 3) + 2( b + 3) = 2( a + b ) + 12 = 2 S + 12 . Difficulty: Medium NCTM Standard: Number and Operations Standard for Grades 9–12: Understand meanings of operations and how they relate to one another. Mathworld.com Classification: Number Theory > Arithmetic > General Arithmetic
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What is the maximum number for the possible points of intersection of a circle and a triangle? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 2001 AMC 10, Problem #4 “The circle can intersect at most two points of each side of the triangle.” Solution (E) The circle can intersect at most two points of each side of the triangle, so the number can be no greater than six. The figure shows that the number can indeed be six. Difficulty: Medium-easy NCTM Standard: Geometry Standard for Grades 9–12: Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them. Mathworld.com Classification: Geometry > Plane Geometry > Circles Geometry > Plane Geometry > Triangles > Special Triangles > Other Triangles > Triangle
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How many of the twelve pentominoes pictured below have at least one line of symmetry?
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