2008 AMC 10 B, Problem #22—“Note that there are6!/(3!2!1!) = 60distinguishableorders of the beads on the line.” Solution Answer (C): There are 6! / (3!2!1!) = 60 distinguishable orders of the beads on the line. To meet the required condition, the red beads must be placed in one of four configurations: positions 1, 3, and 5, positions 2, 4, and 6, positions 1, 3, and 6, or positions 1, 4, and 6. In the first two cases, the blue bead can be placed in any of the three remaining positions. In the last two cases, the blue bead can be placed in either of the two adjacent remaining positions. In each case, the placement of the white beads is then determined. Hence there are 2 · 3 + 2 · 2 = 10 orders that meet the required condition, and the requested probability is 10 60 = 1 6 . Difficulty: Medium-hard NCTM Standard: Data Analysis and Probability Standard: understand and apply basic concepts of probability. Mathworld.com Classification: Probability and Statistics > Probability > Probability
A rectangular floor measures a feet by b feet, where a and b are positive integers with b > a . An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width 1 foot around the painted rectangle and occupies half the area of the entire floor. How many possibilities are there for the ordered pair ( a, b ) ? (A)1(B)2(C)3(D)4(E)5 2008 AMC 10 B, Problem #23—2008 AMC 12 B, Problem #16—“Because the area of the border is half the area of thefloor, the same is true of the painted rectangle.Thepainted rectangle measuresa-2byb-2feet.” Solution Answer (B): Because the area of the border is half the area of the floor, the same is true of the painted rectangle. The painted rectangle measures a - 2 by b - 2 feet. Hence ab = 2( a - 2)( b - 2) , from which 0 = ab - 4 a - 4 b + 8 . Add 8 to each side of the equation to produce 8 = ab - 4 a - 4 b + 16 = ( a - 4)( b - 4) . Because the only integer factorizations of 8 are 8 = 1 · 8 = 2 · 4 = ( - 4) · ( - 2) = ( - 8) · ( - 1) , and because b > a > 0 , the only possible ordered pairs satisfying this equation for ( a - 4 , b - 4) are (1 , 8) and (2 , 4) . Hence ( a, b ) must be one of the two ordered pairs (5 , 12) , or (6 , 8) . Difficulty: Hard NCTM Standard: Geometry Standard: analyze properties and determine attributes of two- and three-dimensional objects. Mathworld.com Classification: Geometry > Plane Geometry > Rectangles > Rectangle
A B C D M A B C D O Quadrilateral ABCD has AB = BC = CD , ∠ ABC = 70 ◦ , and ∠ BCD = 170 ◦ . What is the degree measure of ∠ BAD ?
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