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DUSO MATHEMATICS LEAGUE INDIVIDUAL QUESTIONS - MEET #3 DECEMBER 3, 2008 1. EL. ALG (MATH A) 4 MINUTES If four times my age four years ago is subtracted from four times my age four years from now, the result will be my current age in years. How old am I now? 2. GEOMETRY (MATH A) 4 MINUTES In convex quadrilateral ABCD, if the area of ∆CAD is 6, the area of ∆BAD is 8, and the area of ∆ABC is 10, what is the area of ∆DBC ? Express your answer in simplest form. 3. IntALG./TRIG (MATH B) 8 MINUTES Find the values of a, b, and c in the equation of the parabola, y ax 2 bx c , such that the parabola passes through each of the following points: ( 2, 6), ( 1, 1) and ( -1, 3) Express the values of a, b, and c as an ordered triple of the form (a, b, c). DUSO MATHEMATICS LEAGUE INDIVIDUAL QUESTIONS - MEET #3 DECEMBER 3, 2008 4. GEOMETRY (MATH B) 5 MINUTES Two circles are externally tangent. A diameter of the larger circle is 6. A tangent segment of length 4 connects the center of the smaller circle to its point of tangency on the larger circle. What is the area of the smaller circle? Express your answer in simplest form. 5. EL. ALG (MATH A) 5 MINUTES In how many zeroes does 6. IntALG./TRIG x 2 and terminate? (MATH B) 1 If 125! x 8 MINUTES 1 2 x Express your answer in simplest form. 4 , what is the value of x 12 ? DUSO MATHEMATICS LEAGUE RELAY TEAM QUESTION- MEET #3 DECEMBER 3, 2008 1). In triangle ABC, the base is one-sixth as long as the height. The area of the triangle is 108. How long is the base? 2). TNYWR is the number of wooden posts needed to build a fence. The posts are each 15 feet apart, and there is a post at each end of the fence. How many feet long is the fence? 3). Find the slope of a line containing the points ( 21, TNYWR) and ( g, 39), given that g is the number of prime numbers between 12 and 44. Express your slope to the nearest whole number. 4). An angle is 6 1 2 times as big as its complement. Find the smaller of the two angles and round it to the nearest whole number. Add this to TNYWR. 5). The squares of four consecutive integers have a sum of 2606. Find the largest of these integers (the largest integer, not its square), and add that integer to TNYWR. DUSO MATHEMATICS LEAGUE SOLUTIONS - MEET #3 DECEMBER 3, 2008 Answers for RELAY TEAM QUESTION 1). 6 2). 75 3). 3 4). 15 5). 42 (Solutions for the Relay question) 1). 1 2 bh 1 2 2). 6 x x 108 Given the number of posts is 6, 15 3 x 108 x 2 36 x 6, 2 15 15 15 15 Each is 15 feet apart, so the fence will be 75 feet long. (15 x 5 = 75) 6x x 3). There are 9 primes between 12 and 44. ( 13,17,19.23.29,31,37,41,43) y So slope is m qq x 75 39 21 9 4) 36 12 3 x 6.5x 90 7.5x 90 x = 12 12 + TNYWR = 12 + 3 = 15 5). Let the integers be x-1, x, x+1, x+2 x 12 x 2 x 12 x 22 2606 x 2 2 x 1 x 2 x 2 2 x 1 x 2 4 x 4 2606 4 x 2 4 x 6 2606 4 x 2 4 x 2600 0 x 2 x 650 0 x 26x 25 0 x = -26 or x = 25 largest = x + 2 = 27 242 252 262 272 = 576 = 625 = 676 = 729 2606 27 + TNYWR = 27 + 15 = 42 Individual Questions written by Steve Conrad www.mathleague.com; edited by Dan Flegler & J.S. Relay team question written for DUSO by J.A.; DUSO Editor J.S. DUSO MATHEMATICS LEAGUE SOLUTIONS - MEET #3 DECEMBER 3, 2008 Answers for INDIVIDUAL QUESTIONS 1). 32 2). 8 3). (2, -1, 0) 4). 4π 5). 31 6). 1/2 Solutions for the Individual questions) 2) 1 ) Let x = current age B 4x 4 4x 4 x 4x 16 4x 16 x 32 x Draw diagonal AC C 10 A 6 D 32 + 4 = 36 32 – 4 = 28 B 4(36) = 144 4(28) = 112 32 Now Draw diagonal BD 8 (2,6) : 6 4a 2b c (1,1): 1 a b c (-1,3): 3 a b c 5). 1 a 1 c or 2 a c So then 6 3a a=2 which means c = 0 Thus (a, b, c) is (2, -1, 0) The numbers of ending zeroes, indicates the number of factors of 10. Since 10 = 5 x 2, the number of factors of 10 will be governed by the number of factors of 5, because there will be more factors of 2’s than 5’s in a factorial. So for 125! the number of factors of 5 = 125 125 125 = 25 + 5 + 1 =31 5 5 2 53 Area triangle BCD = 16 - 8 = 8 4). Tangent BC is perpendicular to radius Triangle ABC is a right triangle, so AB = 5 AC C Using the last two equations, -2 = 2b, so b = -1 Now using the first two equations, where b = -1 We get 6 4a 2 c or 8 4a c and But since Area quad ABCD =16 and Area triangle BAD = 8 D 3) Area quad ABCD = 16 C A Since Area triangle ACD = 6 Area triangle ABC = 10 3 4 A 3 B 3 Thus the radius of the smaller circle is 5-3 or 2. The area of this circle is ( 22)π or 4π 1 6) x 1 4 2 x (2 x) x 4 x(2 x) 2 4 x 2 8 x 4 x 2 8x 2 0 2x 2 4x 1 0 4 or alternatively 4 8 2 x x 2 2 x 1 / 2 4 42 2 x x 2 2x 1 1 1/ 2 4 x 12 2 x 1 ( x 1) 2 2 2 2 2 2 4 1 2 1/ 2