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Download Level 2: Part B The 25th Math Bowl March 6, 2010 1. Anne, Bob
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The 25th Level 2: Part B Math Bowl March 6, 2010 Write your answers in the Blue Book. Print your names & write the number of students taking this test in the upper right corner of the Blue Book. Put this test & the Blue Book in the provided envelope. For credit and partial credit show your work. 1. Anne, Bob, Carol, and Don are rescued from a desert island by a pirate who forces them to play a game. Each of the four, in alphabetical order by first name, has to roll two dice. If the total on the dice is either 8 or 9, the person rolling the dice has to walk the plank. The game stops as soon as one player loses or after all have rolled the dice once. What is the probability that Don survives? Solution: Don does not survive if Anne, Bob, and Carol all survive, but he loses. The chance that any individual player loses on his or her turn is 41 , since there are nine ways to roll an 8 or a 9, out of 27 , and the chance 36 possible rolls. Therefore the chance that Don does not survive is 43 · 34 · 43 · 14 = 256 27 229 that he survives is 1 − 256 = 256 . 2. Find all three-digit numbers with the property that the sum of the three digits is equal to the product of the three digits. Solution: Let the three-digit number be abc = a · 100 + b · 10 + c, where 1 ≤ a ≤ 9, 0 ≤ b ≤ 9, 0 ≤ c ≤ 9. We want a + b + c = a · b · c. Since a 6= 0, we see that also b 6= 0, c 6= 0, since otherwise the product would be 0 but the sum is not. Let k be the largest of the three numbers; then the sum of the three numbers is at most 3k, which means that the product of the remaining two numbers is at most 3, so the remaining two numbers are 1 and 1, 1 and 2, or 1 and 3. In the first case, the sum of the numbers is then k + 2, so we would have k + 2 = k · 1 · 1, or 2 = 0, an impossibility. In the second case, the sum of the numbers is then k + 3, so we would have k + 3 = 2k, so {a, b, c} = {1, 2, 3} is a possibility. In the third case, k + 4 = 3k, so 4 = 2k and k = 2, contradicting the assumption that k is the largest of the three numbers. Thus the set of three-digit numbers with this property is {123, 132, 213, 231, 312, 321}. 3. If a + 1 a = 5, find a2 + 1 a2 and a4 + 1 a4 . 1 1 1 1 1 Solution: (a + )2 = a2 + 2 + 2 = 25, so a2 + 2 = 25 − 2 = 23. Then (a2 + 2 )2 = a4 + 4 + 2 = a a a a a 1 232 = 529, so a4 + 4 = 529 − 2 = 527. a 4. Four distinct positive integers a, b, c, d form an arithmetic sequence1 . The integers a, b, d also form a geometric sequence.2 The sum of the four integers is a perfect cube. Find the least possible value for a, and describe all possible values for a. Solution: Let k be the common difference for the arithmetic sequence and let r be the common ratio for the geometric sequence. Then r = ab = db , so b2 = da. But b = a + k, d = a + 3k, so (a + k)2 = (a + 3k)a. Then a2 + 2ka + k 2 = a2 + 3ka, giving ka = k 2 , so a = k. Then b = a + k = 2a = ar so r = 2. The sum of the four integers is then a + 2a + 3a + 4a = 10a = n3 for some integer n. Then we must have a = 102 m3 for some integer m, and any such a will satisfy the given condition. The least possible value occurs when m = 1 and a = 100. 1 Each 2 Each term is formed by adding a constant to the proceeding term. term is formed by multiplying a constant times the proceeding term. Page 1 of 2 Please continue Math Bowl The 25th Level 2: Part B March 6, 2010 5. The horizontal rectangle below has length a and height b. Find the area and perimeter of the slanted rectangle. a b Solution: Draw a perpendicular from the diagonal of the horizontal rectangle to the lower right corner of the horizontal rectangle. This divides the slanted rectangle into two smaller rectangles, whose diagonals are formed by the sides of the first rectangle. This makes it clear that the triangle formed by the diagonal of the first rectangle and these two sides make up exactly half of the slanted rectangle, so that its area is exactly equal to that of the first rectangle, or a · b square units. To find the perimeter, √ let h be the length of the shorter side of the slanted rectangle. The longer side has length a2 + b2 , p √ 2ab = , and the perimeter of this square is 2 a2 + b2 + √ and h · a2 + b2 = a · b, so h = √aa·b 2 +b2 a2 + b 2 2(a2 + b2 + ab) √ . a2 + b 2 Page 2 of 2 -End of Level 2: Part B-