1. Triangle ABC is an obtuse, isosceles triangle. Angle A measures 20 degrees. What is the measure of the largest interior angle of triangle ABC? degrees 1. ________________ 2. Quentin spent $480 to purchase 30 books. Using the same average price per book, how much will 45 books cost? $ 2. ________________ 3. In Clara county, 25% of households earn less than $30,000 per year, and 65% of households earn less than $80,000 per year. What is the largest possible percent of households that could earn between $30,000 and $80,000 per year? percent 3. ________________ 4. For what value of x is the following equation true: 3000 + x 2000 = 1500 + 1000? 4. ________________ 5. How many cubic feet are in one cubic yard? One yard is equal to three feet. cu feet 5. ________________ 1 cu ft 1 cu yd 6. Forty-eight congruent parallelograms with sides of length 62 feet and 20 feet are placed in a chevron pattern forming hexagon ABCDEF, as shown. What is the perimeter of hexagon ABCDEF? A B F E feet 6. ________________ C D Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Sprint Round 7. The mean of four distinct positive integers is 5. If the largest integer is 13, what is the smallest integer? 7. ________________ 8. Congruent segments are used to form equilateral triangles in this sequence so that each figure contains one more triangle than the preceding figure. Figure 3 of the sequence is made with seven congruent segments. How many congruent segments must be used to make Figure 25 of the sequence? segments 8. ________________ Figure 1 Figure 2 Figure 3 Figure 4 9. What is the sum of the odd integers from 11 through 39, inclusive? 9. ________________ 10. The average amount of money spent by a person who attended a local sporting event in 2000 was $8.00, of which 75% was the ticket price. In 2005, the average amount of money spent by a person who attended a local sporting event increased by 50%, but the ticket price did not increase. By how many dollars did the non-ticket costs of 2000 increase to become the non-ticket costs of 2005? $ 10. ________________ 11. The I-Pick-Up messenger service delivers packages using the following rate structure: $1 per ounce of the packages’ weight plus $5 for each distinct drop-off site. Then a 4% service fee per order is added to the subtotal of the weight and drop-off site charges. Chen Li places the order below. How much should I-Pick-Up charge for Chen Li’s order? (There are 16 ounces in one pound.) 1 four-ounce package to Imatrin 1 two-pound package to Imatrin 1 eight-pound package to Storyville $ 11. ________________ 12. A line contains the points (1, 6), (6, k) and (20, 3). What is the value of k? 12. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Sprint Round 13. A particular convex polygon with seven sides has exactly one right angle. How many diagonals does this seven-sided polygon have? diagonals 13. ________________ 14. The product of three consecutive odd integers is 1287. What is the sum of the three integers? 14. ________________ 15. In square ABCD, point M is the midpoint of side AB and point N is the midpoint of side BC. What is the ratio of the area of triangle AMN to the area of square ABCD? Express your answer as a common fraction. 15. ________________ 16. How many non-congruent triangles are there with sides of integer length having at least one side of length five units and having no side longer than five units? triangles 16. ________________ 17. What is the value of the following expression: 17. ________________ 1 1 1 1 1 ? Express your 3 9 27 81 243 answer as a common fraction. 18. A customer ordered 15 pieces of gourmet chocolate. The order can be packaged in small boxes that contain 1, 2 or 4 pieces of chocolate. Any box that is used must be full. How many different combinations of boxes can be used for the customer’s 15 chocolate pieces? One such combination to be included is to use seven 2-piece boxes and one 1-piece box. combinations 18. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Sprint Round 19. The value of [x] is the greatest integer less than or equal to x. What is the arithmetic mean of the 10 members of the set [;P=, ; =, ;0=, ; =, ;0.689=, ¨ª 1 2 1 2 P 4 19. ________________ ·¹, ¨ª P3 ·¹, ;2=, ¨ª 5 ·¹ , ;P=]? Express your answer as a common fraction. 20. The summary of a survey of 100 students listed the following totals: 59 students did math homework 49 students did English homework 42 students did science homework 20 students did English and science homework 29 students did science and math homework 31 students did math and English homework 12 students did math, science and English homework How many students did no math, no English and no science homework? students 20. ________________ 21. Given that 6x + y = 15, the value of 3x can be written in terms of y as ay + b for some numbers a and b. What is the simplified value of a + b? 21. ________________ 22. Suelyn counts up from 1 to 9, and then immediately counts down again to 1, and then back up to 9, and so on, alternately counting up and down (1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1,2,3,4,...). What is the 1000th integer in her list? 22. ________________ 23. The positive three-digit integer N yields a perfect square when divided by 5. When N is divided by 4, the result is a perfect cube. What is the value of N? 23. ________________ 24. Grady rides his bike 60% faster than his little brother Noah. If Grady rides 12 miles further than Noah in two hours, how fast does Noah ride? miles per hour 24. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Sprint Round 25. The length of a diagonal of a square is 2 3 units. What is the area of the square? Express your answer in simplest a a form as c , where is a common fraction and c has no b b perfect square factors other than 1 . sq units 25. ________________ 26. Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius? inches 26. ________________ 27. Consider this pattern where the positive, odd integers are arranged in a triangular formation. The 1st through 4th rows are shown; each row has one more entry than the previous row. What is the sum of the integers in the 15th row? 1 3 7 13 27. ________________ 5 9 15 11 17 19 28. Four couples are at a party. Four people of the eight are randomly selected to win a prize. No person can win more than one prize. What is the probability that both members of at least one couple win a prize? Express your answer as a common fraction. 28. ________________ 29. The points of this 3-by-3 grid are equally spaced horizontally and vertically. How many different sets of three points of this grid can be the three vertices of an isosceles triangle? sets 29. ________________ 30. In parallelogram ABCD, AB = 16 cm, DA = 3 2 cm, and sides AB and DA form a 45-degree interior angle. In isosceles trapezoid WXYZ with WX ≠ YZ, segment WX is the longer parallel side and has length 16 cm, and two interior angles each have a measure of 45 degrees. Trapezoid WXYZ has the same area as parallelogram ABCD. What is the length of segment YZ? centimeters 30. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Sprint Round 1. The symbols !, ", # and $ each represent a distinct digit that has not been used already in the subtraction problem below. Whenever a symbol appears more than once, it represents the same digit each time. What is the digit that ! represents in the following subtraction problem? 1. ________________ 6 " # !! 8 # 1 $ " 2. An 8-inch by 8-inch square is folded along a diagonal creating a triangular region. This resulting triangular region is then folded so that the right angle vertex just meets the midpoint of the hypotenuse. What is the area of the resulting trapezoidal figure? sq inches 2. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Target Round 3. The distance between two cities on a map is 4 centimeters. If the scale is 0.5 cm = 1 km, how far apart are the actual cities? kilometers 3. ________________ 4. Six years ago a vacant lot was turned into a park. At that time 46 trees were planted. Three years ago 50 trees were planted in the park, and 60 trees were planted in the park today. Each tree was planted as a seed. Assuming that all of these planted trees survive and no other trees are added in the next 10 years, what will be the average age of the trees in the park 10 years from today? Express your answer to the nearest whole number. years 4. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Target Round 5. In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan’s measurement has an error of up to 20%. What is the largest possible percent error in Juan’s computed area of the circle? percent 5. ________________ 6. A quadrilateral in the plane has vertices at (1, 3), (1, 1), (2, 1) and (2006, 2007). What is the area of the quadrilateral? sq units 6. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Target Round 7. Six boys and six girls are seated randomly in a row of 12 chairs. What is the probability that no two boys are seated next to one another and no two girls are seated next to one another? Express your answer as a common fraction. 7. ________________ 8. Dr. Lease leaves his house at exactly 7:20 a.m. every morning. When he averages 45 miles per hour, he arrives at his workplace five minutes late. When he averages 63 miles per hour, he arrives five minutes early. What speed should Dr. Lease average to arrive at his workplace precisely on time? Express your answer as a decimal to the nearest tenth. miles per 8. ________________ hour Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Target Round 1. The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? 1. ________________ 2. The square quilt block shown is used in a larger quilt. The four small squares in the block are congruent, and the four small vertical rectangles in the block that are not squares are also congruent. The total area of the shaded regions is what fraction of the total area of the block? Express your answer as a common fraction. 2. ________________ 3. The ages of the 27 students signed up for the community center’s Spanish I class are given below. In the stem and leaf plot shown, 3 2 represents 32 years old. For this data, the mean is x years, the median is y years, and the mode is z years. What is the value of x(y z)? 3. ________________ 1 2 3 4 9 0 0 5 9 0 1 1 2 4 5 5 6 7 8 8 0 0 1 2 2 4 4 5 8 9 8 4. If 40♦ represents a three-digit positive integer with a ones digit of ♦ and 1♦ is a two-digit positive integer with a ones digit of ♦, what value of ♦ makes the equation 40♦ ÷ 27 = 1♦ true? 4. ________________ 5. A particular right pyramid has a square base, and each edge of the pyramid is four inches long. What is the volume of the pyramid? Express your answer as a decimal to the nearest hundredth. cu inches 5. ________________ Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Team Round 6. The diagram shown is the net of a regular dodecahedron. In a regular dodecahedron, three edges come together at each vertex. When the net of this dodecahedron is put together, the solid has x vertices and y edges. What is the value of x + y? 6. ________________ 7. On the first day, Barry Sotter used his magic wand to make an object’s length increase by 12 ; on the second day he increased the object’s longer length by 13 ; on the third day he increased the object’s new length by 14 ; and so on. On the nth day of performing this trick, Barry will make the object’s length exactly 100 times its original length. What is the value of n? 7. ________________ 8. A cube is sliced with one straight slice which passes through two opposite edges. The result is two solids, as shown. The area of the largest face on one of these two solids is 242 2 square units. What was the exact surface area of the original cube? Express your answer in simplest form. sq units 8. ________________ 9. Using the digits 2, 3, 4, 7 and 8, Carlos will form five-digit positive integers. Only the digit 2 can be used more than once in any of Carlos’ five-digit integers. How many distinct fivedigit positive integers are possible? integers 9. ________________ 10. Bricklayer Ben places 42 bricks per hour. Bricklayer Bob places 36 bricks per hour. Bricklayer Bob worked twice as many hours as Bricklayer Ben, and the two of them placed a total of 1254 bricks. How many bricks did Bricklayer Ben place? 10. ________________ bricks Copyright MATHCOUNTS, Inc. 2006. All rights reserved. 2007 Chapter Team Round

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