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MATH COUNTS 2008 Chapter Competition Countdown Round
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1. The forecasted maximum wind speeds, in mph, for the next five days are 13, 6, 10, 5 and 8. What is the range of the wind speeds, in miles per hour?
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3. A segment with endpoints at A(2, 2) and B(14, 4) is extended through B to point C. If BC = ( )AB, what are the coordinates for point C? Express your answer as an ordered pair.
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4. If a snack-size tin of peaches has 40 calories and is 2% of a person’s daily caloric requirement, how many calories fulfill a person’s daily caloric requirement?
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5. If Ellen chooses a day of the week at random, what is the probability that its English spelling uses the letter “n”? Express your answer as a common fraction.
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6. A catalogue sells pencils two ways: (1) 50 cents each and (2) $2 per pack of 12. What is the least expensive purchase price for exactly 30 pencils?
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7. The five highest kick return averages, in yards, in the NFL’s history were 25.0, 24.3, 24.1, 23.8 and 23.4. What is the median of these five averages? Express your answer as a decimal to the nearest tenth.
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10. Margo walks to her friend’s house in 10 minutes. Using the same route, it takes Margo 20 minutes to get back home. If her average walking rate for the entire trip is 4 miles per hour, how many total miles did she walk?
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11. Adjacent sides of Figure 1 are perpendicular. Four sides of Figure 1 are removed to form Figure 2. What is the total length, in units, of the segments in Figure 2?
12. What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?
13. What is the area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 6 units and lateral edges measuring 5 units?
16. A 5×5×5 cube is formed by assembling 125 unit cubes. Nine unit squares are painted on each of the six faces of the cube according to the pattern shown. How many of the 125 unit cubes have no paint on them?
19. A rocket traveled 12,000 miles in the first four hours. Then the rocket traveled 8000 miles in the next four hours. What was the rocket’s average speed, in miles per hour, for these 8 hours?
21. Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book?
22. The first four triangular numbers are 1, 3, 6 and 10 where each successive number can be found by adding a row of dots with one more dot than the previous row. What is the sixth triangular number?
23. John has 5 quarters and David has 5 dimes. John owes David 35 cents. John should give David m quarters and John should get n dimes from David so that John has paid his debt exactly. What is the value of m + n ?
26. Betty draws one disc from a jar that contains 2 red discs, 5 green discs and 8 yellow discs. What is the probability that her selection is not a red disc? Express your answer as a common fraction.
30. Quadrilateral ABCD is a square with area 16 square inches. The figure represents the pieces of a Chinese tangram in which all the triangles are isosceles and piece “e” is a square. What is the area of the gray piece, in square inches?
31. Anna purchased 1 pound of fusilli pasta for $3.40 and a 12-ounce package of spaghetti for $2.20. There are 16 ounces in 1 pound. What was the average price per ounce, in cents, for her total purchase?
34. An 80-pound box of apples will have 10% of the apples spoil. At the same rate, how many pounds of spoiled apples will there be for every 2160 pounds of unspoiled apples?
35. The four midpoints of adjacent sides of a square are joined to form a smaller square. What is the ratio of the area of the smaller square to the area of the original square? Express your answer as a common fraction.
36. What is the height of Jack’s house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number.
38. A recent study found that 60% of men and 80% of women surveyed support increased funding for particular medical research. The study surveyed 100 men and 900 women. What was the overall percent of the people surveyed who supported increased funding?
40. A bag of grapes is to be distributed evenly to 5 kids in a class, and the grapes that are left over will be thrown out. If each student receives the greatest possible number of grapes, what is the greatest possible number of grapes that could be thrown out?
42.What is the smallest possible area, in square units, of a triangle whose side lengths are all integers? Express your answer as a common fraction in simplest radical form.
44. A dose of a shrinking potion makes a person 50% smaller than they were before they drank it, and eating a growth cake makes a person 50% larger than they were before they ate it. After drinking 2 doses of shrinking potion and eating 1 growth cake, what is the ratio of Alice’s final size to her original size? Express your answer as a common fraction.
45. If you continue this pattern in which each line-segment extremity is replaced by a gradually smaller Y in the next figure, in the manner shown, how many endpoints will Figure 5 have?
46. A right triangle has leg lengths of 8 and 15 units. What is the length, in units, of the median to the hypotenuse? Express your answer as a common fraction.
47. An 8.5-by-11-inch piece of paper is folded in half repeatedly (never being unfolded), each time shortening the then longer side. What is the length of the longest side, in inches, immediately after the second fold? Express your answer as a decimal to the nearest tenth.
52. A square with side length 5 inches and a 3-by-6-inch rectangle are each drawn on the coordinate plane such that there are 12 square inches of overlap. What is the area, in square inches, of the portion of the square that is not overlapping the rectangle?
53. One cookie and two brownies cost the same as two cookies and one brownie. One cookie and one brownie together cost $1. What is the cost, in cents, of one cookie?
54. An isosceles, obtuse triangle has one angle with a degree measure that is 50% larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a decimal to the nearest tenth.
55. Consecutive powers of 3 are added to form this sequence: 3 0 , 3 0 + 3 1 , 3 0 + 3 1 + 3 2 , and so on. What is the simplified value of the fourth term of the sequence?
56. Five years ago, a computer that worked half as fast as a current computer cost twice as much money. The current computer’s speed to price ratio is what percent of the older computer’s speed to price ratio?
58. A rectangular garden shares one side with one side of a house. An adjacent side of the garden is eight feet long. If the area of the garden is 184 square feet, what is the length, in feet, of the common side?
60. A right triangle has leg lengths of 7 and 24 units. A second triangle is similar to the first triangle and has a hypotenuse of 100 units. What is the length, in units, of the shorter leg of the second triangle?
61. There is a total of 20 quarters stacked in four piles. The first pile has 3 fewer than the second pile. The second pile has 2 more than the third pile. The fourth pile has twice as many as the second pile. How many quarters are in the fourth pile?
62. A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of h units. What is the value of h ?
65. Simone has a collection of coins consisting of 1 quarter, 1 nickel and 1 penny. Using at least two coins from the collection, how many different sums of money are possible?
67. In a music collection, there are 100 songs with an average length of exactly 180 seconds. One song is added and the new average length is exactly 181 seconds. How long, in seconds, was the added song?
72. Apartment rentals in Fairview run approximately $0.90 per square foot. Jillian has determined that she can afford $630 per month for rent. What is the largest apartment, in square feet, she should consider at the given rate?
75. The digits 1, 3 and 5 are each used once to form each of the possible three-digit positive integers. The three-digit integers are listed from greatest to least. Which integer is listed fifth?
79. Charlie rides her bike on a route that is 22 miles long. It takes her 165 minutes to complete the ride. What is her resulting average speed, in miles per hour?
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