2005 Chapter Competition Countdown Round Problems 1 80

Transcription

1 005 Chapter Competition Countdown Round Problems 1 80 This section contains problems to be used in the Countdown Round. Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics National Sponsors ADC Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Texas Instruments Incorporated 3M Foundation 004 MATHCOUNTS Foundation, 140 King Street, Alexandria, VA 314

2 1. A 3-ounce can of tomato sauce costs $1.68. In cents, what is the price per ounce?. How many two-digit prime numbers can be formed by choosing two different digits from the set {, 7, 8, 9} to be used as the tens digit and units digit? 3. Five is added to four times an integer to get a result of 77. What was the integer? 4. Connie is starting an exercise program. On June 1, she will do 5 sit-ups. Each day after that, she will increase her number of sit-ups by four. On which date during the month of June will Connie first do more than 100 sit-ups in one day? 5. What is the area of the parallelogram shown, in square feet? 6. How many diagonals are in the fifth polygon of this sequence of regular polygons below? Polygon # of diagonals 0 5 9? 7. What is the next term in the geometric sequence, 6x, 18x, 54x 3,? Express your answer in terms of x. 8. What is the value of 5!!? 3! 9. Monica, who is 5 feet tall, is standing next to a pine tree that casts a shadow 34 feet long. If Monica s shadow is feet long, how tall is the pine tree, in feet? 10. If a + 4b = 33 and 6a + 3b = 51, what is the value of a + b? 11. The first four terms of a sequence are 1, 1, 0 and 1, respectively. Each subsequent term is the sum of the previous four terms. What is the sixth term in the sequence? 1. A triangular region is bounded by the two coordinate axes and the line given by the equation x + y = 6. What is the area of the region, in square units? 13. What is the sum of the median and the mean of the set of numbers 1, 1, 6, 11 and 30?

3 14. The measures of the three angles of a triangle form an arithmetic sequence. What must be the measure, in degrees, of at least one of its angles? 15. Express 5% of 147 as a decimal to the nearest hundredth. 16. What is the greatest possible quotient of any two distinct members of the set { 1 5,,5,10 }? 17. What is the least number of colors needed to shade the tessellation shown, such that no two tiles sharing a side are the same color? 18. The mean of five numbers is 5. If one of the numbers is removed from the group, the mean of the remaining four numbers is 4.5. What number was removed from the group? 19. Javier is biking at 15 miles per hour. There are 580 feet in a mile. How many feet does he travel in one minute? 0. What is the greatest common factor of 30, 90 and 75? 1. Joe multiplies a number by 4, adds 1, and then divides by 3, getting a result of 7. Sue divides the same original number by 3, adds 1, and multiplies by 4. What result does she get? Express your answer as a common fraction.. A line is drawn through the points (, 6) and ( 4, ). What is the y-coordinate of the point where this line intersects the y-axis? 3. Simplify ( ) If a b = and b c = 3 4, what is the value of c a? Express your answer as a common fraction. 5. The area of square ABCD is 100 square centimeters, and AE = cm. What is the area of square EFGH, in square centimeters? 6. On a Tuesday, a magician said, I made my wife disappear 31 days ago. What day of the week did he make her disappear? 7. Sam is hired for a 0-day period. On days that he works, he earns $60. For each day that he does not work, $30 is subtracted from his earnings. At the end of the 0-day period, he received $660. How many days did he not work?

4 8. Twenty percent of 30 equals 50% of what number? 9. What integer is closest to 500? 30. A teacher finds that when she offers candy to her class of 30 students, the mean number of pieces taken by each student is 5. If every student takes some candy, what is the greatest number of pieces one student could have taken? 31. What is the greatest multiple of 3 that is less than 500 and is also a multiple of 4? 3. When Bob exercises, he does jumping jacks for five minutes and then walks the track at four minutes per lap. If he exercised for 73 minutes on Monday, how many laps did he walk? 33. The area of a square plot of land is 35 square meters. What is the perimeter of the square, in meters? Express your answer in simplest radical form. 34. After eating lunch together, Jay and Paul start walking in opposite directions. Jay walks 0.75 miles every 15 minutes and Paul power walks.5 miles every 30 minutes. In miles, how far apart are they after 1.5 hours? 35. If the numbers 4, 5 and 6 are each used exactly once to replace the letters in the expression A ( B C ), what is the least possible result? 36. What number is 17 less than its negative? Express your answer as a decimal to the nearest tenth. 37. The sum of the first N positive odd numbers is 11. What is the value of N? 38. What is the value of the reciprocal of as a decimal to the nearest tenth. 39. On this 5 by 5 grid of points, what fraction of the larger square s area is inside the shaded square? Express your answer as a common fraction. 5? Express your answer 40. How many distinct values are possible for the expression if one set of parentheses may be placed in the expression?

5 41. If x 7 = 7x +, what is the value of x? Express the product 1 10 as a mixed number It takes a window washer one hour to wash 10 windows. How many hours will it take him to wash all of the windows in a 36-story building if each story has 0 windows? 44. How much greater than 19 is 31? 45. In square ABCD with sides of length 4 cm, N is the midpoint of side BC and M is the midpoint of side CD. What is the area of triangle AMN, in cm? 46. The sum of four numbers is one-half. What is the mean of the four numbers? Express your answer as a common fraction. 47. One out of every five Americans suffers from allergies. In a random sample of 50 Americans, how many would you expect to suffer from allergies? 48. A criminal in Florida was sentenced to pay $180 million at the rate of $500 per month. At this rate, in how many years will this debt be paid in full? 49. What is the sum of all integers x for which 30 < x < 6? 50. What is the arithmetic mean of the integers from 4 through 5, inclusive? Express your answer as a decimal to the nearest tenth. 51. Calculate the sum If = N, what is the value of N? 53. Without seeing each other s number, Stephen and Carla will each write down a positive integer less than 100. What is the greatest possible difference between Carla s integer and Stephen s integer? 54. Calculate 75x x 14x. Express your answer in simplest radical form in terms of x. 55. What is the only integer whose square is less than its double? 56. What is the greatest common factor of 99 and 100?

6 57. The arithmetic mean of four different two-digit numbers is 60. What is the greatest possible value of any of these four numbers? 58. Patty ordered a pizza to be delivered. The bill was $1.00. Patty pays $15 and tells the delivery person to keep the change as a tip. What percent tip did she give? 59. Twelve students in Mrs. Stephenson s class have brown eyes. Twenty students in the class have a lunch box. Of Mrs. Stephenson s 30 students, what is the least possible number of students who have brown eyes and a lunch box? 60. Evaluate The ratio of boys to girls in a room is 6:5. If three boys leave the room, the ratio is 1:1. How many girls are in the room? 6. An elevator is at a height of 00 feet. It is descending at a constant rate of 8 feet per second. What is the height, in feet, of the elevator after 13 seconds? 63. What is the percent of increase in the rent for Bob s apartment if it went from $560 to $616? 64. What is the sum of the prime numbers between 10 and 0? 65. If you flip a fair coin four times, what is the probability of flipping at least three heads in a row? Express your answer as a common fraction. 66. In 1996, there were 500 cases of measles reported in the U.S. In 1960 there were 450,000 cases reported. What is the ratio of the number of cases in 1996 to the number of cases in 1960? Express your answer as a common fraction. 67. Juan takes a number, adds to it, multiplies the answer by, subtracts from the result, and finally divides that difference by. If his answer is 7, what was the original number? 68. A domino consists of two congruent squares placed side by side. The perimeter of the domino is 60 units. What is the area of the domino, in square units? 69. What is the tens digit of 5 005?

7 70. Evaluate As shown, a square is constructed on the hypotenuse of a right triangle whose legs have lengths 3 units and 4 units. What is the area of the pentagon ABCDE, in square units? 7. How many positive factors of 4000 are multiples of 100? 73. How many positive multiples of 5 that are less than 100 have a units digit of 5? 74. If 5x 3 = 1, what is the value of 5x + 3? 75. Eleven is the fifth prime number. What is the tenth prime number? 76. If f (n) = n + n + 17, what is the value of f (11)? 77. The notation [x] stands for the greatest integer that is less than or equal to x. Calculate [ 1.]. 78. A positive integer is 3 more than a multiple of 4 and 4 more than a multiple of 5. What is the least integer it could be? 79. If y x = 7, what is the value of x y? 80. If N is a positive integer and N = 91,04, what is the value of N?