MATHCOUNTSÒ School Competition Sprint Round Problems 1 30 DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

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1 MATHCOUNTSÒ 2002 School Competition Sprint Round Problems 1 30 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, slide rules, books or other aids during this round. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the right-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers. Total Correct Scorer s Initials Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics National Sponsors The Dow Chemical Company Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Phillips Petroleum Company Texas Instruments Incorporated 3M Foundation Toyota USA Foundation 2001 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314

2 MATHCOUNTSÒ 2002 School Competition Target Round Problems 1 and 2 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of eight problems, which will be presented in pairs. Work on one pair of problems will be completed and answers will be collected before the next pair is distributed. The time limit for each pair of problems is six minutes. The first pair of problems is on the other side of this sheet. When told to do so, turn the page over and begin working. Record your final answer in the designated space on the problem sheet. All answers must be complete, legible and simplified to lowest terms. This round assumes the use of calculators, and calculations may also be done on scratch paper, but no other aids are allowed. If you complete the problems before time is called, use the time remaining to check your answers. Total Correct Scorer s Initials Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics National Sponsors The Dow Chemical Company Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Phillips Petroleum Company Texas Instruments Incorporated 3M Foundation Toyota USA Foundation 2001 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314

3 MATHCOUNTSÒ 2002 School Competition Target Round Problems 3 and 4 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Scorer s Initials Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics National Sponsors The Dow Chemical Company Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Phillips Petroleum Company Texas Instruments Incorporated 3M Foundation Toyota USA Foundation 2001 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314

4 MATHCOUNTSÒ 2002 School Competition Target Round Problems 5 and 6 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Scorer s Initials Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics National Sponsors The Dow Chemical Company Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Phillips Petroleum Company Texas Instruments Incorporated 3M Foundation Toyota USA Foundation 2001 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314

5 MATHCOUNTSÒ 2002 School Competition Target Round Problems 7 and 8 Name DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. Total Correct Scorer s Initials Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics National Sponsors The Dow Chemical Company Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Phillips Petroleum Company Texas Instruments Incorporated 3M Foundation Toyota USA Foundation 2001 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314

6 MATHCOUNTSÒ 2002 School Competition Team Round Problems 1 10 DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of ten problems which the team has 20 minutes to complete. Team members may work together in any way to solve the problems. Team members may talk during this section of the competition. This round assumes the use of calculators, and calculations may also be done on scratch paper, but no other aids are allowed. All answers must be complete, legible and simplified to lowest terms. The team captain must record the team s official answers on his/her own problem sheet. If the team completes the problems before time is called, use the remaining time to check your answers. Team Members, Captain Total Correct Scorer s Initials Founding Sponsors CNA Foundation National Society of Professional Engineers National Council of Teachers of Mathematics National Sponsors The Dow Chemical Company Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Phillips Petroleum Company Texas Instruments Incorporated 3M Foundation Toyota USA Foundation 2001 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314

7 MATHCOUNTSÒ 2002 School Competition Countdown Round Problems 1 60 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 The Dow Chemical Company Foundation General Motors Foundation Lockheed Martin National Aeronautics and Space Administration Phillips Petroleum Company Texas Instruments Incorporated 3M Foundation Toyota USA Foundation 2001 MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314

8 Sprint Round Answers (segments) 2. ($) (dollars) (square centimeters) (numbers) (cents) (pins) (minutes) (triangles) ($) (dollars) (minutes) (eggs) (centimeters) (digits) (participants) (cents or cents per ounce) (seconds) (coins) Target Round Answers (percent) ($) 25,500 (dollars) (minutes) (degrees) (inches) (feet) Team Round Answers 1. 3 (sets) 2. ($) 3,276,000 (dollars) (pounds) (pizzas) (square units) (cubic centimeters) MATHCOUNTS Foundation: 2002 School Answer Key

9 Countdown Round Answers 1. 21, (eighths) (degrees) (degrees) (dollars) (square inches) (integers) (inches) (one-point game pieces) (meters) (degrees) (hours) (comic strips) (students) (square meters) (dollars) (seats) (ounces) 24. 9π (square inches) (percent) (combinations) (percent) (cubic units) (dollars) (square feet) , (ounces) (square units) Tuesday (cubic feet) (degree Celsius) (feet) (tiles) (stamps) MATHCOUNTS Foundation: 2002 School Answer Key

10 What is the value of ? 2. What is the value obtained by dividing the number of square inches in a square yard by the number of square inches in a square foot? 3. What is the number of square inches in the area of a rectangular tablecloth that measures 4 feet by 5 feet? What is the value of ? 5. How many different positive integers can be represented as a difference of two distinct members of the set {1, 2, 3, 4, 5, 6 }? 6. How many hours will it take Katina to drive 495 miles at 45 miles per hour? 7. Ten students stand on a circle. No two adjacent students have brown hair. What is the maximum number of these students that could have brown hair? 5 8. The bill for dinner was $63. Joanna agreed to pay of the bill, 9 since she had dessert. How many dollars should Joanna pay? 9. John s diet requires him to eat, at most, 30 grams of carbohydrates at snack time. If he selects chips with 24 grams of carbohydrates per ounce, how many ounces of chips may he eat at snack time? Express your answer as a decimal to the nearest hundredth. 10. What single percent discount from the original price is equivalent to a 40% discount on an item already on sale at a 20% discount from the original price? 11. The volume of a cone is given by the formula V = 1 Bh, where B is 3 the area of the base and h is the height. The area of the base of a cone is 30 square units, and its height is 6.5 units. What is the number of cubic units in its volume? 12. If 25% of n is 18, what is 125% of n? 13. If 3x = 8y, 12y = 7z and kz = 72x, then what is the value of k? 14. Ashley s old dresser has nine drawers. Six of the drawers each have a volume of 2 ft 3, and three drawers each have a volume of 1 ft 3. Her new dresser has the same total volume in five equal-sized drawers. What is the volume in cubic feet of each drawer of her new dresser? 15. Bryan has some 3 stamps and some 4 stamps. What is the least number of stamps he can combine so the value of the stamps is 33? MATHCOUNTS Foundation: 2002 School Countdown Round

11 16. If a * b = 2a + 3b for all a and b, then what is the value of 4 * 3? 17. How many degrees are in the complement of a 54 angle? 18. The sum of two numbers is 352. One number is three times the other. What is the value of the smaller number? 19. A couple plans to have three children. The birth of a boy or girl is equally likely. What is the probability that all three children will be the same sex? Express your answer as a common fraction. 20. What is the positive difference, in inches, between the greatest cabin height and the least cabin height of the following business jets? Business Jet Learjet 31A Learjet 60 Cessna Citation V Gulfstream V Cabin Height 4 feet, 5 inches 5 feet, 8 inches 4 feet, 7 inches 6 feet, 2 inches 21. Charles Schultz drew 18,000 comic strips over a 50-year period. What is the mean number of comic strips he drew each year? 22. The mean of ten numbers is 7.2. When an 11 th number is added to the set, the mean becomes 5. What is the 11 th number? 23. Each row of the Misty Moon Amphitheater has 30 seats. Rows 12 through 22 are reserved for a special group. How many seats are reserved for this group? 24. A circular pizza of diameter 12 inches is cut into 4 pieces of equal area. What is the number of square inches in the area of each piece? Express your answer in terms of π. 25. On a survey, students must give exactly one of the answers provided to each of these three questions: a) Were you born before 1990? (Yes / No) b) What is your favorite color? (Red / Green / Blue / Other) c) Do you play a musical instrument? (Yes / No) How many different answer combinations are possible? 26. What is the remainder when the sum of the first five primes is divided by the sixth prime? 27. What is the positive difference between one-million and seventy-two thousand? 28. The value of 400,000 squared can be written as 1.6 x 10 n. What is the value of n? MATHCOUNTS Foundation: 2002 School Countdown Round

12 29. On five consecutive days, the low temperatures in Sioux City, SD were 2, -9, 9, 7 and -4, measured in degrees Celsius. How many degrees Celsius was the mean of these low temperatures? 30. What is the greatest possible quotient using two distinct numbers from the set { 0.2, 0.5, 0.8, 3.2 }? 31. How many eighths are in 5.75? 32. Compute: 98 x The table below represents the male population of students attending Walker Middle School. One boy is selected at random from the male students. What is the probability that he is in seventh grade? Express your answer as a common fraction th Grade 8 th Grade 9 th Grade Males Marcia has a total of 65 game pieces worth 89 points. Each game piece is worth one or two points. How many one-point game pieces does she have? 35. The sum of three numbers is 90. One of the numbers is 22. What is the arithmetic mean of the other two numbers? 36. For what value of n does 3 8 = 9 n? 37. The longest side of a right triangle is 5 meters, and the shortest side is 3 meters. What is the area of the triangle in square meters? 1 x What integer x satisfies < <? Compute: What percent of a 48-minute basketball game is over if the time remaining is one-third the time played? 41. If carpet costs $18 per square yard, how many dollars will it cost to completely carpet a 15 foot by 18 foot rectangular floor? 42. Alexa has a mixture consisting of 8 ounces of red jelly beans and 13 ounces of green jelly beans. What is the minimum number of ounces of white jelly beans she must buy so that, when added to the old mixture, her new mixture is an integer number of pounds? MATHCOUNTS Foundation: 2002 School Countdown Round

13 43. April 1, 2002 is a Monday. What day is April 23, 2002? 44. The area of a triangle is 600 ft 2. If the length of the base is 30 feet, then what is the number of feet in the altitude? 45. For what positive integer n is n 2 = 2 6? 46. In parallelogram ABCD, angle B measures 110. What is the number of degrees in the measure of angle C? 47. Together, Larry and Lenny have $35. Larry has two-fifths of Lenny s amount. How many more dollars than Larry does Lenny have? 48. An airplane travels at 360 kilometers per hour. How many meters does it travel in one second? 49. For what value of n does = 4.8 x 10 n? 50. How many degrees is the measure of the supplement of an angle whose complement measures 50? 51. By how much does 3.5 exceed its reciprocal? Express your answer as a common fraction What is of 200? 2 % 53. A rectangular sheet of paper with dimensions 9 inches by 13 inches is cut into n identical notecards, each of which is 3 inches by 2.6 inches. What is the maximum value of n? What is the ratio of + to + +? Express your answer as a common fraction. 55. What is the value of 169.? Express your answer as a decimal to the nearest tenth. 56. How many square feet are in a square yard? 57. What is the number of square units in the area of a triangle whose sides are 9, 12 and 15 units? a 58. If and, then what is the value of? b =12 b c =20 a c 59. What is the least number of 2 cm x 5 cm rectangular tiles that will completely cover a 30 cm x 35 cm rectangle? 60. Two percent of half a number is 5. What is the number? MATHCOUNTS Foundation: 2002 School Countdown Round

14 Calculate: +. Express your answer as a common fraction Midtown Taxi charges $1.60 plus $0.25 per one-eighth mile. The distance from the hotel to the airport is miles. Two passengers will evenly split the fare. How many dollars will each passenger pay? Express your answer to the nearest hundredth Calculate: 4 + ( -7 ) ( -7 ) ( ) Anjali has 230 quarters, 300 dimes and 165 nickels. Niki has 210 quarters, 316 dimes and 173 nickels. How many more cents does Anjali have than Niki? A space shuttle travels from Earth towards the moon at a speed of 30,000 kilometers per hour for 12 hours. The total distance to the moon is 400,000 kilometers. At the current velocity, how many additional minutes are needed to reach the moon? 5. N 6. A whole number, N, is chosen so that is strictly between 7.5 and 3 8. What is the value of N? MATHCOUNTS Foundation: 2002 School Sprint Round

15 7. Morgan has 3 hockey shirts, 2 football shirts and 7 baseball shirts in her closet. If she randomly selects one of these shirts, what is the probability that it will not be a baseball shirt? Express your answer as a common fraction =. What is the value of x? 2 3 x What is the greatest possible number of digits in the product of a 4-digit whole number and a 3-digit whole number? The price of a 12 ounce box of cereal is $2.88 and an 18 ounce box of the same cereal sells for $3.78. How many cents per ounce are saved by buying the larger box? Simplify:. Express your answer as a common fraction A cube has a volume of 343 cubic cm. What is the number of square centimeters in the surface area of the cube? MATHCOUNTS Foundation: 2002 School Sprint Round

16 13. What is the greatest three-digit multiple of 33 that can be written using three different digits? John divided his souvenir hat pins into two piles. The two piles had an equal number of pins. He gave his brother one-half of one-third of one pile. John had 66 pins left. How many pins did John originally have? The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers? Working 22 hours in the second week of June, Xenia was able to earn $47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth Of the eggs produced by salmon, 80% hatch, and of those, 25% survive to migrate to the ocean. How many eggs are needed to produce 100 salmon that migrate to the ocean? Julian has 800 identical 1 cm by 1 cm by 1 cm cubes. What is the number of centimeters in the length of one edge of the largest solid cube that Julian could build with these cubes? MATHCOUNTS Foundation: 2002 School Sprint Round

17 19. The ratio of girls to boys participating in intramural volleyball at Ashland Middle School is 7 to 4. There are 42 girls in the program. What is the total number of participants? Four seconds pass between the first and third flash of a strobe light. The rate at which the strobe flashes is constant. How many seconds will pass between the first and the twelfth flash of the same light? The segments below form 26 congruent squares. What is the maximum number of segments, both vertical and horizontal, that a straight line, p, could intersect if line p is not collinear with any of the grid s segments? How many odd whole numbers are factors of 180? Calculate: Adele can mow her entire yard in three hours. Her sister, Brooke, can mow 3/4 of the same yard in two hours. Using two identical mowers, what part of the yard can they mow in one hour working together? Express your answer as a common fraction MATHCOUNTS Foundation: 2002 School Sprint Round

18 25. Congruent equilateral triangles are used to tessellate the plane. How many triangles share a common vertex? While practicing for a running race, Nakimbe averaged 4 mph for the first 4 miles, 3 mph for the next 5 miles and 2 mph for the last mile. What was the total number of minutes that he ran? A bag contains 16 congruent blocks, 4 of which are blue and 12 of which are red. Two blocks are randomly pulled from the bag at the same time. What is the probability that both blocks are blue? Express your answer as a common fraction What is the greatest prime factor of the sum of the arithmetic sequence ? A number x is twice its reciprocal. What is x 6? What is the minimum number of United States coins Samantha needs (pennies, nickels, dimes, quarters, half-dollars) to ensure she is capable of making change for any amount of money from one cent to 99 cents? MATHCOUNTS Foundation: 2002 School Sprint Round

19 1. Abigail earned 72% of the first 800 possible points in history class. To achieve an overall average of 80%, what percent of the remaining 400 points must she earn? Ms. Beller graded 63 quizzes from her algebra class in 49 minutes. If she continues grading at the same rate, how many minutes will it take her to grade the remaining 45 quizzes? MATHCOUNTS Foundation: 2002 School Target Round

20 3. The sum of four consecutive multiples of 5 is 110. What is the greatest of the four numbers? The pie chart below shows a recent distribution of the populations of the three largest countries in North America. At this time the population of the United States was 275 million, Canada s population was 32 million and the population of Mexico was 98 million. Compute the degree measure of angle m for Mexico s sector. Express your answer to the nearest degree MATHCOUNTS Foundation: 2002 School Target Round

21 5. A contestant on a quiz show wins $100 for answering the first question correctly, $200 for answering the second question correctly, and so on, winning twice as much for each CORRECT answer as for the previous CORRECT answer. Alvin answered eight consecutive questions correctly. What is the number of dollars he won? Two right circular cylinders have equal volume. The first cylinder has a radius of 6 inches and a height of 12 inches. What is the number of inches in the height of the second cylinder if its radius is 8 inches? Express your answer as a mixed number MATHCOUNTS Foundation: 2002 School Target Round

22 7. What is the least whole number divisible by four different prime numbers? A square piece of land with side length 1/4 mile has an area of 40 acres. A second square piece of land has an area of one acre. There are 5280 feet in one mile. What is the number of feet in the length of a side of this second piece of land? Express your answer to the nearest whole number MATHCOUNTS Foundation: 2002 School Target Round

23 1. A set of magnetic letters contains two of each consonant and three of each vowel. Only a complete set of letters can be purchased. How many complete sets of magnetic letters must be purchased to make a sign which reads: MATHCOUNTS COMPETITION TODAY? Lemon Ice sold 3.9 million CD s at $14.00 each. Lemon Ice received a royalty of 6% for each CD sold. How many dollars did Lemon Ice receive? Fill in the two blanks in the expression with two different operation symbols from this list:, +,. What is the greatest possible value of this expression? Express your answer as a decimal to the nearest tenth Aaron s weight is 10 pounds greater than twice Levi s weight. If the sum of their weights is 118 pounds, how many more pounds is Aaron s weight than Levi s weight? Mikka wants to order a pizza with two different toppings. He has 8 different toppings to choose from. How many different pizzas could he order? MATHCOUNTS Foundation: 2002 School Team Round

24 6. A triangular corner region is sliced off of a rectangular region as shown. What is the number of square units in the area of the pentagonal region ABEFD that remains? Find a value for x so that the list 100, 102, 104, x, x, 106, 104, 118, 113, 101 has the same mean and mode In the Multiplication Magic Square shown, 8. A B 9 C D the product of the three numbers in any row and in any column is the same number. What is the sum A + B + C + D? 9. An equilateral triangle of side 14 centimeters is rotated about an altitude to form a cone. What is the number of cubic centimeters in the volume of the cone? Express your answer to the nearest whole number The sum of the reciprocals of three consecutive integers is. 60 What is the sum of the three integers? MATHCOUNTS Foundation: 2002 School Team Round