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Natl2008

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  • 1. MATH COUNTS 2008 National Competition Countdown Round 
  • 2. Given the information shown in the table, what percent of the human body is made up of elements other than oxygen, carbon and hydrogen? Element Percent Carbon 65% Oxygen 18% Hydrogen 10%
  • 3. Answer: 7 (percent)
  • 4. The values of a , b , c and d are 1, 2, 3 and 4, but not necessarily in that order. What is the largest possible value of the sum of the four products ab , bc , cd and da ?
  • 5. Answer: 25
  • 6. Alexia designed a logo 2 inches wide and 1.5 inches tall to be used on her school’s notebooks. The school wants the logo on the notebook to be 8 inches wide. How tall, in inches, will it be if it is enlarged proportionately?
  • 7. Answer: 6 (inches)
  • 8. Tickets to a concert cost $16 for adults and $13 for children. A group of adults and children spent $154 for its tickets. How many adults were in the group?
  • 9. Answer: 8 (adults)
  • 10. A square of area 6 square units is split into rectangles of areas 3, 2 and 1 square units, as shown. What is the ratio of the length of the longer side of the gray rectangle to the length of the shorter side of the gray rectangle? Express your answer as a common fraction. 2 1 3
  • 11. Answer:
  • 12. What real number x satisfies the equation (3 x )(4 x ) = 7 x ?
  • 13. Answer: 0
  • 14. BoatWorks built 3 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of March of this year?
  • 15. Answer: 21 (canoes)
  • 16. Bob can run 1 mile in the same time that Gill can run of a mile. Cindy can run of a mile in the same time that Gill can run 1 mile. If each person runs at those rates, how many miles will Cindy run in the time it takes Bob to complete an 8-mile run? 5 6 3 4
  • 17. Answer: 5 (miles)
  • 18. Each of the four students on each of 16 MATHCOUNTS teams wins a math book valued at $125. What is the total value, in dollars, of all of the math books received by the students?
  • 19. Answer: 8000 or 8000.00 (dollars)
  • 20. For integers a and b , what is the smallest possible positive value of 8 a + 12 b ?
  • 21. Answer: 4
  • 22. The Norman H.S. math club has twice as many male members as female members. Twenty percent of the male members and 30% of the female members participated in a math contest. What fraction of those members participating in the math contest were female? Express your answer as a common fraction.
  • 23. Answer:
  • 24. The first three terms of an arithmetic sequence are 1, 10 and 19, respectively. What is the value of the 21st term?
  • 25. Answer: 181
  • 26. Forty-nine of the first 50 positive integers are in a list whose sum is 1257. Which one of the first 50 positive integers is missing from the list?
  • 27. Answer: 18
  • 28. What is the slope of a line perpendicular to the line whose equation is ? Express your answer as a common fraction.
  • 29. Answer:
  • 30. What is the sum of all possible solutions of ?
  • 31. Answer: 0
  • 32. When Julian spends his weekly allowance at the bookstore, his father lets him buy a book whose regular price is double his allowance. However, he gets a 10% discount on the price of the book. What fraction of the final purchase price is Julian’s weekly allowance? Express your answer as a common fraction.
  • 33. Answer:
  • 34. Sandy is thinking of a two-digit perfect square, x . The sum of the digits of x is also a perfect square. What is the sum of the possible values of x ?
  • 35. Answer: 117
  • 36. Each term of a sequence, after the first term, is inversely proportional to the term preceding it, and the constant of proportionality stays the same. If the first term is 2 and the second term is 5, what is the 12th term?
  • 37. Answer: 5
  • 38. If 4 daps = 7 yaps and 5 yaps = 3 baps, how many daps equal 42 baps?
  • 39. Answer: 40 (daps)
  • 40. Math City has eight streets, all of which are straight. No street is parallel to another street. One police officer is stationed at each intersection. What is the greatest number of police officers needed?
  • 41. Answer: 28 (police officers)
  • 42. The equation x 2 – kx – 12 = 0 has only integer solutions for certain positive integers k . What is the sum of all such values of k ?
  • 43. Answer: 16
  • 44. Paint is to be mixed so that the ratio of red paint to white paint is 3 to 2. If Ben wants to make 30 cans of the mixture and all cans hold the same volume of paint, how many cans of red paint will he need?
  • 45. Answer: 18 (cans)
  • 46. Kyle’s birthday party in 2007 cost d dollars per person and p people attended. In 2008, the cost per person doubled and the number of people attending the party also doubled. What is the ratio of the total cost of the party in 2007 to the total cost of the party in 2008? Express your answer as a common fraction.
  • 47. Answer:
  • 48. An integer is randomly selected from the integers 0 through 99, inclusive. What is the probability that the integer is in the range 10 through 20, inclusive? Express your answer as a common fraction.
  • 49. Answer:
  • 50. The two solutions of the equation x 2 + bx + 18 = 0 are in the ratio of 2 to 1 for some values of b . What is the largest possible value of b ?
  • 51. Answer: 9