MATH COUNTS ®

1. MATHCOUNTS® 2000 National Competition Countdown Round

3. In the Village League, the team to win two of three softball games becomes the champion. If the probability of Team Alpha beating Team Beta is 60% for every game, what is the probability that Beta wins the championship? Express your answer as a common fraction.

4. 44 125 Answer:

5. Circles A and B are congruent. Circle A is rolled around circle B, remaining tangent at all times. Circle A rolls around circle B exactly once. How many times will circle A revolve around its own center before the radii are lined up again as shown? A B

6. Answer: 2 (times)

7. A line that passes through (-5,-8) and (-3,-4) will cross the y-axis at what y-coordinate?

8. Answer: 2

9. What is the number of inches in the radius of a circle whose area is one-half the area of a circle with radius 4 inches? Express your answer in simplest radical form.

10. Answer: (inches) 2 2

11. A unicycle has a wheel with a radius of 1 foot. How many complete revolutions will the wheel make when the unicycle rolls 100 feet? Express your answer to the nearest whole number.

12. Answer: 16 (revolutions)

13. What is the value of - + - + - + 1 2 3 4 5 6 . . . - + - ? 98 99 100

14. Answer: –50

15. A 3-foot high tree was planted and grows by an equal number of feet each year. At the end of the seventh year, it is 1/9 taller than at the end of the sixth year. How many feet tall will it be at the end of the 13th year?

16. Answer: 16 (feet)

17. x, y and z are positive odd integers. What is the remainder when is divided by 4? + + 2 2 2 x y z

18. Answer: 3

19. In how many consecutive zeroes does the product end? ´ ´ ´ ´ 115 116 117 201 L

20. Answer: 23 (zeroes)

21. What is the total number of different committees that can be formed by selecting one or more persons from a group of six people?

22. Answer: 63 (committees)

23. In ABC, AB = 5 cm, BC = 10 cm, and the altitude drawn to AB is 8 cm. What is the number of centimeters in the length of the altitude to BC?

24. Answer: 4 (centimeters)

25. When four numbers are added three at a time, the four sums are 42, 43, 47 and 48. What is the sum of the four numbers?

26. Answer: 60

27. The average of 11 consecutive even integers is 24. What is the greatest of these integers?

28. Answer: 34

29. What is the fewest possible number of units in the perimeter of a triangle with side lengths that are relatively prime integers?

30. Answer: 12 (units)

31. Four distinct digits are used to make 2 two-digit numbers. What is the greatest possible product of the two numbers formed?

32. Answer: 8352

33. How many whole numbers from 10 to 99 have a units digit greater than the tens digit?

34. Answer: 36 (numbers)

35. A 200% increase is the same as a 50% increase followed by what other percent increase?

36. Answer: 100 (percent)

37. What percent of the first 200 prime numbers have reciprocals less than 0.05?

38. Answer: 96 (percent)

39. What is the probability that the product of two numbers chosen randomly from the set of all positive integers is divisible by 2? Express your answer as a common fraction.

40. 3 Answer: 4

41. A circle has a radius of 10 centimeters and a chord of the circle is 16 centimeters long. How many centimeters is the midpoint of the chord from the center of the circle?

42. Answer: 6 (centimeters)

43. How many miles per hour is 1298 feet per second?

44. Answer: 885 (mph)

45. The ratio of length to width to height of a rectangular prism is 3:2:1. If the surface area of the prism is 198 m2, how many cubic meters are in its volume?

46. Answer: 162 (cubic meters)

47. In 1984, July 4 fell on a Wednesday. On what day of the week did July 4, 1990, fall?

48. Answer: Wednesday

49. The measures of the angles of a triangle are in the ratio of . What is the number of degrees in the measure of the largest angle? : : 1 1 1 3 4 6

50. Answer: 80 (degrees)