MATH COUNTS 

1. MATHCOUNTS 2003 National Competition Countdown Round

2. 1. If 23x = 7, evaluate 8x+1.

3. Answer: 56

4. 2. Sixty percent of all cereal boxes have a prize inside. What is the probability that two randomly selected cereal boxes both contain a prize? Express your answer as a common fraction.

5. Answer:

6. 3. A 4' by 2' rectangular flag is made of a 2' by 2' black square bordered by a 1' by 2' white rectangle on each of two opposite sides of the black square. A white symbol covers one-third of the area of the black square. What fraction of the total area of the front of the flag is white? Express your answer as a common fraction.

7. Answer:

8. 4. How many positive integers n satisfy 200 < n2 < 900?

9. Answer: 15 (integers)

10. 5. The points (1, 2), (1, 4), (3, 4) and (3, 2) are the midpoints of the sides of a square. How many square units are in the area of the square?

11. Answer: 8 (square units)

12. 6. What is the greatest prime factor of 221?

13. Answer: 17

14. 7. The length of one leg of a right triangle is 9 meters. The lengths of the other two sides are consecutive integer numbers of meters. What is the number of meters in the perimeter of the triangle?

15. Answer: 90 (meters)

16. 8. A biker completes of a trip in 24 minutes. If the biker continues at the same average rate, how many more minutes are required to complete the trip?

17. Answer: 12 (minutes)

18. 9. This shape is built from six unit cubes. How many square units are in the surface area of the shape?

19. Answer: 26 (square units)

20. 10. A three-digit number “abc” has the property that the product of a and b is equal to c. What is the greatest possible value of the three-digit number?

21. Answer: 919

22. 11. Compute:

23. Answer: 16

24. 12. A six-faced die is weighted so that the probabilities of rolling a 1, 2 and 3 are equally likely; the probabilities of rolling a 4, 5 and 6 are equally likely; and the probability of rolling a 5 is four times the probability of rolling a 3. What is the probability that a 2 or a 5 occurs on a single roll of the die? Express your answer as a common fraction.

25. Answer:

26. 13. How many distinct numbers can be written as the sum of two or more distinct members of the set {0, 1, 2, 3, 4, 5}?

27. Answer: 15 (numbers)

28. 14. Angle A is the complement of angle B and the supplement of angle C. The measure of angle C is 106 degrees. How many degrees are in the measure of angle B?

29. Answer: 16 (degrees)

30. 15. How many positive integers n satisfy (n + 8)(n – 3)(n – 12) < 0 ?

31. Answer: 8 (integers)

32. 16. Compute: .

33. Answer: 90

34. 17. The decimal expansion of is a repeating decimal. What is the least number of digits in a repeating block of ?

35. Answer: 2 (digits)

36. 18. How many integers n are there such that 3  n 10 and 121 base n (or 121n) is a perfect square?

37. Answer: 8 (integers)

38. 19. Triangle PQR is isosceles and the measure of angle R is 40°. The possible measures of angle P are x, y or z degrees. What is the value of the sum x + y + z ?

39. Answer: 210 (degrees)

40. 20. Compute: 93 + 3·92 + 3·9 + 1

41. Answer: 1000

42. 21. In a class of 50 students, 28 participate in MATHCOUNTS, 21 participate in science club and 6 students participate in neither. How many students participate in both MATHCOUNTS and science club?

43. Answer: 5 (students)

44. 22. What is the 100th digit after the decimal point of the decimal representation of ?

45. Answer: 8

46. 23. Solve for x: 32x + 19 = 10x .

47. Answer: 2

48. 24. How many non-overlapping 9-inch by 9-inch square tiles does it take to cover a 12-foot by 15-foot room?

49. Answer: 320 (tiles)

50. 25. What is the ratio of the volume of cylinder A to the volume of cylinder B, if all measurements are in inches? Express your answer as a common fraction. 40 60 20 30 A B