MATHCOUNTS

2002 National Competition

Countdown Round

1. A rectangular field is half as wide as it is long, and it is completely enclosed by 54 meters of fencing. What is the number of square meters in the area of the field?

Answer: 162 (square meters)

2. What is the sum of the distinct prime factors of 735?

Answer: 15

3. Compute: .

15

25

35

95

105 ...

Answer: 11

4. How many square inches are in the area of a square inscribed in a circle of radius 6 inches?

Answer: 72 (square inches)

5. If 7 is placed to the right of a three-digit number to form a four-digit number, the new number is 7000 greater than the original number. What was the original number?

Answer: 777

6. What is the 87th odd positive integer?

Answer: 173

7. Simplify:

Express your answer as a common fraction.

2 2 2

2 2

4

3

n n

n

( )

( ).

Answer: 7

8

8. A rectangle having integer length and width has a perimeter of 100 units. What is the number of square units in the least possible area?

Answer: 49 (square units)

9. What is the units digit of

?20 21 22 23 24 25

1000

Answer: 2

10. The Badgers play the Cougars in a series of seven basketball games. Each team has an equal chance of winning each game. What is the probability that the Badgers will win at least four games? Express your answer as a common fraction.

Answer:1

2

11. How many prime numbers less than 100 have a units digit of 3?

Answer: 7 (numbers)

12. Brad is younger than 30. His age is a multiple of 5, and next year his age will be a multiple of 7. Brad is how many years old?

Answer: 20 (years)

13. How many prime numbers between 30 and 65 have a prime remainder when divided by 10?

Answer: 4 (numbers)

14. What is the smallest integer

value of n such that ? 21

1000 n

Answer: 10

15. The degree measures of the interior angles of a pentagon form an arithmetic sequence. What is the middle term of this sequence?

Answer: 108 (degrees)

16. How many numbers can be expressed as the sum of two or more distinct elements of the set {0, 1, 2, 4, 8, 16} ?

Answer: 31 (numbers)

17. A photograph measuring 16 inches by 20 inches is reduced uniformly so that the greater measure becomes 5 inches. What is the number of inches in the perimeter of the reduced photo?

Answer: 18 (inches)

18. How many distinct, natural-number factors does have?4 5 63 4 2

Answer: 135 (factors)

19. What is the sum of all integers x that satisfy ?1 1 10 ( ) x

Answer: 10

20. Compute: .15 35 21

Answer: 105

21. A standard die is rolled six times. What is the probability that the result of each roll is odd? Express your answer as a common fraction.

Answer:1

64

22. What is the sum of all values of x that are solutions to the equation ?x x2 7 12

Answer: 7

23. What is the smallest four-digit whole number divisible by 9 which has two even and two odd digits?

Answer: 1089

24. If the endpoints of one side of a square are at (2, 3) and (5, 4), then how many square units are in the area of the square?

Answer: 58 (square units)

25. The lengths of the sides of are 3 cm, 4 cm and 6 cm. Determine the number of centimeters in the least possible perimeter of a triangle similar to which has one side of length 12 cm.

ABC

ABC

Answer: 26 (centimeters)