COUNTDOWN ROUND

STATE1998 - 1999

1. How many of the first 100 positive integers are neither perfect squares nor perfect cubes?

88

2. A palindrome is a number that reads the same forwards and backwards. How many palindromes are between 1,000 and 10,000?

90

3. The surface of a 4” x 5” x 6” block is painted yellow. If the block is separated into one-inch cubes, what is number of cubes with exactly one yellow face?

52

4. The lengths of two sides of a right triangle are 2 11 meters and 6 3 meters. The number of meters in the length of the third side is a whole number. How many meters are in the third side of the triangle?

8

5. A bag contains yellow and blue marbles. The probability of selecting a yellow marble from the bag is 8/17. If there are 27 blue marbles in the bag, what is the number of yellow marbles?

24

6. A uniform piece of lumber is 12ft. long and weighs 30 lb. If a 5ft. length is removed, what is the number of pounds in the remaining piece? Express your answer as a mixed number.

17 ½

7. If q, r, and s, each represent an integer 0-9, what is the value of s?

qqqq rrrr

+ssss rsssq

8

8. Compute: (262 - 242)

10,000

9. What number is 11 1/9% of 13,239?

1471

10. For what whole number n is 10n < 695000 < 10n+1?

5

11. What is the ratio of 0.16 to 0.83? Express your answer as a common fraction.

Michael D. Aide:

.16 6 repeats and .83 3 repeats

Michael D. Aide:

.16 6 repeats and .83 3 repeats

1 5

12. The measures of the angles of a pentagon are in the ratio of 3:3:3:4:5. What is the number of degrees in the measure of the largest angle?

150

13. A digital clock shows hours and minutes. How many times during a twelve hour period will the sum of the digits on the clock be 3?

12

14. It takes 6 days for a team of 6 workers to carpet a house. At the same rate, what is the minimum number of additional workers needed to carpet the same house in just five days?

2

15. If a + b = 12 and a – b = 6, what is the value of a2 + 2a + 2b + b2?

114

16. John has some loaves of bread in a bag. If he had 6 more loaves, he would have three times as many. How many loaves does John have in the bag?

3

17. If a two-digit prime number is selected a random, what is the probability that the sum of its digits is 9?

0

18. In a right ABC, CAB is a right angle. Point M is the midpoint of BC. What is the number of centimeters in the length of median AM? Express your answer as a decimal to the nearest tenth.

C M

3 cm A 4cm B

2.5

19. What is the number of degrees in the exterior angle of the largest angle of a triangle if the interior angles are in the ratio of 2 : 3 : 5?

90??? (144)

20. A 12 feet by 18 feet rectangular Persian rug is centered within a rectangular room so that a region of constant width surrounds the carpet. Given that 96 square feet of flooring is not covered by the rug, what is the number of feet in the measure of the longer dimension of the room?

16

21. What is the least positive integer that is not a factor of 7!?

11

22. A right circular cone is inscribed in a right cylinder. The volume of

the cylinder is 72π cubic centimeters. What is the number of cubic centimeters available inside the cylinder but outside of the cone?

Express your answer in terms of π

48π

23. Compute 12 ½% of 240.

30

24. In an isosceles trapezoid, the lengths of the bases are 8cm and 16cm. The length of a diagonal is 13cm. What is the number of square centimeters in the area of the trapezoid?

60

25. What is the greatest two-digit whole number that can be expressed as the product of two consecutive whole numbers?

90

26. What is the greatest three-digit whole number that is divisible by 6?

996

27. If the measure of each side of a square is decreased by 10%, by what percent is the area decreased?

19

28. What is the units digit of 2345 + 6789?

0

All squares and circles have the same value. What is the sum of the three circles?

+ + + + = 21+ + + + = 19

+ + = ?

9

30. Six less than a number is multiplied by 6. Eight less than the same number is multiplied by 8. The two results are equal. What is the number?

14

31. When Sarah turns 36 years old, her father will be 62. Currently, Sarah is 1/3 of her father’s age. How many years old is Sarah now?

13

32. The sum of seven consecutive even numbers is 13,902. What is the smallest of these numbers?

1980

33. Eleven coins total $1.19, but no combination of the coins will give a value of exactly $1. How many of the coins are quarters?

3

34. Jimi is taking a bicycle trip and he and needs to arrive by 6 o’clock. If he travels an average rate of 15 mph, he’ll arrive at his destination one hour early, but if he travels at an average rate of 10 mph, he’ll arrive one hour late. How many miles away is his destination.

60

35. Mark baked cookies for the neighborhood children. He gave each child 6 cookies and he had 7 cookies remaining. He was going to give another cookie to each child, but he was 1 cookie short. How many cookies had Mark baked?

55

36. An eight-digit number contains two 5’s, two 6’s, two 7’s, and two 8’s. The 5’s are separated by one digit, the 6’s are separated by two digits, the 7’s are separated by three digits, and the 8’s are separated by four digits. What is the greatest possible value for the number?

85,756,876

37. Calculate: (243)3/5

27

38. Calculate and express your answer as a common fraction:

2

2 + 2 + 2

3 5 7

105 71

39. What is the least positive integer n such that < n? 16

n

5

40. How many different three-member committees can be chosen from a group of 10 people?

120

41. A rectangle with length 8 units and width 5 units is centered at the origin of a coordinate plane, and its longer side parallel to the x-axis. What percent of the rectangular region is to the right of x = -2?

75

42. What is the smallest natural number that has exactly five factors?

16

43. For what value of n does 5n – 5 =1/125

2

44. What is the sum of the real values of x for which the following expression is undefined? Express your answer as a common fraction.

8 5 _ 2

x 3

15 2

45. A 3” x 5” photograph can be enlarged to a panoramic view that is 5” x 10”. What is the area of the original photograph to the area of the enlargement? Express your answer as a common fraction.

3 10

46. The sum of two numbers is 9, and their difference is 7. What is the positive difference of the squares of these two original numbers?

63

47. One diagonal of a rhombus is 48 inches. Given that one side of the rhombus has length 26 inches, what is the number of square inches in the area of the rhombus?

480

A centipede climbs a 40-foot tree. Each day he climbs 5 feet, and each night he slides down 2 feet. In how many days will the centipede reach the top of the tree?

X