1. MATHCOUNTS 
2007 National Competition
Countdown Round 1-25

2. Eighty percent of adults drink coffee and 70% drink tea
Eighty percent of adults drink coffee and 70% drink tea. What is the smallest possible percent of adults who drink both coffee and tea?

3. Answer: 50 (percent)

4. A pyramid with volume 40 cubic inches has a rectangular base
A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by 50%, what is the volume of the new pyramid, in cubic inches?

5. Answer: 360 (cubic inches)

6. For positive integers x and y, what is the least possible value of x if x3 = y2 + 2?

7. Answer: 3

8. Cory has 3 apples, 2 oranges and 2 bananas
Cory has 3 apples, 2 oranges and 2 bananas. If Cory eats one piece of his fruit per day for a week and the pieces of fruit within each category are indistinguishable, in how many orders can Cory eat the fruit? One such order is AAAOOBB.

9. Answer: 210 (orders)

10. If A:B:C = 2:1:4, what is the value of (3A + 2B)  (4C – A)
If A:B:C = 2:1:4, what is the value of (3A + 2B)  (4C – A)? Express your answer as a common fraction.

11. Answer:

12. Define the operation $ as a $ b = (a)(b) – a
Define the operation $ as a $ b = (a)(b) – a. What is the value of (3 $ 5) $ (5 $ 3)?

13. Answer: 108

14. What is the least possible product obtained from multiplying three distinct members of the set {- , 9, 5, 12, } ?

15. Answer: -54

16. A set of five integers has unique mode 7, median 9, and arithmetic mean 11. What is the greatest possible value in the set?

17. Answer: 22

18. Each of four students hands in a homework paper
Each of four students hands in a homework paper. Later the teacher hands back the graded papers randomly, one to each of the students. In how many ways can the papers be handed back such that every student receives someone else’s paper? The order in which the students receive their papers is irrelevant.

19. Answer: 9 (ways)

20. Given that x, , y, , z and are all integers, how many distinct values of x + y + z are possible?

21. Answer: 4 (values)

22. A 1 x 1 x 1 wire frame cube is to be made by gluing together pieces of wire. The wire can be bent to form corners of the cube. If exactly 12 units of wire is used to make the frame, what is the fewest number of pieces of wire that can be used to make the frame?

23. Answer: 4 (pieces)

24. A family of five has Momma, Poppa and three children
A family of five has Momma, Poppa and three children. Each child generates 2 loads of laundry per week while each parent generates loads per week. How many loads of laundry are generated by this family in 52 weeks?

25. Answer: 468 (loads)

26. Jane and her brother each spin this spinner once
Jane and her brother each spin this spinner once. The spinner has five congruent sectors. If the non-negative difference of their numbers is less than 3, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction. 1 2 3 4 5

27. Answer:

28. What is the ordered pair of real numbers (x, y) which satisfies the equation x + y – 7 + 4x – y + 12 = 0 ?

29. Answer: (-1, 8)

30. In the figure, the visible gray area within the larger circle is equal to three times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle? Express your answer as a common fraction.

31. Answer:

32. How many integers are there in the solution set of x – 2  5.6?

33. Answer: 11 (integers)

34. The three-digit positive integer N has a ones digit of 3
The three-digit positive integer N has a ones digit of 3. What is the probability that N is divisible by 3? Express your answer as a common fraction.

35. Answer:

36. Pizzas are sized by diameter
Pizzas are sized by diameter. What percent increase in area results if Chantel’s pizza increases from a 10-inch pizza to a 12-inch pizza?

37. Answer: 44 (percent)

38. Figure A is made from three unit squares, and will be called “a trio
Figure A is made from three unit squares, and will be called “a trio.” Figure B is a 4 by 4 grid of unit squares. Without overlap, what is the maximum number of trios that can be placed on the grid such that each trio covers exactly three of the unit squares in the grid?
Figure A
Figure B

39. Answer: 5 (trios)

40. While standing in line to buy concert tickets, Kit moved feet closer to the ticket window over a period of 30 minutes. At this rate, how many minutes will it take her to move the remaining 70 yards to the ticket window?

41. Answer: 105 (minutes)

42. On a number line, what is the coordinate of the point that is between point A at and point B at and is one-third the distance from A to B? Express your answer as a common fraction in terms of x.

43. Answer:

44. Sue has 122 crates of the same size
Sue has 122 crates of the same size. Each crate contains at least 90 apples and at most apples. Crates containing the same number of apples are stacked one on top of another in their own stack. Not all of the stacks have the same number of crates. What is the least possible number of crates in the tallest stack of crates?

45. Answer: 12 (crates)

46. Here are two functions:. f(x) = 3x2 – 2x + 4
Here are two functions: f(x) = 3x2 – 2x + 4 g(x) = x2 – kx – 6 If f(10) – g(10) = 10, what is the value of k?

47. Answer: -18

48. All positive integers whose digits add up to 11 are listed in increasing order: 29, 38, 47, … . What is the eleventh number in that list?

49. Answer: 137

50. How many subsets of {1, 2, 3, 4, 5} with two or more elements have the property that the sum of their elements is a positive, even number?

51. Answer: 13 (subsets)