MATHCOUNTS 2019 School Competition Countdown Round Please note

MATHCOUNTS 2019 School Competition Countdown Round

Please note that videotaping, photographing, reproducing or publishing the following questions or answers is strictly prohibited. A sample question follows that you are allowed to reproduce.

Sample Question If the price of a stamp is 33¢, what is the maximum number of stamps that could be purchased with $32?

Sample Question Answer: 96 (stamps)

MATHCOUNTS 2019 School Competition Countdown Round

1. What is the units digit when 4 5 is expressed as an integer?

Answer: 5

2. What is the quotient when 3 is divided by its reciprocal?

Answer: 9

3. The complement of angle M has measure 10 degrees. What is the degree measure of angle M?

Answer: 80 (degrees)

4. Stephen has four blue socks and six gray socks in a drawer. What is the least number of socks he must select, at random, to guarantee having at least two socks of the same color?

Answer: 3 (socks)

5. In the land of Ink, the money system is unique. One Trinket is equal to 4 Blinkets, and 3 Blinkets are equal to 7 Drinkets. In Trinkets, what is the value of 56 Drinkets?

Answer: 6 (Trinkets)

6. An integer is tripled, and the result is divided by 2. When the new result is reduced by 1, the final result is 11. What was the original integer?

Answer: 8

7. The acute angles of a particular isosceles trapezoid each measure 40 degrees. What is the measure, in degrees, of each obtuse angle in the trapezoid?

Answer: 140 (degrees)

8. How many distinct prime divisors does 56 have?

Answer: 2 (divisors)

9. The quotient of two positive 5 integers is and their product 2 is 160. What is the value of the larger of the two integers?

Answer: 20

10. During a year when Thanksgiving is on Thursday, November 23, on what day of the week does December 23 occur?

Answer: Saturday

11. A particular convex pentagon has two congruent acute angles. The measure of each of the other interior angles is equal to the sum of the measures of the two acute angles. What is the common measure of the large angles, in degrees?

Answer: 135 (degrees)

12. Define the operation ☼ as K ☼ L = (K + L)(K − L) for all integers K and L. What is the value of 6 ☼ 5?

Answer: 11

13. Margaret started a stamp collection. She collected 8 stamps the first day. Each subsequent day she collected 8 more stamps than she collected the previous day. If she collected stamps for 5 consecutive days, what was the average number of stamps she collected per day?

Answer: 24 (stamps)

1 14. A 2 -cup mixture is flour 3 2 and cornmeal. If 1 cup of flour 3 is added to the 2 -cup mixture, what fraction of the new 3 -cup mixture is flour? Express your answer as a common fraction.

5 Answer: 9

15. What is the value of 8(− 6) + 8(10) − 8(4)?

Answer: 0

16. On a certain farm, each chicken has two feet and each rabbit has four feet. If the combined number of chickens and rabbits on the farm is 100 and there a total of 260 feet on these animals, how many chickens are there?

Answer: 70 (chickens)

17. Angie’s class has 2 girls for every 3 boys. If there are 20 students in the class, how many girls are in Angie’s class?

Answer: 8 (girls)

18. What is the area, in square units, of a rectangle with side lengths 15 and 16 units?

Answer: 240 (units 2)

19. Ivory successfully shot 7 free throws in 15 free-throw attempts. How many additional successful free throws, without a miss, must she make in order to attain a success rate of 75%?

Answer: 17 (free throws)

20. Jeremiah is riding in a car that is traveling 60 mi/h. At this rate, how many minutes will it take him to travel 20 miles?

Answer: 20 (minutes)

21. How many unique diagonals can be drawn in a five-sided convex polygon?

Answer: 5 (diagonals)

22. What is the units digit of 313 + 133?

Answer: 8

23. What is the smallest integer side length, in units, of a square whose area is numerically greater than its perimeter?

Answer: 5 (units)

24. If 3 x = 8 y and 5 y = 15 z, x what is the value of ? z

Answer: 8

25. The owner of a house worth $120, 000 pays $3, 000 in taxes. At this rate, how much, in dollars, will the owner of a house worth $160, 000 pay?

Answer: 4000 (dollars)

26. How many positive divisors of 72 are perfect cubes?

Answer: 2 (divisors)

27. A sequence starts with the term 2222. Each succeeding term is found by adding 1010 to the previous term. What is the sum of the sixth and seventh terms?

Answer: 15, 554

28. A cone and a cylinder are of equal height and have congruent bases. The volume 3 of the cylinder is 30 in. What is the volume of the cone, in cubic inches?

Answer: 10 (in 3)

29. Five workers paint 4 houses in 6 days. Working at the same rate as these workers, how many workers are needed to paint 12 houses in 3 days?

Answer: 30 (workers)

30. How many positive two-digit integers are divisors of both 100 and 150?

Answer: 3 (integers)

31. When writing the integers from 1 through 97, how many times is the digit 3 written?

Answer: 20 (times)

32. In January, a wrestler weighed 180 pounds, and in March he weighed 171 pounds. What percent of his weight in January had the wrestler lost by March?

Answer: 5 (percent)

33. What are the coordinates of the point on the line 5 x − 9 y = 42 whose x and y coordinates are additive inverses of each other? Express your answer as an ordered pair.

Answer: (3, − 3)

34. If 2 x > 415 and x is a positive integer, what is the least possible value of x?

Answer: 31

35. What is the area, in square inches, of the square inscribed in a circle of radius 10 inches?

Answer: 200 (in 2)

36. What is the probability of getting a sum of 3 when three fair six-sided dice are rolled? Express your answer as a common fraction.

1 Answer: 216

37. If N is a three-digit integer with units digit 4, what is the probability that N is divisible by 6? Express your answer as a common fraction.

1 Answer: 3

38. What is the value of m for m − 5 m + 5 = which ? m + 5 m − 5

Answer: 0

39. What is the sum of the prime divisors of 91?

Answer: 20

40. Roberto has four pairs of pants, seven shirts and three jackets. How many different outfits can he put together that consist of a pair of pants, a shirt and a jacket?

Answer: 84 (outfits)

41. The mean of one set of five numbers is 13, and the mean of a separate set of six numbers is 24. What is the mean of the set of all eleven numbers?

Answer: 19

42. What is the value of x in x 4 4 4 the equation 9 + 9 = 3 ?

Answer: 9

43. The ratio of the degree measures of two complementary angles is 3: 2. What is the measure, in degrees, of the smaller angle?

Answer: 36 (degrees)

44. In a certain dormitory, there are 72 rooms for 131 students. If each room is occupied by either one or two students, how many rooms are occupied by just one student?

Answer: 13 (rooms)

45. If you can trade 7 beeps for 3 bops and you can trade 4 bops for 5 bings, how many beeps can you trade for 60 bings?

Answer: 112 (beeps)

46. What is the area, in square units, of a circle with a circumference of 8π units? Express your answer in terms of π.

Answer: 16π (units 2)

47. The sum of the digits of a two-digit positive integer is seven. Subtracting 45 from this integer yields another two-digit integer with the same digits, but in reversed order. What is the original integer?

Answer: 61

48. What is the largest integer that is less than the square root of 250?

Answer: 15

49. Jordan ran 2 miles in half the time it took Steve to run 3 miles. If it took Steve 24 minutes to run 3 miles, using the same rates, how many minutes would it take Jordan to run 5 miles?

Answer: 30 (minutes)

50. Hadley scored 92, 73, 79 and 87 points on the first four tests of the term. There is one test remaining. What is the minimum number of points that Hadley must score on the final test in order to have a mean of 80 points for the five tests?

Answer: 69 (points)

51. A freshly painted circular emblem on a football field is completely covered by the smallest possible square tarp. The tarp covers an area of 196 ft 2. In square feet, what is the area of the circular emblem? Express your answer in terms of π.

Answer: 49π (ft 2)

52. What is the sum of the distinct prime divisors of 315?

Answer: 15

53. The sum of the numerator and the denominator of a fraction is 216. The fraction is 2 equivalent to . What is the 7 value of the denominator?

Answer: 168

54. What is the sum of the first 10 odd positive integers?

Answer: 100

55. A square region of side length 1 foot is to be covered completely using 2 -inch by 2 -inch square stickers. How many of the stickers are needed to cover the region exactly without overlap?

Answer: 36 (stickers)

56. Two triangles are similar. The ratio of their areas is 1: 4. If the height of the smaller triangle is 3 cm, what is the corresponding height of the larger triangle, in centimeters?

Answer: 6 (cm)

57. Wilma has 3 red pens, 3 blue pens, 4 black pens and 5 green pens in her school bag. If she randomly selects one pen from her bag, what is the probability that she selects a green pen? Express your answer as a common fraction.

1 Answer: 3

58. How many positive integers, x, satisfy x − 4 < 3?

Answer: 6 (integers)

59. Zachary paid for a $1 burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used?

Answer: 17 (nickels)

60. The sum of three consecutive one-digit, positive, odd integers is one-seventh of the product of the same three integers. What is the middle integer when the three integers are listed in ascending order?

Answer: 5