MATHCOUNTS 2003 National Competition Countdown Round
№Бе
1. If 23x = 7, evaluate 8x+1.
Answer: 56
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.
9 Answer: 25
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.
2 Answer: 3
4. How many positive integers n satisfy 200 < n2 < 900?
Answer: 15 (integers)
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?
Answer: 8 (square units)
6. What is the greatest prime factor of 221?
Answer: 17
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?
Answer: 90 (meters)
2 3
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?
Answer: 12 (minutes)
9. This shape is built from six unit cubes. How many square units are in the surface area of the shape?
Answer: 26 (square units)
10. A three-digit number â&#x20AC;&#x153;abcâ&#x20AC;? has the property that the product of a and b is equal to c. What is the greatest possible value of the threedigit number?
Answer: 919
6
6
6
6
11. Compute: 2 + 2 + 2 + 2 .
Answer: 16
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.
1 Answer: 3
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}?
Answer: 15 (numbers)
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?
Answer: 16 (degrees)
15. How many positive integers n satisfy (n + 8)(n â&#x20AC;&#x201C; 3)(n â&#x20AC;&#x201C; 12) < 0 ?
Answer: 8 (integers)
6!5! 16. Compute: 6 ! + 5 ! + 5 ! .
Answer: 90
17. The decimal expansion of 8 is a repeating decimal. 11
What is the least number of digits in a repeating block of 8 11 ?
Answer: 2 (digits)
18. How many integers n are there such that 3 â&#x2030;¤ n â&#x2030;¤10 and 121 base n (or 121n) is a perfect square?
Answer: 8 (integers)
19. Triangle PQR is isosceles and the measure of angle R is 40째. The possible measures of angle P are x, y or z degrees. What is the value of the sum x + y + z ?
Answer: 210 (degrees)
20. Compute: 3 2 9 + 3路9 + 3路9 + 1
Answer: 1000
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?
Answer: 5 (students)
22. What is the 100 digit after the decimal point of the 1 decimal representation of 7 ? th
Answer: 8
23. Solve for x: 2x x 3 + 19 = 10 .
Answer: 2
24. How many nonoverlapping 9-inch by 9-inch square tiles does it take to cover a 12-foot by 15-foot room?
Answer: 320 (tiles)
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 A
60 30
20 B
2 Answer: 3