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?

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MATHCOUNTS?
2003 National Competition Countdown Round
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1. If 23x 7, evaluate 8x1.
3
Answer 56
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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.
5
Answer
6
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.
7
Answer
8
4. How many positive integers n satisfy 200 lt
n2 lt 900?
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Answer 15 (integers)
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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?
11
Answer 8 (square units)
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6. What is the greatest prime factor of 221?
13
Answer 17
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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?
15
Answer 90 (meters)
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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?
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Answer 12 (minutes)
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9. This shape is built from six unit cubes. How
many square units are in the surface area of the
shape?
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Answer 26 (square units)
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10. A three-digit number abc has the property
that the product of a and b is equal to c. What
is the greatest possible value of the three-digit
number?
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Answer 919
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11. Compute
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Answer 16
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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.
25
Answer
26
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?
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Answer 15 (numbers)
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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?
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Answer 16 (degrees)
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15. How many positive integers n satisfy (n
8)(n 3)(n 12) lt 0 ?
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Answer 8 (integers)
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16. Compute .
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Answer 90
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17. The decimal expansion of is a
repeating decimal. What is the least number of
digits in a repeating block of ?
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Answer 2 (digits)
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18. How many integers n are there such that 3 ?
n ?10 and 121 base n (or 121n) is a
perfect square?
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Answer 8 (integers)
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19. Triangle PQR is isosceles and the measure of
angle R is 40. The possible measures of angle P
are x, y or z degrees. What is the value of the
sum x y z ?
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Answer 210 (degrees)
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20. Compute 93 392 39 1
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Answer 1000
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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?
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Answer 5 (students)
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22. What is the 100th digit after the decimal
point of the decimal representation of ?
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Answer 8
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23. Solve for x 32x 19 10x
.
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Answer 2
48
24. How many non-overlapping 9-inch by 9-inch
square tiles does it take to cover a 12-foot by
15-foot room?
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Answer 320 (tiles)
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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
60
20
30
A
B
51
Answer