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MATHCOUNTS

2003 National Competition
Countdown Round
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
2
200 < n < 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 “abc”
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
11. Compute: 2  2  2  2 .
6
6
6
6
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 – 3)(n – 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  n 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
40°. 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 + 3·9 + 3·9 + 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 100th digit
after the decimal point of the
1
decimal representation of 7 ?
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