MATHCOUNTS

2001 National Competition

Countdown Round

The NASA Space Shuttle transports material to the International Space Station at a cost of $22,000 per kilogram. What is the number of dollars in the cost of transporting a 250g control module?

Answer: 5500 (dollars)

What is the value of

? Express your

answer as a common fraction.

( )( )( )( )23

34

45

56

Answer: 13

Two rectangles have integer dimensions, and both have a perimeter of 144 cm. What is the greatest possible difference between the areas of two such rectangles?

Answer: 1225 (square centimeters)

The expression

can be written as . What

is the value of ?

125 10 14.

1

2 5m n

m n

Answer: 29

How many ways can change for $16.00 be made using quarters and dimes if at least one of each type of coin must be used?

Answer: 31 (ways)

For how many different values of does ? x x x x5 4 372

Answer: 3 (values)

What is the sum of the odd integers between 46 and 64?

Answer: 495

It takes 64 identical wooden cubes to fill a cubical box. How many of these cubes does it take to make the bottom layer of the box?

Answer: 16 (cubes)

Find such that

.

n

2 3 8! 5 2 n

Answer: 140

At Makettle's, Jim bought30 oranges at a price of threefor 59¢. At Paduka's, he then bought 20 oranges at a price of two for 41¢. What was the number of cents in the average price of an orange?

Answer: 20 (cents)

If each edge of a cube is increased by 50% , what is the percent increase in the surface area of the cube?

Answer: 125 (percent)

What is the value of

?3 3 35 5 53

Answer: 9

How many isosceles triangles exist which have a perimeter of 10 and integer side lengths?

Answer: 2 (triangles)

The prime factorization

of 2160 is . How

many of its positive integer

factors are perfect squares?

2 3 5 4 3

Answer: 6 (factors)

If f (x) = x 4 + x 2 + 5x, evaluate f (5) - f (-5).

Answer: 50

A piece of lumber feet long

is cut into 9 pieces. If the waste

per cut is inch, how many

inches of lumber are wasted?

51

4

1

8

Answer: 1 (inch)

The product of three consecutive multiples of 5 is 26,250. What is the largest of these three factors?

Answer: 35

What is the value of the arithmetic series 28 + 30 + 32 + ... + 86 ?

Answer: 1710

What is the probability that a two-digit number selected at random has a tens digit less than its units digit? Express your answer as a common fraction.

Answer:2

5

What is the sum of all possible values of for which is the square of a binomial?

kx k x x2 12 16

Answer: 24

What is the maximum number of square centimeters in the area of a triangle whose perimeteris 30 cm? Express your answer in simplest radical form.

Answer: 25 (square centimeters)

3

How many digits are in the decimal representation of the product of and ?2 12 5 7

Answer: 9

Two different integers from1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction.

Answer: 14

95

What is the remainder when

is divided by 8?5137

Answer: 5

Find the distance between the points (1, 1) and (4, 7) . Express your answer in simplest radical form.

Answer: 3 (units)5