MATHCOUNTS1993-1994

State CompetitionCountdown Round

A man digs a hole 6 inches deep for a post to hold his mailbox. If the square base of the post is 4 in. x 4 in., how many cubic inches of dirt will be displaced by the post?

96 (cubic inches)

If 2x = 8 what is 3x?

27

At a certain intersection, 30% of the cars turn left, 40% turn right, and 30% go straight. What is the probability that the next car turns left and the following one turns right?

3 25

Expressin simplest form

6666 2222

(26 + 26+ 26+ 26)

16

What is the sum of the two values that will make the

following expression undefined?

x – 2 x2 – 6x + 8

6

Find the largest integer less than or equal to 2-3 ·33.

3

Seven equal 10-inch-long pieces are cut from an eight-foot-long piece of lumber. If the waste per cut is 1/8 inch, how many inches long is the piece that is left over? Express your answer as a mixed number.

25 1/8 (inches)

If 5 technicians can produce 64 surfboards in 4 days,

how many surfboards can 8 technicians produce in 10

days?

256 (surfboards)

What digit is used only once in the decimal

representation of(1,111,111)2

7

The school population has increased from 3,200 to 3,600 this year. If this percent rate of increase remains constant, what will be the school population be at the end of next year?

4,050 (students)

In an arithmetic sequence the 113th term is 786 and the 125th term is 870. Find the 150th term

1,045

A rectangle has length 15 inches and width 10 inches. If these dimensions are both increased by 20%, by how

many square inches will the area of the rectangle be

increased?

66 (square inches)

If f(x) = 2x + 3 and g(x) = 3x – 2, find f(g(f(2)))

g(f(g(2)))Express your answer in the

form a .

b

41 31

What is the sum of the distinct prime factors of 3, 780

17

Two interior angles A and B of pentagon ABCDE are 60º

and 85º. Two remaining angles, C and D, are equal and the fifth angle is 15º more than twice C. Find the measure

of the largest angle.

205 (degrees)

Simplify the following and express the result without using negative exponents.

(3a-2b3c-1)2 · (√2ab-1c2)

2 3

3√2b5

2a3

Evaluate 1312 – 2(131)(31) + 312

10,000

Segment AC has three semi-circles with radii 4cm, 6cm, and 10 cm. In terms of π, how many square centimeters are in the shaded area?

A C

24π (sq. cm.)

Find )5)(20)(18)(8(

120

Find the volume, in cubic inches, of a cube whose surface area is 96 square inches.

64 (cubic inches)

Given that ab = (a2 - b2),

abexpress 62 as a common

fraction.

8 3

A candy bar weighs 4.48 oz. If a person eats half the remaining candy bar with each bite, how many bites have been taken when there is exactly 0.07 oz remaining?

6 (bites)

Evaluate (35)(23) – (24)(34)

648

If the sides of a right triangle are 14cm, 48 cm,

and 50 cm, how many square centimeters are in

its area?

336 (sq. cm.)

Central angles a, b, and c separate the circle into regions which are 1/3, 1/4, and 1/6 of the area. How many degrees are there in central angle d?

a b c

d

90 (degrees)

What is the value of (81)0.04 • (81) 0.21 ?

3

Find the sum of all possible values for C which will make

49C345 divisible by 15.

15

If X > 3 find Y such that Y = |X| - |3 – X|

3

How many positive integers less than 575 are divisible

by 2, 3, and 5?

19 (integers)

One bus company has a bus arrive at the Parkview bus stop every 16 minutes, while another bus company has a bus arrive at the same stop every 20 minutes. If a bus from each company arrives at Parkview bus stop simultaneously at 3:00 PM, what is the next time they will arrive there simultaneously?

4:20 (PM) or 16:20

Find the value of (n + 1)! when n = 100

(n – 1)!

10,100

Find the value of x in the triangle shown. Express your answer in simplest radical form.

45°

16x

x

8√2 (sq. cm.)

Express 0.003 X 0.00033 in scientific notation.

9.9 X 10-7

A girls basketball team has 4 players of heights 6’1”,

5’9”, 5’8”, and 5’11”. If the average height of the 5-girl team is 5’10”, then the fifth

player must be 5’a” tall. Find a.

9

The sum of two numbers is 19 and their difference is 5.

What is their product?

84

Express as a common fraction.

383

3

- 3

2

What is the greatest common factor of (a2 – 9) and (3a2 – 3a –

36)?

( a + 3)

What is the value of x in the diagram?

x 2

60° 45°

√6 (units)

Find the area of the shaded cross section of the magnet, where OE = 2, OF = 4, OP = 4, and both arcs shown are semicircles. Leave your answer in terms of π.

E 2

F 2

o

4

2 P 2

16 + 6π (sq. units)

Find the product of (1/4)-3(8)-

2

1

What is the value of 6!

(6 – 4)!4!

15

If a light bulb has an expected life of 1,000 hours, for how many

complete days of continuous use can it be

expected to work?

41 (days)

Laws in certain states require a 60% approval rate from voters before an issue is “passed.” If 1,140 people voted in favor of the law, and this represents 57% of the total number of voters, how many voters were there in this election?

2,000 (voters)

If a 25cm X 12cm X 7cm full box of paper towels

contains 175 individual towels, what is the volume

of each towel in cm3?

12 (cm.3)

Given the square of an integer x is 1,521.

What is the value of (x + 1)(x – 1)

1,520

A step pyramid is formed by stacking successively smaller “steps” as illustrated in the drawing. If the edges of each square “step” have lengths 9,7,5,3, and 1, and each “step” is 1 unit high, find the number of cubic units in the volume of the pyramid.

1 3

3 5

5 7

7 9

9

165 (units3)

What is the value of (3/5)2 + 2(3/5)(2/5) +

(2/5)2?

1

The configuration below is built from 27 segments. How many paths are there from A to B using exactly five of these segments?

B

A

6 (paths)

Suppose the estimated $45 billion cost of sending a person to Mars is shared

equally by the 250 million people of the United States.

What is each person’s share?

180 (dollars)

If (x + 1)1/2 = 21/3 find the value of

(x + 1)2 in simplest radical form.

2 ·3√2

How many numbers can be written as the sum of two or more distinct members of the set {0, 1, 2, 3, 4, 5}

15 (numbers)

How many positive integers less than 100 can be

expressed as a power of 3?

5 (integers)

How many ways can 5 distinct paperback books and 1 hardcover book be arranged on a shelf if the hardcover book must be the rightmost book on the

shelf?

120 (ways)

Find the negative root(s) of the equation (x + 1)2 = 2(x

+ 1)

- 1

Given x varies directly as y2 and x = 3 when y = 5, find x when y = 6. Express your answer as a common fraction.

108 25

17 (yards)

√2

54 (diagonals)

2

29 ¼

51

24 (sq. cm.)

18

1 38

1

34 27

7