MATHCOUNTS
1993-1994
State Competition
Countdown 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
x
2 =
8 what is
x
3?
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
6
6
6
6
2 2 2 2
Express (26 + 26+ 26+ 26)
in simplest form
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
-3
3
or equal to 2 ·3 .
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
2
(1,111,111)
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.
-2
3
-1
2
-1
2
(3a b c ) · (√2ab c )
2
3
5
3√2b
3
2a
Evaluate 1312 – 2(131)(31) + 312
10,000
Segment AC has three semicircles 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 (8)(18)(20)(5)
120
Find the volume, in cubic inches,
of a cube whose surface area is
96 square inches.
64 (cubic inches)
2 - b2),
(a
Given that ab =
ab
express 62 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
5
3
(3 )(2 )
–
4
4
(2 )(3 )
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°
16
x
x
8√2 (sq. cm.)
Express 0.003 X 0.00033 in
scientific notation.
9.9 X
-7
10
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
3
Express 3  3 as a common fraction.
8
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
F2
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
5
7
9
3
5
7
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
2
y
Given x varies directly as
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