MATHCOUNTS
2000 State Competition
Countdown Round
®
How many different sets of three
numbers, each number chosen
from the integers 1-16, could be
the side lengths of a right
triangle?
Answer: 4 (sets)
BC is parallel to the segment
through A, and AB = BC. What is
the number of degrees
represented by x?
A
124°
x°
B
C
Answer: 28 (degrees)
What is the difference between
32% of 24 and 24% of 32?
Answer: 0
4
5
A ball rebounds to of its
previous height. What is the
number of feet in the height of
the third bounce if the ball was
dropped from a height of 5 feet?
Express your answer as a decimal
to the nearest tenth.
Answer: 2.6 (feet)
A square is drawn inside a circle
so that one vertex lies at the center
of the circle and another vertex lies
on the circumference. The area of
the circle is 72 square inches.
How many square
inches are in the area
of the square?
Answer: 36 (square inches)
A positive number is entered on a
calculator. Jamie pressed
2
the x key and got a result of
0.0625. What answer would have
been displayed had Jamie pressed
the square root key? Express your
answer as a decimal to the
nearest tenth.
Answer: 0.5
What is the least number greater
than 9000 that is divisible by 11?
Answer: 9009
Joel is three times as old as he
was 6 years ago. In how many
years will he be twice as old as
he is now?
Answer: 9 (years)
Circle C has radius 6 cm. How
many square centimeters are in
the area of the largest possible
inscribed triangle having one side
as a diameter of circle C?
Answer: 36 (square centimeters)
Sam drove 25 kilometers farther
than Susan. If Susan had driven
twice as far as she did, Sam
would have driven only
10 kilometers farther. How
many kilometers did Sam drive?
Answer: 40 (kilometers)
The three sides of a right
triangle have integer lengths.
The hypotenuse is 1 foot longer
than the longer leg, and the
shorter leg is 9 feet. What is the
number of feet in the perimeter
of the triangle?
Answer: 90 (feet)
What is the remainder when the
product of the first 25 prime
numbers is divided by 4?
Answer: 2
For positive numbers x and y,
2
2
x + 2xy + y = 36. What is the
mean of x and y?
Answer: 3
The circumference of a circle is
increased by 10%. By what
percent will the area of the circle
increase?
Answer: 21 (percent)
At a picnic of at least three
people, a plate of 26 cookies is
passed around. Each person takes
one cookie and passes the plate to
the next person. Jana gets the first
cookie and the next-to-last
cookie. What is the median of all
possible numbers of people at the
picnic?
Answer: 7 (people)
Equilateral triangles are
constructed on each side of a
right triangle. The areas of the
triangles constructed on the legs
2
of the right triangle are 36 cm
2
and 64 cm . What is the number
of square centimeters in the area
of the triangle constructed on the
hypotenuse?
Answer: 100 (square centimeters)
For how many two-digit positive
integers is the tens digit twice the
units digit?
Answer: 4 (integers)
Farmer Flagg has 400 feet of
fencing to enclose a circular
garden and a second 400 feet of
fencing to enclose a square
garden. What is the ratio of the
area of the circular garden to the
area of the square garden?
Express your answer as a
common fraction in terms of .
Answer:
4

In a game of chance, there is a
1
chance of winning a car, a 4
chance of winning a sofa, and a
1
chance
of
winning
a
goat.
2
What is the probability that a
player will not win anything?
Express your answer as a
common fraction.
1
8
Answer:
1
8
The tens digit of a two-digit
number exceeds the units digit
by 1. If the number is divided by
the sum of its digits, the quotient
is 6. What is the number?
Answer: 54
How many solutions does the
equation |x + 2| = 2x have?
Answer: 1 (solution)
The average of 15 consecutive
positive integers is 12. What is
the greatest of these integers?
Answer: 19
A six-question test has three
true/false questions and three
multiple choice questions with
four choices each. If Ojeni
guesses at all the answers, what is
the probability that she has all
answers correct? Express your
answer as a common fraction.
Answer:
1
512
What is the smallest natural
number greater than
1
1
1
1
1
1 + 3 + 6 + 10 + 15 + 21 ?
Answer: 2
For what value of x will a line
which passes through the
points (5,3) and (x,7) have a
1
slope of - ?
3
Answer: -7
Each term of a sequence is
generated by adding the two
previous terms. What is the sum
of the values of the four missing
terms in the following sequence?
-12, __, -5, __, -3, __, -4, __, -9, -14, -23
Answer: 3
A red number cube and a white
number cube, each with faces
numbered 1 to 6, are rolled. What
is the probability that the red
cube shows a 4 or a 5 and that the
sum of the numbers shown on the
two cubes is greater than 9?
Express your answer as a
common fraction.
Answer:
1
12
For what value of x does
128
x+2
=
x
1
( 64 ) ?
Express your
answer as a common fraction.
Answer: -
14
13
Two whole numbers have a
sum of 25. What is the greatest
possible product of the two
numbers?
Answer: 156
What is the number of centimeters
in the length of AB in the right
rectangular prism shown?
A
9 cm
12 cm
B
20 cm
Answer: 25 (centimeters)
What is the probability that a
randomly selected two-digit
whole number is not divisible
by 2 or 3? Express your answer
as a common fraction.
Answer:
1
3
Find the positive difference
between the solutions to
x
|
2
- 6| = 20.
Answer: 80
For what value of m does
(3!)(2!)(m) = 6! ?
Answer: 60
A sphere is cut into four congruent
wedges. The circumference of the
sphere is 12 inches. What is the
number of cubic inches in the
volume of one wedge? Express
your answer in terms of .
Answer: 72 (cubic inches)
What is the number of square
units in the triangular region
bounded by the lines y = 2x,
y = -2x, and x = 5?
Answer: 50 (square units)
2
Given that f (x) = x - 2x + 5 and
g(x) = x + 3, what is the value of
f (g(5)) - g( f (5))?
Answer: 30
Two binary operations are
defined by the rules
3
3
a  b = a - b and
3
a  b = (a + b) . What is the
value of (2  3)  9?
Answer: -1000
Mae ran one mile in
10.5 minutes. How many
minutes per mile must she
average for the next four miles to
run five miles in 47.5 minutes?
Express your answer as a
decimal to the nearest hundredth.
Answer: 9.25 (minutes per mile)
Six bippitties equal three bops,
and six bops equal two boos.
How many boos are there in one
bippitty, one bop and one boo?
Express your answer as a
common fraction.
Answer:
3
2
Twelve colored balls are placed
in a bag. Six of the balls are red,
four are white, and two are green.
If two balls are drawn from the
bag without replacement, what is
the probability that at least one of
them is white? Express your
answer as a common fraction.
Answer:
19
33
Choose a 3  3 array of dates from the
February calendar shown. In the array,
what is the positive difference between
the product of the dates in the
upper-right and lower-left corners
and the product of
FEBRUARY
the dates in the
1
2
3
4
5
6
7
8
9 10 11 12 13 14
upper-left and
lower-right corners? 15 16 17 18 19 20 21
22
23
24
25
26
27
28
Answer: 28
Suli will walk to school along the
paths shown. The numbers on the
grid represent the time in minutes
required to travel along each
path. What is the least
6
School
number of minutes it
7
2
will take Suli to get
5
5
5
to school?
2
4
3
Home
5
Answer: 14 (minutes)
How many CDs which cost $7.98
each plus 4% sales tax can Kylie
buy with $84?
Answer: 10 (CDs)
An isosceles trapezoid has the
dimensions shown. How many
square centimeters are in the area
of the trapezoid?
50 cm
39 cm
80 cm
Answer: 2340 (square centimeters)
There are 7 boys and 9 girls in
the school choir. Two of the
students are chosen at random to
sing a duet. What is the
probability that both students
chosen are girls? Express your
answer as a common fraction.
Answer:
3
10
Anna and Bjorn each choose an
integer 1-6. What is the probability
that Anna’s number is less than
Bjorn’s number? Express your
answer as a common fraction.
Answer:
5
12
A bag contains three red marbles,
two blue marbles, and one white
marble. Two marbles are randomly
drawn without replacement. What
is the probability of drawing one
red and one blue marble? Express
your answer as a common fraction.
Answer:
2
5
The lengths of the legs of a
right triangle are 12 meters
and 16 meters. What is the
number of meters in the
altitude to the hypotenuse?
Express your answer as a
common fraction.
Answer:
48
5
John jogged from his house to the
gym at 6 miles per hour and
walked back along the same route
at 4 miles per hour. If the entire
trip took him one hour, how
many miles is his house from the
gym? Express your answer as a
decimal to the nearest tenth.
Answer: 2.4 (miles)
The painted area of a rectangular
2
canvas is 72 in , and there is a
4-inch unpainted border on all
sides. The dimensions of the
canvas are whole numbers. What
is the number of inches in the
least possible perimeter of the
border?
Answer: 66 (inches)
The rectangle is inscribed in the
circle, and the rhombus is
inscribed in the rectangle. If the
radius of the circle is 4 ft, how
many feet are in the
perimeter of the
rhombus?
Answer: 16 (feet)
A nine-digit number uses the
digits 1-9 once each. What is the
probability that the number is
prime?
Answer: 0
A bin in the shape of a
rectangular prism is 18 feet long,
1
15 2
feet wide and
There are
1
14
3
34
feet deep.
cubic feet in a
bushel. How many bushels are in
the bin?
Answer: 837 (bushels)
Jim has the same number of
nickels, dimes and quarters in his
pocket. If he has $4.80, how
many dimes does he have?
Answer: 12 (dimes)
For each 10 bottles recycled,
1 new bottle can be produced.
Over time, how many new bottles
will be produced by recycling
1000 bottles, if all recycled
bottles are again recycled?
Answer: 111 (bottles)
If a = b - 1, then what is the
value of |a - b| + |b - a|?
Answer: 2
One watch loses 2 minutes
every hour, and the other gains
2 minutes every hour. If they start
at the same time, in how many
hours will the fast watch be one
hour ahead of the slow watch?
Answer: 15 (hours)
If x is 25% of y, and y is 80%
of z, what percent of z is x?
Answer: 20 (percent)
A circle with a radius of
5 centimeters is inscribed in an
equilateral triangle. What is the
number of centimeters in the
altitude of the triangle?
Answer: 15 (centimeters)
What is the arithmetic mean of
the consecutive natural numbers
from 1 to 201 inclusive?
Answer: 101
Stetson runs one lap on a circular
track which is 10 meters wide.
What is the difference in the
number of meters traveled if he
runs on the outside of the track
rather than on the inside of the
track? Express your answer in
terms of .
Answer: 20 (meters)
The perimeter of a rectangle with
integer side lengths is 14 units.
What is the positive difference
between the number of square
units in the greatest and least
possible areas for the rectangle?
Answer: 6 (square units)
What is the fifth term of the
geometric sequence whose first
term is
is
1
100
1
4
and whose third term
? Express your answer as a
common fraction.
Answer:
1
2500
A triangle with integer side
lengths has two sides that
measure 7 cm and 12 cm. What is
the number of centimeters in the
greatest possible perimeter for
the triangle?
Answer: 37 (centimeters)
The average of five numbers
is 42. The first number is
increased by 1, the second
number is increased by 2, and
so on, until the fifth number is
increased by 5. What is the
average of the new numbers?
Answer: 45
Three boys, all working at the
1
same rate, can mow 3 of the
town soccer fields in 8 hours.
How many hours will it take four
boys, mowing at the same rate, to
mow the rest of the fields?
Answer: 12 (hours)
A rectangular prism whose
dimensions are whole numbers
has a volume of 300 cubic units.
What is the smallest possible
whole number that could not be
one of the dimensions?
Answer: 7
Jennifer is thinking of an integer
from 1 to 17 inclusive. Rob will
guess an integer, and Jennifer
will tell him if his guess is higher
or lower than her number. What
is the least number of guesses
Rob must make to guarantee that
he will know Jennifer's number?
Answer: 4 (guesses)
In which quadrant is the
intersection of the lines
2x + 3y = 15 and 3x - 2y = -6?
Answer:(Quadrant) I (or first)
3n +1
Solve for n: 3
=
1
243
.
Answer: -2
What is the probability that a
randomly-chosen positive integer
less than 50 is a perfect cube or a
perfect square?
Answer:
9
49
What is the probability that a
randomly-chosen three-digit
number contains the digit 4
exactly twice? Express your
answer as a common fraction.
Answer:
13
450
x
If 25 = 36, what is
x
the value of 5 ?
Answer: 6
What is the sum of all the
2
solutions of (x + 8) = (
1 -4
)
?
2
Answer: -16
The median to the hypotenuse of a
right triangle is 6.5 cm long. What
is the number of centimeters in the
length of the hypotenuse?
Answer: 13 (centimeters)