MATHCOUNTS
1990-1991
State Countdown Round
How many of the even counting
numbers less than 100 are
prime?
1
The probability of rain on any
given day in Atlanta is 20%.
After how many days would you
expect it to have rained on 30
days?
150 (days)
Two fair tetrahedral (four-sided)
dice are thrown. If the faces of
each are numbered 3, 5, 7, and 9,
what is the probability that the sum
of the numbers on the bottom faces
is a prime?
0
What is the shortest distance that can be
traveled if you start at any point, A, B, C, or
D and visit the other three points once?
A
6
5
D
6
4
C
B
3
5
13
How many positive integer pairs (x, y)
satisfy A = x + y,
3
when A = 2?
5 (pairs)
If the digit 7 is placed to the right
of a three-digit number, forming a
four-digit number, the value of the
original number increases by
7,000. What was the original
number?
777
Fifty cards numbered 1 to 50 are
placed in a box. If a card is
selected at random, what is the
probability that the card is a
prime number and a multiple of
seven? Express your answer as
a common fraction.
1
50
What is the last digit in the
product of all natural numbers
between 1 and 100?
0
If 5! is expressed as the product of
its prime factors in exponential
form, what is the sum of its
exponents?
5
This table shows the distribution of scores of Mr.
Sampson’s 10-point quiz. What is the median score?
Test Scores
10
9
8
7
6
5
4
3
2
Frequency
I
II
III
IIII
IIII
I
II
7
A certain type of 8 ½“ by 11” paper
has a thickness of 0.02 cm. If
15,000 sheets of paper are stacked
on top of each other, what is the
height, in meters, of the stack?
3 (meters)
Points A, B, C, and D are located on AB
such that AB = 3AD = 6BC. If a point is
selected at random, what is the
probability that it is between C and D?
Express your answer as a common
fraction.
A
D
C
B
1
2
Three identical squares are
placed side by side to form a
rectangle with a perimeter of 104
inches. What is the area, in
square inches, of each square?
169 (sq. in.)
Brad is less than 30 years old. His
age is a multiple of 5. Next year
his age will be a multiple of 7.
How old is Brad now?
20 (years old)
How many different triangles are
in the figure?
13 (triangles)
How many even numbers are
between 202 and 405?
101
What is the positive square root of
16x16?
8
4x
When n is divided by 5, the
remainder is 1. What is the
remainder when 3n is divided by
5?
3
Express 10% of 30% of 50 as a
decimal.
1.5
A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, …}.
What fractional part of the numbers
in set A are not divisible by 3?
2
3
Six softball teams are to play
each other once. How many
games are needed?
15 (games)
Two ropes, 18 meters in length
and 24 meters in length, need to
be cut into pieces which are all the
same length. What is the greatest
possible length of each piece?
6 (meters)
What value of n makes this
sentence true?
n = n+1
6
9
2
If f(x) = 3x + 2 and g(x) = (x –
what is f(g(-2))?
2
1) ,
29
If March 1 is a Monday, what day of
the week will it be 270 days later?
Friday
Given x = 3 and y = 2, simplify:
2x3 – 3y2
6
7
Simplify:
7!
4!3!
35
In how many different orders can
a group of six people be seated
around a round table?
120 (ways)
How many different arrangements
are there using the letters in the
word PARALLEL?
3,360 (arrangements)
What is √200 to the nearest tenth?
14.1
What is the positive geometric
mean of 8 and 18?
12
Find the midpoint of the line segment
with endpoints (5, -5) and ( -5, 5).
(0, 0)
Give the decimal expression for q% of q.
0.01q2
Solve for x: 2x + 2 = 128
5
The area of a triangle is 600 square
feet. Find the altitude, in feet, of the
triangle if the length of the base is
30 feet.
40 (meters)
The longest side of a right triangle is
5 feet and the shortest side is 3 feet.
What is the area of the triangle in
square meters?
6 (sq. meters)
If each side of a square is increased
by 50%, what is the percent of
increase in the area of the square?
125%
Four lines are drawn in a plane.
What is the maximum number of
distinct regions in which the lines
could divide the plane?
11
Simplify: .6 + .3
.9
(all numbers are repeating decimals)
1
What percent of 80% of a number
will lave the number unchanged?
125%
Find the number of square units in
the area of the shaded region
(0, 30)
(20, 30)
(30, 30)
(30, 20)
(0, 10)
(0, 0)
(10, 0)
(30, 0)
500 (sq. units)
If y = -7 + √(5 – x), find x when y = 3
- 95
In quadrilateral ABCD, angle BAD and
angle CDA are trisected as shown.
What is the degree measure of angle
AFD?
B
C
F
E
x
A
x
X
y
y
y
D
80 (degrees)
Find the number of square units in
the area of the triangle.
(-2, 6)
(-6, 2)
x
(0, 0)
y
32 (sq. units)
What is the smallest integer that
satisfies |2x – 3|< 8?
-2
How many two-digit whole
numbers are divisible by 14?
7
What percent of the whole
numbers less than 100 have no
remainders when divided by 5?
20 (%)
What is the 100th odd whole
number?
199
ABCD is a square 4 inches on a side,
and each of the inside squares is
formed by joining the midpoints of
the outer square’s sides. What is the
area of the shaded region?
4 (sq. inches)
If all multiples of 3 and all
multiples of 4 are removed from
the list of numbers 1 through
100, how many whole numbers
are left?
50
Divide 4abc by 2a2b
3d2
6cd2
a
What is the simplest radical form
of 3√4  6√2
√2
Two fair cubical dice are tossed.
What is the probability, expressed
as a common fraction, that the
sum of the numbers showing on
the dice will be four?
1
12
If a*b is defined as a + b,
2
what is the value of 6*(3*5)?
5
If x and y are positive integers
such that xy = 108, what is the
least possible value of x + y?
21
Find √√256,000
40
Express the value of
1 of 10% of 0.001 in scientific notation.
100
1 or(1.0) X 10-6
Find the sum of all solutions to
this equation:
x2 + 62 = 102
0
What is the sum of the lengths, in
centimeters, of the two legs of a 3060-90 right triangle, if the length of
the hypotenuse is 2√6 cm.?
√6 + 3 √2
How many diagonals can be
drawn for a hexagon?
9 (diagonals)
FIN!