Jake writes a list of all the four-digit positive integers with the property that the product of the four digits is 5. What is the arithmetic mean of the integers in his list?

1
MATHCOUNTS?
2005 National Competition Countdown Round
2
Gigi and Hector just began new jobs at annual
salaries of 19,000 and 17,500, respectively.
Gigi will receive annual raises of 2000 while
Hector will receive annual raises of 2500.
After how many years will they each be making the
same salary?
3
Answer 3 (years)
4
If 4x1 2, express 4x-1 as a common fraction.
5
Answer
6
Jake makes a list of all the four-digit positive
integers with the property that the product of
the four digits is 5. What is the arithmetic
mean of the integers on his list?
7
Answer 2222
8
In how many patterns can six identical L-shaped
pieces, each consisting of three unit squares, be
placed to perfectly cover a 3 by 6 board? One
such pattern is shown.
9
Answer 8 (patterns)
10
Coplanar squares ABGH and BCDF are adjacent, with
CD 10 units and AH 5 units. Point E is on
segments AD and GB. What is the area of triangle
ABE, in square units? Express your answer as a
common fraction.
11
Answer (square units)
12
The students at Sherry Curtis School were able
to vote for one or more events for their field
day. Below are the top three choices. List the
choices from most popular to least
popular.Event Kickball Picnic Softball
Fraction of students who like the event
13
Answer Softball, Kickball, Picnic
14
What is the least positive integer multiple of 30
that can be written with only the digits 0 and 2?
15
Answer 2220
16
The measure of angle ACB is 40 degrees. If ray
CA is rotated 480 degrees about point C in a
clockwise direction, what will be the positive
measure of the new acute angle ACB, in
degrees?
A
C
B
17
Answer 80 (degrees)
18
Contractor Steve agreed to complete a job in 30
days. After 6 days he found that the 8 people
assigned to the work had already done of the
job. If everyone works at the same rate, what is
the least number of people he must keep on the
job to ensure that the job will be completed on
time?
19
Answer 4 (people)
20
Melinda will roll two standard six-sided dice and
make a two-digit number with the two numbers she
rolls. For example, if she rolls a 6 and 3, she
can either form 36 or 63. What is the
probability that she will be able to make an
integer between 10 and 20? Express your answer
as a common fraction.
21
Answer
22
Four positive integers A, B, C and D have a sum
of 36. If A 2 B 2 C 2 D 2, what
is the value of the product A B C D?
23
Answer 3840
24
In the two concentric circles shown, the radius
of the outer circle is twice the radius of the
inner circle. What is the area of the gray
region, in square feet, if the width of the gray
region is 2 feet? Express your answer in
terms of p.
2
25
Answer 12p (square feet)
26
An airplane has three sections First Class (24
seats), Business Class (25 of the total number
of seats) and Economy ( of the total number of
seats). How many seats does the plane have?
27
Answer 288 (seats)
28
The square of an integer is 182 greater than the
integer itself. What is the sum of all integers
for which this is true?
29
Answer 1
30
A flag is made of three horizontal strips of
fabric, each of a solid color, either red, white,
blue or green. If no two adjacent strips can be
the same color, how many distinct flags are
possible? These two flags are different.
31
Answer 36 (flags)
32
From each vertex of a regular octahedron of edge
length 6 units, a square-based pyramid of
height1 unit will be cut away. How many
vertices will the new solid have?
6
33
Answer 24 (vertices)
34
How many perfect squares are factors of 2 4
6 8 10 12?
35
Answer 12 (squares)
36
A rectangular box is 4 cm thick, and its square
bases measure 16 cm by 16 cm. What is the
distance, in centimeters, from the center point P
of one square base to corner Q of the opposite
base? Express your answer in simplest terms.
P
Q
37
Answer 12 (centimeters)
38
A MATHCOUNTS competition is held in a room with
many tables, and there are 4 stools at each
table. Each stool has 3 legs, and each table has
4 legs. If there is a total of 368 legs on all
the tables and stools in the room, how many
tables are in the room?
39
Answer 23 (tables)
40
If 4x2 x16, what is the value of x ?
41
Answer 4
42
Let a, b and c be positive numbers satisfying ab
11.25 ac 20 and bc 36. What is the
value of c?
43
Answer 8
44
A cylindrical water tank is full. If three
liters were added, the tank would be full.
How many liters does the tank hold when it is
full?
45
Answer 60 (liters)
46
How many distinct three-letter sequences with at
least one T can be formed by using three of the
six letters of TARGET? One such sequence is
T-R-T.
47
Answer 48 (sequences)
48
What is the minimum number of equilateral
triangles, of side length 1 unit, needed to cover
an equilateral triangle of side length 10 units?
49
Answer 100 (triangles)
50
The average age of the 10 females in a choir is
30 years. The average age of the 15 males in
the same choir is 35 years. What is the average
age, in years, of the 25 people in the choir?
51
Answer 33 (years)