Transcript countdown state1998 - 1999 with solutions

COUNTDOWN
ROUND
STATE
1998 - 1999
1. How many of the first 100
positive integers are neither perfect
squares nor perfect cubes?
88
2. A palindrome is a number that
reads the same forwards and
backwards. How many palindromes
are between 1,000 and 10,000?
90
3. The surface of a 4” x 5” x 6”
block is painted yellow. If the block
is separated into one-inch cubes,
what is number of cubes with
exactly one yellow face?
52
4. The lengths of two sides of a right
triangle are 2 11 meters and 6 3
meters. The number of meters in the
length of the third side is a whole
number. How many meters are in
the third side of the triangle?
8
5. A bag contains yellow and blue
marbles.
The probability of
selecting a yellow marble from the
bag is 8/17. If there are 27 blue
marbles in the bag, what is the
number of yellow marbles?
24
6. A uniform piece of lumber is 12ft.
long and weighs 30 lb. If a 5ft.
length is removed, what is the
number of pounds in the remaining
piece? Express your answer as a
mixed number.
17 ½
7. If q, r, and s, each represent an
integer 0-9, what is the value of s?
qqqq
rrrr
+ssss
rsssq
8
8. Compute: (262 - 242)
10,000
9. What number is 11 1/9% of
13,239?
1471
10. For what whole number n is
10n < 695000 < 10n+1?
5
Michael D. Aide:
.16 6 repeats and
.83 3 repeats
11. What is the ratio of 0.16 to 0.83?
Express your answer as a common
fraction.
1
5
12. The measures of the angles of a
pentagon are in the ratio of 3:3:3:4:5.
What is the number of degrees in the
measure of the largest angle?
150
13. A digital clock shows hours and
minutes. How many times during a
twelve hour period will the sum of
the digits on the clock be 3?
12
14. It takes 6 days for a team of 6
workers to carpet a house. At the
same rate, what is the minimum
number of additional workers
needed to carpet the same house in
just five days?
2
15. If a + b = 12 and a – b = 6, what
is the value of a2 + 2a + 2b + b2?
114
16. John has some loaves of bread in
a bag. If he had 6 more loaves, he
would have three times as many.
How many loaves does John have in
the bag?
3
17. If a two-digit prime number is
selected a random, what is the
probability that the sum of its digits
is 9?
0
18. In a right
ABC,
CAB is a
right angle. Point M is the midpoint
of BC. What is the number of
centimeters in the length of median
AM? Express your answer as a
decimal to the nearest tenth.
C
M
3 cm
A 4cm B
2.5
19. What is the number of degrees in
the exterior angle of the largest angle
of a triangle if the interior angles are
in the ratio of 2 : 3 : 5?
90??? (144)
20. A 12 feet by 18 feet rectangular
Persian rug is centered within a
rectangular room so that a region of
constant width surrounds the carpet.
Given that 96 square feet of flooring
is not covered by the rug, what is the
number of feet in the measure of the
longer dimension of the room?
16
21. What is the least positive integer
that is not a factor of 7!?
11
22. A right circular cone is inscribed
in a right cylinder. The volume of
the cylinder is
72π cubic
centimeters. What is the number of
cubic centimeters available inside
the cylinder but outside of the cone?
Express your answer in terms of π
48π
23. Compute 12 ½% of 240.
30
24. In an isosceles trapezoid, the
lengths of the bases are 8cm and
16cm. The length of a diagonal is
13cm. What is the number of square
centimeters in the area of the
trapezoid?
60
25. What is the greatest two-digit
whole number that can be expressed
as the product of two consecutive
whole numbers?
90
26. What is the greatest three-digit
whole number that is divisible by 6?
996
27. If the measure of each side of a
square is decreased by 10%, by what
percent is the area decreased?
19
28. What is the units digit of 2345 + 6789?
0
All squares and circles have the
same value. What is the sum of the
three circles?
+
+
+
+
+
+
+
+
+
+
= 21
= 19
=?
9
30. Six less than a number is
multiplied by 6. Eight less than the
same number is multiplied by 8. The
two results are equal. What is the
number?
14
31. When Sarah turns 36 years old,
her father will be 62. Currently,
Sarah is 1/3 of her father’s age.
How many years old is Sarah now?
13
32. The sum of seven consecutive
even numbers is 13,902. What is the
smallest of these numbers?
1980
33. Eleven coins total $1.19, but no
combination of the coins will give a
value of exactly $1. How many of
the coins are quarters?
3
34. Jimi is taking a bicycle trip and
he and needs to arrive by 6 o’clock.
If he travels an average rate of 15
mph, he’ll arrive at his destination
one hour early, but if he travels at an
average rate of 10 mph, he’ll arrive
one hour late. How many miles
away is his destination.
60
35. Mark baked cookies for the
neighborhood children. He gave
each child 6 cookies and he had 7
cookies remaining. He was going to
give another cookie to each child,
but he was 1 cookie short. How
many cookies had Mark baked?
55
36. An eight-digit number contains
two 5’s, two 6’s, two 7’s, and two
8’s. The 5’s are separated by one
digit, the 6’s are separated by two
digits, the 7’s are separated by three
digits, and the 8’s are separated by
four digits. What is the greatest
possible value for the number?
85,756,876
37. Calculate: (243)3/5
27
38. Calculate and express your
answer as a common fraction:
2
2+2+2
3 5 7
105
71
39. What is the least positive integer
16
n such that
< n?
n
5
40. How many different threemember committees can be chosen
from a group of 10 people?
120
41. A rectangle with length 8 units
and width 5 units is centered at the
origin of a coordinate plane, and its
longer side parallel to the x-axis.
What percent of the rectangular
region is to the right of x = -2?
75
42. What is the smallest natural
number that has exactly five factors?
16
43. For what value of n does
n
–
5
5
=1/125
2
44. What is the sum of the real
values of x for which the following
expression is undefined? Express
your answer as a common fraction.
8
5 _ 2
x
3
15
2
45. A 3” x 5” photograph can be
enlarged to a panoramic view that is
5” x 10”. What is the area of the
original photograph to the area of the
enlargement? Express your answer
as a common fraction.
3
10
46. The sum of two numbers is 9,
and their difference is 7. What is the
positive difference of the squares of
these two original numbers?
63
47. One diagonal of a rhombus is 48
inches. Given that one side of the
rhombus has length 26 inches, what
is the number of square inches in the
area of the rhombus?
480
A centipede climbs a 40-foot tree.
Each day he climbs 5 feet, and each
night he slides down 2 feet. In how
many days will the centipede reach
the top of the tree?
X