1. What is the greatest prime factor of 99

1
MATHCOUNTS?
2006 Chapter Competition Countdown Round
2
(No Transcript)
3
1. Mishas favorite number is even. Her
favorite number is also non-negative and less
than 1. What is Mishas favorite number?
4
(No Transcript)
5
Answer 0
6
(No Transcript)
7
2. A car is averaging 50 miles per hour. If the
car maintains this speed, how many minutes less
would a 450-mile trip take than a 475-mile trip?
8
(No Transcript)
9
Answer 30 (minutes)
10
(No Transcript)
11
3. Three coplanar circles intersect as shown.
What is the maximum number of points on the
circles that a line passing through all three
circles can touch?
12
(No Transcript)
13
Answer 6 (points)
14
(No Transcript)
15
4. An equilateral triangle and a square have the
same perimeter of 12 inches. What is the ratio
of the side length of the triangle to the side
length of the square? Express your answer as a
common fraction.
16
(No Transcript)
17
Answer
18
(No Transcript)
19
5. If 2x y 8, what is the value of 4x 2y?
20
(No Transcript)
21
Answer 16
22
(No Transcript)
23
6. Climbing the first flight of stairs takes
Jimmy 20 seconds, and each following flight takes
5 seconds more than the preceding one. How many
total seconds does it take to climb the first
five flights of stairs?
24
(No Transcript)
25
Answer 150 (seconds)
26
(No Transcript)
27
7. If a b c and b c 11 and c 8,
what is the value of a?
28
(No Transcript)
29
Answer 5
30
(No Transcript)
31
8. What is the greatest three-digit integer in
which the product of the digits is 0?
32
(No Transcript)
33
Answer 990
34
(No Transcript)
35
9. Steve will write the list of consecutive
integers, starting with 1, at the speed of 5
digits per second. What will be the sum of the
final two digits he writes on his list at the end
of 10 seconds?
36
(No Transcript)
37
Answer 12
38
(No Transcript)
39
10. The fenced area of a yard is a 15-foot
by12-foot rectangular region with a 3-foot by
3-foot square cut out, as shown. What is the
area of the region within the fence, in square
feet?
40
(No Transcript)
41
Answer 171 (square feet)
42
(No Transcript)
43
11. When 700 students from Genius M.S. played
SCOOZ, 90 of them scored proficient or above.
What number of students at Genius M.S. did not
score proficient or above?
44
(No Transcript)
45
Answer 70 (students)
46
(No Transcript)
47
12. If the operation Z is defined as a Z b b
10a - a2, what is the value of 2 Z 6 ?
48
(No Transcript)
49
Answer 22
50
(No Transcript)
51
13. Aditi can walk 1.7 miles in half an hour. At
the same constant speed, how many hours would it
take her to walk 17 miles?
52
(No Transcript)
53
Answer 5 (hours)
54
(No Transcript)
55
14. A right hexagonal prism has a height of 3
feet and each edge of the hexagonal bases is 6
inches. What is the sum of the areas of the
non-hexagonal faces of the prism, in square feet?
56
(No Transcript)
57
Answer 9 (square feet)
58
(No Transcript)
59
15. What common fraction is halfway between
and on a number line?
60
(No Transcript)
61
Answer
62
(No Transcript)
63
16. A rectangular quilts length is twice the
length of a rectangular picture, and the quilts
width is three times the width of the same
picture. The area of the picture is 2 square
feet. What is the area of the quilt, in square
feet?
64
(No Transcript)
65
Answer 12 (square feet)
66
(No Transcript)
67
17. Each year, starting with 1999, five new state
quarter designs were introduced. How many
different state quarter designs were introduced
by the end of 2005?
68
(No Transcript)
69
Answer 35 (designs)
70
(No Transcript)
71
18. In a convex quadrilateral, the measure of the
largest angle is twice the measure of the
smallest angle, and the other two angles are both
right angles. How many degrees are in the
largest angle?
72
(No Transcript)
73
Answer 120 (degrees)
74
(No Transcript)
75
19. What is the sum of the six positive integer
factors of 18?
76
(No Transcript)
77
Answer 39
78
(No Transcript)
79
20. Lyndy conducted a survey of the 300 students
in her school. She found that 60 students own
dogs, while only 30 students own cats. What
percent of the students own cats?
80
(No Transcript)
81
Answer 10 (percent)
82
(No Transcript)
83
21. Max must take 10 steps to go the same
distance as three of his dads steps. His dad
takes 30 steps to walk down the hall. How many
steps must Max take to walk down the same hall?
84
(No Transcript)
85
Answer 100 (steps)
86
(No Transcript)
87
22. The area of a rectangle is 80 sq ft. If the
width is increased by 1 foot and the length is
decreased by 1 foot, the new rectangle has area
90 sq ft. What is the perimeter of the original
rectangle, in feet?
88
(No Transcript)
89
Answer 42 (feet)
90
(No Transcript)
91
23. In a round-robin tournament with n teams, the
number of games that must be played is
. How many teams are in a round-robin
tournament in which 55 games are played?
92
(No Transcript)
93
Answer 11 (teams)
94
(No Transcript)
95
24. The arithmetic mean of 7, 2, x and 10 is 9.
What is the value of x?
96
(No Transcript)
97
Answer 17
98
(No Transcript)
99
25. A certain right rectangular prism has a
height of 6 feet and a rectangular base with
sides measuring 3 feet and 5 feet. What is the
volume of this prism, in cubic feet?
100
(No Transcript)
101
Answer 90 (cubic feet)
102
(No Transcript)
103
26. Charlies mom just had a baby. He notices
that they use an average of nine diapers per day.
At this rate, how many total diapers will they
use during June and July of this year?
104
(No Transcript)
105
Answer 549 (diapers)
106
(No Transcript)
107
27. If 20 of 10 of a number is 12, what is 10
of 20 of the same number?
108
(No Transcript)
109
Answer 12
110
(No Transcript)
111
28. The sum of the median and the range of a
collection of numbers is 16. If each number in
the collection is increased by 1, what is the sum
of the median and the range of the new collection?
112
(No Transcript)
113
Answer 17
114
(No Transcript)
115
29. Amy works for 36 hours per week for 10 weeks
during the summer, making 3000. If she works
for 30 weeks during the school year at the same
rate of pay and needs to make another 3000, how
many hours per week must she work?
116
(No Transcript)
117
Answer 12 (hours or
hours per week)
118
(No Transcript)
119
30. A gumball machine contains 12 red, 6 blue, 1
white and 7 green gumballs. What is the least
number of gumballs Cho must buy to guarantee
having 3 gumballs of the same color?
120
(No Transcript)
121
Answer 8 (gumballs)
122
(No Transcript)
123
31. A piece of yarn is 60 cm long. The yarn is
cut so that one piece is five times the length of
the other piece. How many centimeters long is
the shorter piece?
124
(No Transcript)
125
Answer 10 (centimeters)
126
(No Transcript)
127
32. The larger circle has center O and passes
through D. The smaller circle has diameter OD.
What percent of the larger circles area is
gray?
O
D
128
(No Transcript)
129
Answer 25 (percent)
130
(No Transcript)
131
33. Sally found that one dozen eggs costs 1.20
at Super-X. However, she could get 2.5 dozen of
the same eggs at Limitless for 3.15. When
buying 5 dozen eggs, how many cents would she
save by shopping at the less expensive store?
132
(No Transcript)
133
Answer 30 (cents)
134
(No Transcript)
135
34. What is the greatest number of distinct
positive integer factors that a positive integer
less than 20 can have?
136
(No Transcript)
137
Answer 6 (factors)
138
(No Transcript)
139
35. Uri buys two burgers and a soda for 2.10,
and Gen buys a burger and two sodas for 2.40.
How many cents does a soda cost?
140
(No Transcript)
141
Answer 90 (cents)
142
(No Transcript)
143
36. Bill travels the 400 miles from San Francisco
to Los Angeles at 50 mph. Sam travels the same
distance at 40 mph. How many more hours than
Bill did it take Sam to travel the 400 miles?
144
(No Transcript)
145
Answer 2 (hours)
146
(No Transcript)
147
37. The sides of this parallelogram measure 7, 9,
8y - 1 and 2x 3 units, consecutively. What
is the value of x y?
7
9
2x 3
8y -1
148
(No Transcript)
149
Answer 4
150
(No Transcript)
151
38. If a2 6 and a4 4b3 and a and b are
positive, what is the simplified value of a2b3?
152
(No Transcript)
153
Answer 54
154
(No Transcript)
155
39. The sum of a positive number and its square
is 156. What is the number?
156
(No Transcript)
157
Answer 12
158
(No Transcript)
159
40. An ice cream shop offers three toppings. If
a customers order may include none, one, two or
all of the toppings, how many combinations of the
toppings can a customer choose?
160
(No Transcript)
161
Answer 8 (combinations)
162
(No Transcript)
163
41. A palindrome is a number that reads the same
from left to right as from right to left. What
is the smallest integer greater than 100 which is
both a palindrome and a perfect square?
164
(No Transcript)
165
Answer 121
166
(No Transcript)
167
42. The square root of x is greater than 3 and
less than 4. How many integer values of x
satisfy this condition?
168
(No Transcript)
169
Answer 6 (values)
170
(No Transcript)
171
43. What is the absolute value of the difference
of the square of the cube of 2 and the cube of
the square of 2?
172
(No Transcript)
173
Answer 0
174
(No Transcript)
175
44. A circular spinner for a game has a radius of
5 cm. The probability of winning on one spin of
this spinner is . What is the area, in sq
cm, of the WIN sector? Express your answer in
terms of p.
WIN
LOSE
176
(No Transcript)
177
Answer 10p (square
centimeters)
178
(No Transcript)
179
45. Kenton watched 2000 adults board a cruise
ship. Half of the adults were women. If 20 of
the women and 9 of the men were wearing
sunglasses, what was the total number of men and
women wearing sunglasses?
180
(No Transcript)
181
Answer 290 (men and women)
182
(No Transcript)
183
46. If , what is the value of a?
184
(No Transcript)
185
Answer 2
186
(No Transcript)
187
47. Four pencils and a pen cost 3.00. If a pen
costs 1, how many dollars will 12 pencils and
two pens cost?
188
(No Transcript)
189
Answer 8 (dollars)
190
(No Transcript)
191
48. A book with 53 pages numbered 1 to 53 has its
pages renumbered in reverse, from 53 to 1. For
how many pages do the page numbers from each set
of numbers share the same units digit?
192
(No Transcript)
193
Answer 11 (pages)
194
(No Transcript)
195
49. What is the sum of all the positive two-digit
integers where one of the digits is four times
the other?
196
(No Transcript)
197
Answer 165
198
(No Transcript)
199
50. Point A is at (0, 0) and point B is on the
line y 4. The slope of segment AB is .
What is the sum of the x- and y-coordinates of
point B?
200
(No Transcript)
201
Answer 10
202
(No Transcript)
203
51. What is the value of the expression for
x 2? Express your answer in simplest form.
204
(No Transcript)
205
Answer 4
206
(No Transcript)
207
52. Sarahs bowling score was 40 points more
than Gregs, and the average of their two scores
was 102. What was Sarahs score?
208
(No Transcript)
209
Answer 122 (points)
210
(No Transcript)
211
53. This pattern is made from toothpicks. If the
pattern is continued by adding two toothpicks to
the previous stage, how many toothpicks are used
to create the figure for the 15th stage?
Stage 1
Stage 2
Stage 3
212
(No Transcript)
213
Answer 31 (toothpicks)
214
(No Transcript)
215
54. How many cubic feet of wood are needed to
build a solid door 7 feet high, 3 feet wide and 2
inches thick? Express your answer as a decimal
to the nearest tenth.
216
(No Transcript)
217
Answer 3.5 (cubic feet)
218
(No Transcript)
219
55. Suppose A is the set of all odd numbers
between 10 and 16, and B is the set of all prime
numbers between 10 and 16. How many distinct
results are possible if a number from set B is
subtracted from a number in set A?
220
(No Transcript)
221
Answer 4 (results)
222
(No Transcript)
223
56. Four consecutive prime numbers have a sum
that is divisible by three. What is the smallest
possible value of this sum?
224
(No Transcript)
225
Answer 36
226
(No Transcript)
227
57. Point D is on side AC of triangle ABC, m? ABD
15 and m? DBC 50. What is the measure of
angle BAD, in degrees?
B
50
15
A
C
D
228
(No Transcript)
229
Answer 25 (degrees)
230
(No Transcript)
231
58. Today a fathers age is five times his sons
age. Exactly three years ago, the sum of their
ages was 30. How old is the son today?
232
(No Transcript)
233
Answer 6 (years)
234
(No Transcript)
235
59. A line parallel to y 4x 6 passes through
(5, 10). What is the y-coordinate of the point
where this line crosses the y-axis?
236
(No Transcript)
237
Answer -10
238
(No Transcript)
239
60. What is the sum of the positive, odd factors
of 24?
240
(No Transcript)
241
Answer 4
242
(No Transcript)
243
61. A 16-ounce carton of milk costs 63 cents,
while a half-gallon jug of milk sells for 2.24.
How much more expensive, in cents per quart, is
the carton of milk compared to the half-gallon
jug of milk? (32 ounces 1 quart
gallon)
244
(No Transcript)
245
Answer 14 (cents or
cents per quart)
246
(No Transcript)
247
62. What is the sum of the negative integers
greater than -5?
248
(No Transcript)
249
Answer -10
250
(No Transcript)
251
63. How many integers satisfy the inequality -39
lt x lt 39?
252
(No Transcript)
253
Answer 77 (integers)
254
(No Transcript)
255
64. What is the slope of a line parallel to the
line 2x - 4y 9? Express your answer as a
common fraction.
256
(No Transcript)
257
Answer
258
(No Transcript)
259
65. If x2y 4 and x 4, what is the value of
y? Express your answer as a common fraction.
260
(No Transcript)
261
Answer
262
(No Transcript)
263
66. What is the arithmetic mean of 14, 22 and 36?
264
(No Transcript)
265
Answer 24
266
(No Transcript)
267
67. Two more than three times B is equal to 20.
What is the value of B?
268
(No Transcript)
269
Answer 6
270
(No Transcript)
271
68. A carton of juice with a 50-ounce capacity
contains 49 ounces of juice. If Pete drinks 38
ounces of the juice, what percent of the carton
will be empty?
272
(No Transcript)
273
Answer 78 (percent)
274
(No Transcript)
275
69. When Mr. Krumm purchased a tie he paid 9.27,
which included the 3 sales tax. How many
dollars did the tie cost before the tax was
included?
276
(No Transcript)
277
Answer 9 (dollars)
278
(No Transcript)
279
70. If a _at_ b is defined as a _at_ b 3a - 3b,
what is the value of 3 _at_ 5?
280
(No Transcript)
281
Answer -6
282
(No Transcript)
283
71. Let A be the greatest common factor and let B
be the least common multiple of 8, 12 and 24.
What is the value of A B?
284
(No Transcript)
285
Answer 28
286
(No Transcript)
287
72. A church rings its bells every 15 minutes,
the school rings its bells every 20 minutes and
the day care center rings its bells every 25
minutes. If they all ring their bells at noon on
the same day, at what time will they next all
ring their bells together?
288
(No Transcript)
289
Answer 5 (p.m.)
290
(No Transcript)
291
73. In a single-elimination tournament, each game
is between two players. Only the winner of each
game advances to the next round. In a particular
such tournament there are 256 players. How many
individual games must be played to determine the
champion?
292
(No Transcript)
293
Answer 255 (games)
294
(No Transcript)
295
74. What is the simplified value of 552 - 452?
296
(No Transcript)
297
Answer 1000
298
(No Transcript)
299
75. The diagonal lengths of a rhombus are 24
units and 10 units. What is the area of the
rhombus, in square units?
300
(No Transcript)
301
Answer 120 (square units)
302
(No Transcript)
303
76. What is the value of c if the lines with
equations y 8x 2 and y (2c)x - 4 are
parallel?
304
(No Transcript)
305
Answer 4
306
(No Transcript)
307
77. If 23 4 8 2k, what is the value of k?
308
(No Transcript)
309
Answer 8
310
(No Transcript)
311
78. Ryans truck travels 8 miles every 10
minutes. The speed limit on the road is 40 mph.
By how many miles per hour is Ryans truck
exceeding the speed limit?
312
(No Transcript)
313
Answer 8 (miles per hour)
314
(No Transcript)
315
79. A bookstore sells used books at a 40
discount off the regular price of the book when
it was new. The price of a used book is 18.
What was the regular price of that book when it
was new?
316
(No Transcript)
317
Answer 30 (dollars)
318
(No Transcript)
319
80. In a bag with 20 marbles, five of them are
blue. How many blue marbles must be added to the
bag so that the probability of selecting a blue
marble at random is ?
320
(No Transcript)
321
Answer 10 (marbles)