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MATHCOUNTS
■ 2006 ■
Sprint Round
Problems 1-30
_______________________
Name _________________________________________
School ________________________________________
Chapter _______________________________________
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO
This round of the competition consists of 30 problems.
You will have 40 minutes to complete the problems. You
are not allowed to use calculators, slide rules, books, or
any other aids during this round. If you are wearing a
calculator wrist watch, please give it to your proctor now.
Calculations may be done on scratch paper. All answers
must be complete, legible, and simplified to lowest terms.
Record only final answers in the blanks in the right-hand
column of the competition booklet. If you complete the
problems before time is called, use the remaining time to
check your answers.
_______________________
Total Correct
Scorer’s initials
© 2006 Art of Problem Solving inc. Users
1.
If an angle measures 56 -n degrees and n is not equal to zero. What is
the degree measure of the supplement of this angle?
1.
_________________
2.
A bag contains 4 red balls, 2 green balls, and 4 blue balls. If 2 balls are
removed at random and no ball is returned to the bag after removal,
what is the probabiliy that all 2 balls will be red?. Give your answer as
a common fraction.
2.
_________________
3.
A bag contains 4 red balls, 4 green balls, and 4 blue balls. If 2 balls are
removed at random and each ball is returned to the bag after removal,
what is the probabiliy that all 2 balls will be red. ?
3.
_________________
4.
A set S is defined as follows: An element (a,b) belongs to this set if a
and b are positive integers such that b = (330 -a)/7 . If (a,13) belong to
this set, how much is a ?
4.
_________________
5.
A perfect square number in the form 24ab5 has hundreds digit a and
tens digit b . The sum a + b is equal to:
5.
_________________
6.
If a rectangular prism block of wood has dimension 4 cm x 7 cm x 6
cm and cost $70 , what is the fair price in dollars of a 8 cm x 35 cm x
48 cm block of the same type if price is determined solely by volume?
6.
_________________
7.
Find the arithmetic mean, expressed as common fraction, of all the
7.
_________________
solutions of
8.
S is the set of all points with coordinates (m,n) such that m and n are
positive integers with m < 5 and n < 5 . Two points from S are chosen
at random. What is the probability that the midpoint of the segment of
the two points is also in the S ? Give your answer as a common
fraction in lowest term.
8.
_________________
9.
What integer n has the property that 680 is greater than n60 and 680 is
less than (n+1)60?
9.
_________________
10. Given 5 segments whose length are the elements of the set
S={2,3,5,8,13} , what is the number of distinct triangles that can be
formed using any three of these segments?
10.
________________
11. For what values of x does 1 + 2 +3 + 4 + 5 + ... + x = 300 ?
11.
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12. Mrs. Read can knit one pair of children's mittens with a ball of yarn 5
inches in diameter. How many pairs of identical mittens can she knit
with a ball of yarn 20 inches in diameter? Assume that the balls of
yarn are rolled consistently.
________________
12.
________________
13. A rectangular pool measuring 9 feet by 14 feet is surrounded by a
walkway. The width of the walkway is the same on all four sides of
the pool. If the total area of the walkway and pool is 644 square feet,
what is the number of feet in the width of the walkway?
13.
________________
14. The units digit of a six-digit number is 1 and is removed, leaving a
five-digit number. The removed units digit 1 is then placed at the far
left of the five-digit number, making a new six-digit number. If the
new number is 1/3 of the original number, what is the original
number?
14.
________________
15. What is the sum of the finite series
1+2-3+4+5-6+7+8-9+ ... +121+122-123 ?
15.
________________
16. Triangle ABC is a right triangle in C . Two points D and E are on the
side BC such that AC=CD=DE=EB and AE=9*sqrt(5) in. What is the
number of square inches in the area of the triangle ADB ?
16.
________________
17. What is the remainder of 191997 divided by 25 ?
17.
________________
18. Julie begins counting backward from 2300 by 3 , and, at the same
time, Tony begins counting forward from 1400 by 6 . If they count at
the same rate, what number will they say at the same time?
18.
________________
19. A math teacher wants to curve a set of test grades so that a student
who scored 100 receive a score of 100 , but the student who scored 57
will receive a score of 77 . The teacher wishes to use a linear
19.
________________
© 2006 Art of Problem Solving inc. Users
functional which turns an old grade, x , into a new grade f(x) . Write
your formula in the form of f(x)= m x + b . Find the product of m and
b . Express your answer as a fraction in lowest terms.
20. A thief stole 3/5 of Genevieve's money and spent 3/5 of the money
stolen. The thief was then caught, and the remaining money was
returned to Genevieve. The remaining money was 48 dollars less than
the amount Genevieve had after being robbed. How many dollars did
Genevieve have before the theft?
20.
________________
21. Consider a rectangle ABCD . Let M be a point on the segment AB
such that AM= 8 cm and MB= 12 cm . Let N be a point on the
segment BC such that BN= 4 cm and NC= 8 cm . Let P be a point on
the segment CD such that CP= 8 cm and PD= 12 cm . Let Q be a point
on the segment AD such that DQ= 4 cm and QA= 8 cm . Let O be the
point of intersection of MP and NQ . Find the area of the quadrilateral
MONB .
21.
________________
22. A Mayonnaise jar contains 4 red marbles and 6 blue marbles. A jelly
jar contains 5 red marbles and 8 blue marbles. One marble is randomly
selected from the mayonnaise jar and placed in the jelly jar. A marble
is then selected from the jelly jar. What is the probability that the
selected marble is red? Express your answer as a common fraction in
lowest terms.
22.
________________
23. The point A=(7,-4) is reflected about the x-axis to give the point B ,
then A is reflected about the line y=x to give the point C , and fianlly
A is reflected about the y-axis to give the point D . What is the area of
the quadrilateral ABCD ?
23.
________________
24. A large cube is dipped into red paint and then divided into 343 smaller
congruent cubes. One of the smaller cube is then selected randomly.
What is the probability that cube slected will have at least 25% of its
area painted red? Express your answer as a common fraction in lowest
terms.
24.
________________
25. A fast food restaurant specializes in ham sandwiches. A customer may
chose to add any or none of the following set of goodies: {Mustard,
Cheese, Onion, Hot Sauce, Lettuce} . How many different ham
sandwich combinations are possible?
25.
________________
26. What fraction in the interval 1/2 < x < 7/11 has the smallest
denominator?
26.
________________
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27. What is the sum of the positive whole number divisors of 84?
27.
________________
28. How many unique sets of 3 prime numbers exist for which the sum of
the members of the set is 44 ?
28.
________________
29. Consider the following sequence: 1/3, 1/6, 1/9, 1/12, 1/15 ........, when
the 40 th term is divided by the nth term, the quotient is 4 . What is the
value of n ?
29.
________________
30. The numbers 1,2,3,4, ... , 12 are arranged, one per circle, in the triangle 30.
shown below so that the sum s of the numbers on each side of the
________________
triangle is the same. What is the greatest sum possible?
© 2006 Art of Problem Solving inc. Users
Sprint Round Answers
1.
124+n
11.
24
21.
56
2.
2/15
12.
64
22.
27/70
3.
1/9
13.
7
23.
121
4.
239
14.
42857
24.
68/343
5.
2
15.
2460
25.
32
6.
5600
16.
81
26.
3/5
7.
1/14
17.
14
27.
224
8.
1/5
18.
2000
28.
4
9.
10
19.
46000/1849
29.
160
10.
0
20.
300
30.
37
© 2006 Art of Problem Solving inc. Users