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MATHCOUNTS
� Sprint Round Competition �
Practice Test 1
Problems 1-30
Name
________________________________________________________
Date
_________________________________________________________
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, 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
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1

1. Which number is a prime number? 2001, 2004, 2005, 2007, 2011, and 2013.
2. How many obtuse angles are there among the four angles of 89�, 126�, 180�, and
216�?
3. How many of the numbers from 1 to 2011 are simultaneously divisible by 2, 3, and 5?
4. Alex and Bob are standing in front of a big mirror. Alex looks at the mirror and reads
the number on Bob’s sportswear as shown in the figure. What is the number of Bob’s
sportswear?
5. How many three-digit positive integers are there such that the three digits are in
increasing order (like145) or decreasing order (like 321)?
6. In the sequence 7 1 ,7 2 ,7 3 ,…,7 2011 ,how many terms have 3 as the units digit?
7. How many integer values of x if the expression
2
6x
� 3
represents integers?
2x
-1
8. Ten congruent cubes are put together as shown in the figure. If the cube A
is removed, what is the difference of the surface areas before removing the
cube A and after removing the cube A?

9. According to the pattern below, the positive integer 2011 is in the nth row and mth
column. What is the value of n + m?
10. a1,a2,a3,and a4 are the number of triangles in Figure (1), (2), (3) and (4),
respectively. a1=3.a2=8,a3=15.a3=24. What is a9 ?
11. 2011 students are standing in a row. They are called in the following pattern: 1, 2, 3,
4, 3, 2, 1, 2, 3, 4, 3, 2, ….. What number is the 2011 th student called?
12. At 3:30 P.M, the acute angle formed by the hour and the minute hands on a clock is
x�. Find the value of x.
13. a and b are prime numbers. 3a + 2b is a prime number less than 20. How many pairs
of (a, b) are there?
3

14. A number machine is shown as follows. When input x = 3 and y = - 2, what is the
result?
15. Four distinct balls are randomly put into three distinguishable boxes. Find the
probability that no box is empty. Express your answer as a common fraction.
16. Randomly pick three digits from 1, 2, 3, 4 and 5 to form a three-digit number. Find
the probability that the sum of the digits of the three-digit number is nine. Repetition of
digits is allowed. Express your answer as a common fraction.
17. As shown in the figure, one toy face is obtained by sliding the other. The coordinates
of the eyes of the left toy face are (-4, 2) and (-2, 2), respectively. One of the eyes of the
right toy face is (3, 4). Find the sum of the coordinates of the other eye of the toy face on
the right.
4

18. Less than forty identical small balls can be arranged to form a square. These balls can
also be arranged to form an equilateral triangle. Find the number of small balls.
19. Add a single term to a polynomial 4x 2 +1 to result in a square number. How many
different such single terms are there?
20. Alex wants to make a cylinder shaped container by using a
piece of rectangle shaped metal sheet as shown in the figure. What
is the volume of this container in terms of �?
21. As shown in the figure, Both ABCD and CEFG are squares. Connect AG and AG
meets CE at H. Connect HF. Find the shaded area.
22. As shown in the figure, EABCD is a road along the lake.
When Alex starts to walk along the road, he goes from west
to east (from E to A). He first turn 100� at A, that is �A=
100�. Second time he turns 150� at B, that is �B= 150�.
Third time he turns, at C, he walks from west toward east
again. Find degrees he turns at C, that is find �C.
5

23. Draw a line L passing through point (1, 3). Line L intersects the x-axis at a and the
y-axis at b. How many such lines can be drawn? Both a and b are positive integers.
24. After a math test, Ms. Leighton calculated the average score of 20 students who took
the test and she got 85 points. Alex was late for the test and he made up the test later and
he got exactly 85 points on the test. When Ms. Leighton re-calculated the class average,
what did she get?
25. A six –digit number a2011 b is divisible by 12. How many such six-digit numbers are
there?
26. The sum of n interior angles and one exterior angle of a convex n-gon is 1350�. Find
n.
27. There are four switches on an instrument board. If any two neighboring switches
cannot be turned off at the same time, how many different combinations are there?
28. There are 100 students in 8 th grade at Parrott Academy. 75 of them got perfect scores
on Math and 80 of them got perfect scores on English. At least x students got perfect
scores on both subjects and at most y students got perfect scores on both subjects. Find
the value x+y.
6

29. Find a point P that is on the same plane as the equilateral triangle ABC such that
�PAB, �PAC, and �PBC are all isosceles triangles. How many possible such points of P
are there?
30. A positive number has a decimal part and an integer part. The decimal part, the
integer part, and the positive number itself form a geometric sequence. Find this positive
number.
7

Answer keys to practice test 1
1. 2011
2. 1
3. 67
4. 1105
5. 204
6. 503 (Pattern: 7, 9, 3, 1. Every 4 number has one. 2011= 502 ×4 +3)
7. 4
8. 0
9. 255. (Pattern: every 8 numbers. 2011 = 251 ×8 +3)
10. 99
11. 1 (Pattern: every 6 numbers. 2011 = 335 ×6 +1)
12. 75�
13. 3.
14. -1
15. 4/9
6 � 3 � 3 � 6 �1
19
16. 19/125 (P =
� )
125 125
17. 9
18. 36
19. 5 ( – 4x 2 , – 1, 4x, – 4x, 4x 4 ).
20. 32�
21. 32
22. 130
23. 2
24. 85
25. 6 (a = 3, 6, 9 and b = 2; a= 2, 5, and 8 and b = 6)
26. 9
27. 8
28. 130 (75 + 55)
29. 10
30.
5 � 1 2
b 2 b a
( a � b(
b � a)
� ( ) � �1
� 0 �
2
a a b
8
�
5 �1
� a
�1, b �
2
5 �1
2