GEORGIA SOUTHERN UNIVERSITY
INVITATIONAL MATHEMATICS TOURNAMENT
2013 VARSITY WRITTEN EXAM
Name________________________________________________________
School_______________________________________________________
Directions:
1.
Do not open this test booklet until you are told to do so.
2.
Use only a #2 lead pencil.
3.
No calculators, slide rules, notes or other aids of any kind may be used.
4.
Scratch paper is stapled to the back of the test booklet.
5.
This is a 40 question multiple-choice exam. You will be allotted 90 minutes to complete
the exam.
6.
Geometric figures are not necessarily drawn to scale.
7.
Your score will be determined by the formula 40  4R  W where
R = number of questions answered correctly and W = number of questions answered
wrong. There is no penalty for questions left unanswered.
8.

Tie-breakers will be taken from the written exam in order of difficulty. The
order will be determined by the number of people that answered each question correctly,
with the question(s) correctly answered by the fewest people considered first.
GEORGIA SOUTHERN UNIVERSITY
INVITATIONAL MATHEMATICS TOURNAMENT
2013 VARSITY WRITTEN EXAM
Convert 120˚to radians.
1.
A.
D.
2.

3

6
B.
2
3
E.
None of the above
C.
 Three carpet pieces-in the shapes ofa square, a
4
3

triangle and a semicircle-are attached to one another,
 as shown in the figure, to cover the floor of a room.
If the area of the square is 144 square feet and the
perimeter of the triangle is 28 feet, what is the
perimeter of the room’s floor in feet?
A.
D.
3.
B.
E.
40 + 6π
None of the above
C.
40 + 12π
Fifty boys and fifty girls attended a school dance. Each girl shook hands once with every
boy and girl at the dance, but no boy shook hands with any other boy. How many
handshakes occurred?
A.
D.
4,950
3,725
B.
E.
3,675
None of the above
The mean proportional of two positive numbers m
m x
and n is the positive number x such that
 .
x n
If the area of ABC is 64 square inches and the mean
proportional between sides AB and AC is 12 inches
find the value of sin A.

4.
A.
D.
5.
32 + 12π
52 + 6π
32

3
8
9
C.
3,775
B
c
A
B.
2
9
E.
None of the above
a
C
b
C.
4
9
 Jemima has the same number of brothers

 brother Roland has
as she has sisters, but her

twice as many sisters as he has brothers. How many girls are there in the family?
A.
D.
5
4
B.
E.
2
None of the above
C.
3
6.
7.
A square is inscribed in an equilateral triangle. If
the length of the side of the triangle is x. Find the
square's area in terms of x.
A.
3x 2 (7  4 3)
D.
x 2 33
E.
A.
Q
20
B.
Q
5
D.
10Q
E.
None of the above
C.
x2
4
C.
20Q

 When Maria went to get a passport,she had to give her real date of birth, but under all
other circumstances she refused. When somebody asked how old she was, she said she
was twenty-one, mentally omitting all Sundays. Sundays she did not work, so naturally
she did not get any older. How old was Maria really? Truncate the answer to a whole
number.
A.
D.
9.

3x 2
16
None of the above
 If P percent of 20 is Q, then P equals
 which of the following.?

8.

B.
x
21
26
B.
E.
24
None of the above
C.
25
The denominator of a certain fraction is twice as great as the numerator. If 4 were added
to both the numerator and denominator, the new fraction would be equivalent to
What is the denominator of the fraction?
A.
D.
3
12
B.
E.
6
None of the above
C.

9
5
.
8
10.
The circles shown are tangent to each other and
tangent to the rectangle. If the circles are lids that
have been cut from a rectangular piece of tin, what
percent of the metal has been wasted?
A.
D.

4
25π
1
B.
100π – 100
E.
None of the above
11.  A box contained 31 chocolates. The first day Gabriela ate
C.
100-25π
3
of the number Pam ate the
4
2
of the number Pam ate that day, and the
3
chocolates were all gone. How many chocolates did Gabriela eat?

first day. The second day, Gabriela ate
A.
D.
6
9

B.
E.
13
None of the above
C.
18
12.
If tan  0 and sec  0 in which quadrant does the terminal side of  lie?

A.
D.
13.
B.
E.
II
None of the above
C.
III
Given the hyperbola x 2  y 2  2 and the ellipse 3x2  4y 2  13 find the product of all the
x- coordinates plus the product of all the y-coordinates of all the points of intersection.
A.
D.
14.
I
IV
4
10 
B.

E.
8
None of the above
C.
Find the value of x so that the pair of
triangles is congruent by the Leg-Angle
theorem.
2
(2x + 5)˚
(6x – 11)˚
A.
4
B.
33
D.
16
E.
None of the above
61˚
C.

20
3
15.
Of 60 pairs of socks in a drawer, 40% are blue, while the remaining socks are all gray. If
4 pairs blue socks are removed from the drawer, what is the ratio of gray socks to blue
socks?
A.
D.
16.
D.
18.

9:5
C.
None of the above
3:5
50 3
3
25
B.
50 3
E.
None of the above
C.
25 3
When the product of a real number
p and 3 is decreased by 8, the
result is less than 7.
Which of the following intervals represents all possible values of p?
A.
3, 
B.
3, 
D.
,5
E.
None of the above
C.
,5



In ∆MNP, MN  MP , the measure of angle P is 10 more than 3 times a number, the
 measure of angle N is 8 less than 5 times the same number. Find the measure of angle M.
A.

D.
19.
B.
E.
The angle of depression from the top of one building to the closest point on the top of a
neighboring building is 30˚. If the buildings are 50 feet apart, what is the difference
between the heights of the buildings?
A.
17.
1:2
5:9
48
106
B.
E.
116
None of the above
C.
37
C.
O
What is the 100th letter in the pattern ABBCCCDDDD...?
A.
D.
M
P
B.
E.
N
None of the above
20.
b
The adjacent figure shows a wire
(represented by the dashed line)
connecting the outer edge of a porch roof
to the base of the building’s wall, creating
angle of  degrees at the base of the
building. If a and b are the height of the
building and width of the porch roof,
respectively, the Pythagorean theorem
would give the length of the wire as
a 2  b2 which of the following also
represents the length of the wire?

roof line
a
building

ground
A.
b
cos
B.
a
sin 
D.
bsin 
E.
None of the above
C.
b
sin 


21.
If f x  6 and gx  log 6 x which of the following expressions is equal to f 2gm?


A.
D.

22.
x
2m
6
m
B.
E.
6m
None of the above

If BCD is obtuse which of the following

is true about the figure shown?
A.
C.
E.
mBCD  mA  mB
mBCD  mA  mB  mBCA
None of the above


B
D
A

m2
C.
C
B.
D.
BCD  A
BCD  BCA


23.  I have a very, very generous grandmother. She told
me that starting at midnight on my

birthday she is going to deposit $1000 into my college savings account every time the
hour and minute hands of a clock form a 90-degree angle. How much money will my
grandmother deposit on my birthday?
A.
D.
$44,000
$24,000
B.
E.
$48,000
None of the above
C.
$52,000
The expression sin xsin ycot x  cot ysimplifies to which of the following?
24.
A.
D.

sin x  y
0

B.
E.

cosx  y
None of the above
C.
1
Find the sum of the real roots of (x 1)(x  2)(x  4)(x 1)  72 .
25.
A.
D.
282
138
B.
E.
-3
None of the above
C.

The figure shows a rectangular solid with the
following unit dimensions: QR  3, QS  4 and
ST  5 . How many units long is RT ?
26.



-6
T

S
Q
A.
D.
27.
B.
E.
4 2
4 3
R
6
None of the above
C.
5 2
Cynthia drove for seven hours at an average rate of 50 miles per hour and for one hour at


an average rate of 60 miles per hour. What was her average rate for the entire trip?

1
A.
55 mph
B.
52 mph
C.
57 mph
2
1
D.
E.
None of the above
54 mph
2


Express sin cos1 y as an algebraic expression in terms of y.
28.

A.

D.
y
B.
1 y 2
E.
y 2 1

C.
1
1  y2
None of the above

29.
8
2
f(x)  2
 ?
At what value(s) of x does f(x) have a vertical asymptote if 
x  4x x

A.
D.
x=0
x = 0 and x = 4
B.
E.
x = 0 and x = 2
None of the above

C.
x=4
30.
The circle shown has a center at the point
X and has a diameter of length 16 units.
The measure of ZXY is 80˚. Find the
length of arc ZY.
X

31.
Z
W
Y
A.
640
B.
D.
16π
E.
16
9
None of the above
C.
32
9
The accompanying table represents the number of cell phone minutes unused for one


week by 23 users.
Number of Minutes
71-80
61-70
51-60
41-50
31-40
Number of Users
10
7
2
3
1
Which interval contains the median?
A.
D.
32.
41-50
71-80
B.
E.
51-60
None of the above
C.
61-70
The hiking club set out at 12 noon. The members hiked along a level road at a steady rate
of 4 MPH; up a mountain trail at 3 MPH; immediately back down the mountain trail at 6
MPH, and back home along the level road, again at 4 MPH. They got back at 6 P.M.
Find the total distance the club hiked?
A.
D.
33.
6
24
B.
E.
12
None of the above
C.
21
A spherical bubble lands on a horizontal wet surface and forms a hemisphere of the same
volume as that of the original bubble. If the radius of the hemisphere is 33 2 , find the
radius of the original bubble.


A.
9
B.
3
D.
63 2
E.
None of the above
C.

33 2
2
34.
How many minutes does it take a clock’s hour hand to move through 1 degree of arc?
A.
D.
35.
1
4
B.
E.
2
None of the above
C.
A triangular trough has sides meeting at angles of 60˚. A
ball of radius R is placed in the trough. In terms of R,
what is the radius of the largest ball that will just fit
beneath the bigger ball?
3
R
60˚

36.
A.
1
R
2
B.
1
R
4
D.
1
R
3
E.
None of the above
C.
3R


1
On Bill’s football squad, of the players walk to practice and 25% are driven by their
3
 parents. The remaining 15 players take the bus. How many members are on the football
team?
A.
D.
37.
48
30

B.
E.
24
None of the above
C.
36
An isosceles right triangle is removed from each corner of a square piece of paper, so that
a rectangle remains. The rectangle that remains is not a square. What is the length of the
diagonal of the rectangle if the sum of the areas of the cut-off pieces is 200 square units?
A.
D.
10
20 2
B.
E.
10 2
None of the above
C.
20
C.
2sin x  3
 NOT have a solution?
Which of the following equations does
38.

A.
D.


sin 100x  0.1
1
cos2 x   0
4

B.
tan x 10,000
E.
None of the above

39.

Which of the following are polar coordinates for the rectangular coordinate point 1,
40.

A.
( 2, 60˚ )
B.
( 2, 30˚ )
D.
 3, 60 
E.
None of the above
 3, 30 
C.

The average of twelve different strictly positive integers is 12. What
 is the greatest
possible value of any one of these numbers?
A.
D.
78
144
B.
E.
133
None of the above
C.
66

3 ?
GEORGIA SOUTHERN UNIVERSITY
INVITATIONAL MATHEMATICS TOURNAMENT
2013 VARSITY WRITTEN EXAM ANSWER
1.
B
21.
C
2.
B
22.
A
3.
D
23.
A
4.
D
24.
A
5.
D
25.
B
6.
A
26.
C
7.
E (answer: 5Q)
27.
1
E (answer: 51 mph )
4
8.
B
28.
D
9.
D
29.
C
30.
C
31.
C
32.
D
33.
B
34.
B
35.
D
36.
C
37.
C
38.
C
39.
A
40.
A
10.
C
11.
B
12.
D
13.
D
14.
E (answer: 12)
15.
B
16.
A
17.
C
18.
D
19.
B
20.
C