2009 Marywood Mathematics Contest

Level I

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 21, 2009

Directions:

1. This exam consists of 40 questions on 6 pages. Please check to make sure that you have
all the pages.

2. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

3. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

4. There is a sheet of blank paper on the last page which you can (carefully) tear off and
use as scratch paper. You may also use the back of the pages.

5. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer any
questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
1. How many positive integers below 2009 are divisible by both 4 and 6?

A. 502 B. 334 C. 167 D. 83 E. None of these.

2. What is the distance between the points (4,12) and (-2,4)?

A. 6 B. 8 C. 10 D. 14 E. None of these.

x
3. If f (x) = and h(x) = x1/2 , then f (h(9)) =
2
3 9 31/2 2 E. None of these.
A. B. C. D.
2 2 2 3

4. What term fits in the blank space in the sequence 1/2, 4, 1/8, 16, , 64, · · · ?

A. 1/24 B. 1/32 C. 1/40 D. 1/42 E. None of these.

5. Write an equation in point-slope form of the line that contains the point (−3, 5) and has
the slope −1.

A. y + 3 = −(x − 5) B. y − 3 = −(x + 5) C. y + 5 = −(x − 3)

D. y − 5 = −(x + 3) E. None of these.

6. If the measure of an angle is 78◦ less than the measure of its complement, what is the
measure of the angle?

A. 6◦ B. 51◦ C. 84◦ D. 129◦ E. None of these.

7. The length of one side of a rectangle is 8 cm and its perimeter is 40 cm. Find the area of
the rectangle.

A. 320 cm2 B. 256 cm2 C. 96 cm2 D. 48 cm2 E. None of these.

8. Given the diagram below, what is the measure of ∠2?

2 A. 63◦ B. 71◦

C. 81◦ D. 77◦

E. None of these.
1 45◦ 3
54◦

1
 √
9. a2 =

A. a 1 C. |a| D. a2 E. None of these.

B.
a

c4 d4 e5
10. Evaluate .
c−1 (d−2 e5 )−3
A. c3 d−2 e20 B. c5 d2 e−15 C. c3 d10 e−10

D. c5 d−2 e20 E. None of these.

11. In a movie, an FBI agent has to stop a ticking time bomb within 30 seconds by cutting
one wire from a group of 4 indistinguishable red wires and another wire from a group of
5 indistinguishable green wires. There is exactly one combination that would stop the
bomb from exploding. What is the probability (mathematically, not by the movie script)
that the FBI agent will succeed if he or she randomly cuts a red wire and a green wire?

A. 5% B. 20% C. 25% D. 45% E. 100%

12. Write an equation in point-slope form of the line that contains the points (2, 6) and
(−1, 3).

A. y − 3 = −1 · (x + 1) B. y − 6 = 1 · (x − 2) C. y − 6 = 3(x − 2)

D. y + 3 = 1 · (x − 1) E. None of these.

13. What is the slope of the line 42x + 9y = −7?

A. −6 3 3 14 14
B. C. − D. E. −
14 14 3 3

14. Solve 7x−1 = 494−x for x.

A. 1 B. 2 C. 3 D. 4 E. None of these.

15z 3 − z 2 − 11z − 3
15. Simplify .
3z 2 − 2z − 1
A. 5z − 3 B. 5z + 3 C. 15z + 3 D. 3z + 5 E. None of these.

16. Wendy’s mother is three times as old as Wendy, and twelve years from now she will be
twice as old as Wendy is then. How old is Wendy now?

A. 4 B. 8 C. 12 D. 16 E. None of these.

2
17. Solve for x:
xy
k+t=
v
.
vk v(k + t) v(t − y)
A. x = B. x = C. x =
yt y k
t(v − y) E. None of these.
D. x =
k

18. A rock and a half rock weigh a pound and a half. The rock and the half rock weigh the
same. How much would half the whole rock and the whole of the half rock weigh?
1 1 7 1 E. None of these.
A. 2 B. 1 C. 1 D. 2
2 8 2 4

19. What is the last digit of 72009 ?

A. 1 B. 3 C. 5 D. 7 E. 9

20. Which of the following is a common factor of x2 − 10x + 21 and x2 + 2x − 15?

A. x + 5 B. x − 3 C. x + 3 D. x − 7 E. None of these.

21. Fred and Desi left Steamtown Mall at 9:00 am and began walking in opposite directions.
At 1:00 pm that same day they were 20 miles apart. If Fred walks 0.5 mph slower than
Desi, what is Desi’s speed of walking?

A. 2.25 mph B. 2.75 mph C. 3.5 mph D. 5 mph E. None of these.

22. The area of an equilateral triangle with side a is

1√ 2 1√ 2 1√ 2 1√ 2 E. None of these.
A. 2a B. 3a C. 2a D. 3a
2 2 4 4

23. Expand and simplify the expression (a + b)7 + (a − b)7 .

A. 2a7 + 42a5 b2 + 70a3 b4 + 14ab6 B. 2a7 + 21a5 b2 + 35a3 b4 + 7ab6

C. a7 + 21a5 b2 + 35a3 b4 + 7ab6 D. 2a7 + 42a5 b2 + 35a3 b4 + 14ab6

E. None of these.

3
 √
24. Simplify 128a7 b3 c4 (extract all perfect roots).
√ √ √
A. 8a2 b c 2ab B. 8|a3 | |b| c2 2ab C. 8|a3 | |b| c2 2
√
D. 8|a7 | |b3 | c2 2a E. None of these.

25. If −3b2 = 10b − 48, then b =

8 3 3 1 8 1 E. None of these.
A. or −6 B. or −6 C. or D. or
3 8 8 6 3 6

26. How many liters of water must be added to 15 liters of a 40% acid solution to produce a
solution which is 10% acid?

A. 45 B. 29 C. 55 D. 34 E. None of these.

27. Every Wednesday at the Pizza Express, the manager gives away free slices of pizza and
sodas. Every 6th customer gets a free slice of pizza and every 8th customer gets a free
soda. The Pizza Express served 75 customers last Wednesday. How many customers
received both a free slice of pizza and a free soda?

A. 0 B. 1 C. 2 D. 3 E. 4

28. If a square of side 8 is inscribed in a circle, what is the area of the circle?

A. 64π B. 32π C. 128π D. 64 E. 32

29. How many digits does the number 20082009 have?

A. 2008 B. 2009 C. 2010 D. 6635 E. 6636

30. Given a cube of side length 2, with A being one of the vertices and AB, AC and AD are
the three edges at A. Let E, F and G be the midpoints of AB, AC and AD. We then
cut the corner containing the vertex A off along the plane determined by E, F and G.
What is the volume of the part that was cut off?
C 1 1
A. B.
2 3
F 1 1
B E C. D.
4 5
A 1
E.
6
G

D
4
 5n2 + 10 − 13n + 6n3
31. =
3n − 5
30 30
A. 2n2 + 5n − 4 − . B. 2n2 + 5n − 4 + .
3n − 5 3n − 5
30 30
C. 2n2 + 5n + 4 + . D. 2n2 − 5n + 4 − .
3n − 5 3n − 5
30
E. 2n2 − 5n + 4 + .
3n − 5

32. Tim can paint a fence in 80 minutes by himself while it takes Mike one hour to paint the
same fence alone. If Tim and Mike work together for 10 minutes and Tim then leaves to
attend Algebra class, how many minutes more will it take Mike to finish the job?
6 1 5 2 E. None of these.
A. 23 B. 42 C. 44 D. 39
7 2 8 7

33. A shadow in the sun cast by a tree is 48 ft. At the same time, a shadow cast by a nearby
post is 15 ft. If the line of sight distance from the farthest end of the tree’s shadow to
the top of the tree is 62 ft., what is the line of sight distance from the top of the post to
the farthest end of the post’s shadow? Round your answer to the nearest tenth of a foot.

A. 18.1 ft. B. 19.4 ft. C. 22.9 ft. D. 14.1 ft. E. None of these.

34. Find the surface area of the solid below.

6a A. 96a2 B. 124a2

2a C. 120a2 D. 104a2

E. None of these.

5a
3a
2a 2a

35. Alex orders a round 12 in. (diameter) pizza and it is cut into 12 equal pieces where each
cut goes through the center. He eats a quarter of the whole pizza while his wife eats a
quarter, his son eats a third, and his daughter eats a sixth. He and his daughter do not
eat the crust so they give it to the dog. How many inches of crust (as measured from the
outside) does the dog eat?

A. 5π in. B. 6π in. C. 7π in. D. 8π in. E. None of these.

5
36. Find an equation of the following hyperbola: Foci at (0, −8) and (0, 8) and the difference
of focal radii is 10.
(Note: a focal radius is the distance from a point on the hyperbola to one of the foci.)
x2 y2 y2 x2 x2 y2
A. − =1 B. − =1 C. − =1
39 25 39 25 25 39
y2 x2 E. None of these.
D. − =1
25 39

37. On a line, there are three points J, H, and K (not necessarily in this order). The distance
between J and H is 4x − 15, between H and K is 2x + 3 and between J and K is 48.
There is more than one possible value for x. What is the sum of all these possible values?

A. 10 B. 28 C. 17 D. −5 E. 43

38. In general (a − b)3 6= a3 − b3 . However, for some values of a and b, we indeed have
(a − b)3 = a3 − b3 . If we assume 0 ≤ a ≤ 2009, 0 ≤ b ≤ 2009 and both are integers, how
many different pairs of such (a, b) satisfy (a − b)3 = a3 − b3 ?

A. 2009 B. 2010 C. 4019 D. 6028 E. 6030

39. In the given diagram, the circle has radius equal to 60 while the (short) arc length from
D to E is 80 and the (short) arc length from B to C is 88. What is the measure of ∠BAC
in radians?
7 5
B A. B.
5 7
E 12 5
C. D.
5 12
A E. None of these.

C D

40. What is radius of the largest circle centered at (0, 1) such that no part of the circle is
below the parabola y = x2 ?
√
1 1 3 D. 1 3
A. B. C. E.
4 2 4 2

6
There are no problems on this page and it is intended to be used as scratch paper.
 2009 Marywood Mathematics Contest
Level II

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 21, 2009

Directions:

1. This exam consists of 40 questions on 7 pages. Please check to make sure that you have
all the pages.

2. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

3. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

4. There is a sheet of blank paper on the last page which you can (carefully) tear off and
use as scratch paper. You may also use the back of the pages.

5. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer any
questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
1. 66 + 66 + 66 + 66 + 66 + 66 =

A. 66 B. 366 C. 636 D. 3636 E. 67

2. If 3(4x + 5π) = P , then 6(8x + 10π) =

A. 2P B. 4P C. 6P D. 8P E. 18P

3. If m > 0 and the points (m, 3) and (1, m) lie on a line with slope m, then m =
√ √ √
A. 1 B. 2 C. 3 D. 2 E. 5

4. If a, b, and c are positive integers and a and b are odd, then 3a + (b − 1)2 c is

A. odd for all choices of c. B. even for all choices of c.

C. odd, if c is even; even, if c is odd. D. odd, if c is odd; even, if c is even.

E. odd, if c is not a multiple of 3; even if c is a multiple of 3.

5. A lily pad grows so that each day it doubles its size (area). On the 20th day of its life, it
completely covers the pond. On what day of its life was the pond half covered?

A. 5th B. 10th C. 15th D. 19th E. None of these.

6. What is the diameter (in cm) of the circle whose area (in cm2 ) and circumference (in cm)
have the same numerical value?

A. 1 cm B. 2 cm C. 3 cm D. π cm E. 4 cm

xy y x
7. If x > y > 0, then =
y y xx
 x−y  y−x
A. (x − y)y/x x C. 1 x E. (x − y)x/y
B. D.
y y

8. The ratio of w to x is 4 : 3, of y to z is 3 : 2 and of z to x is 1 : 6. What is the ratio of w

to y?

A. 1 : 3 B. 16 : 3 C. 20 : 3 D. 27 : 4 E. 12 : 1

1
 1 1
9. For all non-zero real numbers x and y such that x − y = xy, − =
x y
1 1 C. 1 D. −1 E. y − x
A. B.
xy x−y

10. A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the
floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total
number of tiles is

A. 121
B. 625
C. 676
D. 2500
E. 2601

11. SuperSneaky Department Store is having a 50% “plus” 40% off sale on everything they
have in store, which entitles you to 50% off the original price first and then a 40% off this
discounted price. How much does a coat originally priced at $150 cost now?

A. $15 B. $30 C. $45 D. $60 E. None of these.

12. The number of positive integers k for which the equation kx − 12 = 3k has an integer
solution for x is

A. 3 B. 4 C. 5 D. 6 E. 7

13. The radii of the three circles below are in the ratio 1 : 2 : 3. What is the probability that
a random shot that hits the target will hit inside the inner most circle?
1 1
A. B.
3 6
1 3
C. D.
9 16
E. None of these.

2
 √
14. Five equilateral triangle, each with side 2 3, are arrange so they are all on the side of a
line containing one side of each. Along this line, the midpoint of the base of one triangle
is a vertex of the next. The area of the region of the plane that is covered by the union
of the five triangular regions is

A. 10 B. 12
√
C. 15 D. 10 3
√
E. 12 3

15. Which of the following equations represents a parabola with y-intercept equal to −6 and
x-intercepts equal to −1 and 2?

A. y = x2 − x − 2 B. y = 3x2 − 3x − 6 C. y = x2 + 5x − 6

D. y = x2 − x − 6 E. None of these.

16. Let y = mx + b be the image when the line x − 3y + 11 = 0 is reflected across the
x-axis. The value of m + b is

A. −6 B. −5 C. −4 D. −3 E. −2

17. How many pairs of positive integers (a, b) with a + b ≤ 100 satisfy the equation

a + b−1
= 13?
a−1 + b
A. 1 B. 5 C. 7 D. 9 E. 13

18. Which of the following equations have the same graph?

x2 − 4
I. y = x − 2 II. y = III. (x + 2) y = x2 − 4
x + 2
A. I and II only. B. I and III only.

C. II and III only. D. I, II, and III.

E. None. All the equations have different graphs.

3
19. Consider the sequence defined recursively by u1 = a (where a is any positive number),
−1
and un+1 = , n = 1, 2, 3, . . . . For which of the following values of n must un = a?
un + 1
A. 14 B. 15 C. 16 D. 17 E. 18

20. An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent
of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What
percent of the objects in the urn are gold coins?

A. 40% B. 48% C. 52% D. 60% E. 80%

√
21. Let i = −1. Define a sequence of complex numbers by z1 = 0, zn+1 = zn2 + i for
n ≥ 1. In the complex plane, how far from the origin is z111 ?
√ √ √ √
A. 1 B. 2 C. 3 D. 110 E. 255

y x + y x
22. If = = for three positive numbers x, y, and z, all different, then
x − z z y
x
=
y
1 3 2 5 E. 2
A. B. C. D.
2 5 3 3

23. Given that f (x) = 3x + 4 and f (g(x)) = x, find an expression for g(x).
1 B. g(x) = 3x − 4 C. g(x) = 4x − 3
A. g(x) = x − 4
3
1 E. None of these.
D. g(x) = x + 4
3

24. The center C and the radius r of the circle represented by x2 + y 2 + 10x − 14y = −38 are

A. C : (−5, 7), r = 6. B. C : (5, −7), r = 36. C. C : (−5, −7), r = 6.

D. C : (−7, 5), r = 6. E. None of these.

25. Express the following product as a reduced fraction.

     
1 1 1 1
1− 1− 1− ··· 1−
2 3 4 2009
2008 1 143 20 E. None of these.
A. B. C. D.
2009 2009 287 41

4
26. Joe walked to the crater at a speed of 1.5 mph. He returned along the same path by
horseback at 9 mph. The roudtrip took 21 hours. How many miles is it to the crater
from where he originally was?

A. 27 B. 3 C. 18 D. 110.25 E. 7

27. The vertex of the parabola represented by y 2 − 8x − 6y + 41 = 0 is

A. (−4, −3) B. (−4, 3) C. (4, 3) D. (4, −3) E. None of these.

28. Sparrows and pigeons sit on a fence. When 5 sparrows leave, there remain 2 pigeons for
every sparrow. Then 25 pigeons leave and the ratio of sparrows to pigeons becomes 3 : 1.
What is the original number of birds?

A. 50 B. 48 C. 75 D. 30 E. None of these.

29. Let [x] be the greatest integer that is less than or equal to x and f (x) = [x]2 − 2[x]. Find
f (2π).

A. 32 B. 16 C. 24 D. 12 E. None of these.

30. If (−2, 8), (8, −15) and (−6, k) are collinear points, then the value of k is

A. 13.5 B. 14.2 C. 18.4 D. 17.2 E. None of these.

31. Given f (x) = (x + 6)3 + 1, then f −1 (x) =

√ √
A. 3 x − 1 − 6. B. 3 4x − 5 − 6. C. (4x − 7)3 − 3
√
D. 5x2 + 9 − 3. E. None of these.

32. The degree measure (to the nearest minute) of the central angle having intercepted arc
measuring 15 ft in a circle of diameter 19 ft is:

A. 15◦ 18′ B. 90◦ 28′ C. 62◦ 15′ D. 78◦ 32′ E. None of these.

33. Given that the terminal side of the angle θ is in Quadrant IV with its initial side being
12
the positive x-axis and csc θ = − , what is the value of cot θ?
7
√ √ √ √
29 95 65 85 E. None of these.
A. B. − C. − D.
7 7 7 7

5
34. The ratio of the radii of two concentric circles is 1:3. If A C is a diameter of the larger
circle, B C is a chord of the larger circle that is tangent to the smaller circle, and A B =
12, then the radius of the large circle is

A. 13 B. 18
B C. 21 D. 24

E. 26
A C

35. Given the rectangle ABCD below and E bisects DC and F bisects AD. Find the area
of the quadrilateral BEDF .

A. 7.5 in2 00
11
0
1
A F D
000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
B. 6.5 in2 0
1
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
in

0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111 E
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
3

C. 6.75 in2 0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
000000000000000000000000000000000000000
111111111111111111111111111111111111111
0
1
0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
000000000000000000000000000000000000000
111111111111111111111111111111111111111
D. 7.75 in2 0
1 000000000000000000000000000000000000000
111111111111111111111111111111111111111
11
00
B C
0
1 0
1
0
1 0
1
111111111111111111111111111111111111111
000000000000000000000000000000000000000
5 in
E. None of these.

36. Beula, a lovable cow, is located 1 mile north of the Pasture River along the fence line.
She wants to get to her barn to take a nap. Her barn is located 3 miles east of the fence
line and 2 miles north of the river. She also wants to stop at the river for a drink along
the way. Where on the river should she stop in order to minimize her total amount of
walking? The answers below are given as coordinates on the x-axis, which is the Pasture
River, and we assume the fence is the y-axis.

1
0 North A. (2, 0)
0
1
0
1
0
1 Barn B. (2.5, 0)
0
1
0
1
0
1 C. (1, 0)
0
1
0
1
0
1 Beula D. (1.5, 0)
0
1
0
1
0
1
0
1
E. None of these.
Fence

1111111111111111111111111111
0000000000000000000000000000
0
1
0
1 East
0
1 Pasture River
0
1
0
1
0
1
0
1
0
1 6
0
1
0
1
0
1
37. How many ways can 10 men and 7 women sit in a row so that no two women are next to
each other?

A. 17! B. (10!)(17!) C. 330(10!)(17!)

D. 120(10!)(17!) E. None of these.

38. The integers from 200 down to 9 are written consecutively to form the large integer

N = 200199198197 . . . 131211109.

If 3k is the highest power of 3 that is a factor of N, then the value of k is

A. 0. B. 1. C. 2. D. 3. E. more than 3.

39. The increasing sequence of positive integers a1 , a2 , a3 , . . . has the property that an+2 =
an + an+1 for all n ≥ 1. If a7 = 120, then a8 is

A. 128 B. 168 C. 193 D. 194 E. 210

40. For each vertex of a solid cube, consider the tetrahedron determined by the vertex of
the midpoints of the three edges that meet at the vertex. The portion of the cube that
remains when these eight tetrahedra are cut away is called a cuboctahedron. The ratio
of the volume of the cuboctahedron to the volume of the original cube is closest to which
of the following?

A. 75% B. 78% C. 81% D. 84% E. 87%

7
There are no problems on this page and it is intended to be used as scratch paper.
 2010 Marywood Mathematics Contest
Level I

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 27, 2010

Directions:

1. This exam consists of 40 questions on 6 pages. Please check to make sure that you have
all the pages.

2. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

3. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

4. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

5. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
  0
4 25
1. Compute 7 + (9)(0 ) − 6 · .
−2 + 7
A. 1 B. 7 C. 10 D. −26 E. None of these.

2. How many pairs of parallel edges are there in the following octohedron?

A. 8
B. 6
C. 4
D. 2
E. None of these.

3. Solve the equation 9x + 2(7x + 5) = 0 for x.

A. x = 12/5 B. x = −10/23 C. x = −5/23 D. x = 15/6 E. None of these.

5 3
4. Find the number which is one fourth of the way from to on the number line.
8 2
27 22 27 21
A. 32
B. 8
C. 31
D. 8
E. None of these.

5. If x and y are non-negative real numbers, then (125x13 y 14 )1/3 =

13 14 13 14
A. 5x 3 y 3 B. 125x 3 y 3 C. 5xy
13 10
D. 25x 3 y 3 E. None of these.

6. Find the equation of the line that has the x and y−intercepts at (−10, 0) and (0, 8).

A. 9x − 11y = −99 B. 7x − 11y = −77 C. −8x − 9y = 72

D. 7x − 9y = −63 E. None of these.

7. Suppose that an operation ♣ is defined as a♣b = a3 − b2 . What is the value of 2♣(3♣4)?

A. −113 B. −15 C. 11 D. −120 E. None of these.

1
 8. If the radius of a circle is doubled, the ratio between its area (measured in cm2 ) and
circumference (measured in cm) will

A. remain the same. B. also double. C. triple.

D. quadruple. E. None of these.

9. Two values of the linear function f (x) are f (3) = −5 and f (6) = 8. If f (x) = mx + b,
what is m − b?
7 213 67 −51 E. None of these.
A. B. C. D.
39 39 3 13

10. A sale at a department store says that customers will receive 40% off their total purchase.
Lucy buys three items: a coat regularly priced at $40, a pair of shoes at $25, and a purse
at $15. What will her final bill be?

A. $32 B. $112 C. $48 D. $54 E. None of these.

11. In a league of seven teams, each team plays each of the other teams twice during the
regular season. The top three teams make the playoffs. The playoffs are single elimination,
and the top team receives a bye for the first round (i.e., the top team skips the first round
and goes to the next round directly). How many games are played altogether?

A. 21 B. 23 C. 42 D. 44 E. None of these.

18x2 + 39x − 70
12. Find the quotient Q(x)and remainder R after simplifying .
−3x − 6
A. Q(x) = −6x − 1, R = −76 B. Q(x) = −6x − 1, R = −64

C. Q(x) = 6x − 1, R = −76 D. Q(x) = 6x − 1, R = −64

E. None of these.

13. The two legs AB and BC of a right triangle are in the ratio of 1 : 3, what is the ratio
AD : DC?
B
A. 1 : 3
√
B. 1 : 10
C. 3 : 10
D. 1 : 9
A D C

E. None of these.

2
14. How many real soultions does the equation x3 − x2 + x − 1 = 0 have?

A. 0 B. 1 C. 2 D. 3 E. None of these.

15. If the two real roots of the equation x2 + 6x − 4 = 0 are x1 and x2 with x1 > x2 , what is
x1 − x2 ?
√
A. −6 B. 2 13 C. 6 D. 2 E. None of these.

16. Gabby and Jen can plant a garden together in 4 days, but Gabby can do it alone in 6
days. How long will it take Jen alone to finish the job after Gabby works on it for 4 days
by herself?

A. 1 day B. 2 days C. 3 days D. 4 days E. None of these.

17. Two planes leave from the same airport at 8AM. One plane is flying north at 600mph,
while the other flies west at the same speed. Approximately how far apart are the planes
from one another at noon?

A. 1700mi B. 2400mi C. 3400mi D. 4800mi E. 7200mi

18. Alfred purchased a combination of candy bars and bags of chips from the grocerey store.
Each candy bar costs $0.70, and a bag of chips costs $0.50. If he bought a total of 7 items
and spent $4.50, how much of the $4.50 was spent on candy bars?

A. $1.40 B. $2.10 C. $2.50 D. $2.80 E. None of these.

19. What is the y-intercept of the line 5x + 3y = 1?

A. 1/3 B. 3 C. 1/5 D. 5 E. None of these.

√
20. ln e3 + 2 ln e1/4 =
3 1 √
A. 7/2 B. ln(e 2 + e 2 ) C. 2 D. 3 + 1/2 E. None of these.

21. There are 2 dimes, 3 nickels, and 5 quarters in a bag. If two coins are selected from the
bag at random, what is the probability of picking one quarter and one nickel?

A. 2/9 B. 2/15 C. 1/3 D. 2/3 E. None of these.

3
22. What is the next number in the sequence 6, 9, 14, 21, 30,...

A. 43 B. 39 C. 47 D. 41 E. None of these.

23. You and your crew, all members of the local math club, the Sand-Reckoners, are walking
down Drinker Street when you come upon your rival mathletes from the high school
across town, the Calculators! You decide to throwdown on each other with weapons of
math destruction. Your group’s three members have three Casio’s, two abacuses and
four TI-89’s. The Calculators have two HP’s, four Sharp’s, three slide rules and one set
of Napier’s bones. What is the ratio of superior (electornic) weapons to inferior (non-
electornic) weapons?

A. 9:10 B. 6:5 C. 8:3 D. 13:6 E. None of these.

24. Corey could save 30% off the original price on any one CD with a coupon. He has another
coupon for 20% off the total purchase price in one transaction regardless of the number of
items, but the two coupons cannot be used together in one transaction. He plans to buy
two CDs, one priced at $18 and the other at $12. What is the absolute lowest amount
that he must spend to buy these two CDs?

A. $24.60 B. $26.40 C. $22.20 D. $19.68 E. None of these.

7
25. Given the line y = x + 1729. What is the slope of a line perpendicular to the given line?
2
7 2 7 2 E. None of these.
A. B. C. − D. −
2 7 2 7

26. Once upon a time, three brothers inherited 750 gold coins from their rich father. The will
states that the youngest should recieve a certain number of these gold coins, the second
youngest should receive 90% of what the youngest brother receives and the oldest son
should receive 2/3 of what the second youngest son receives. How many gold coins would
the oldest son receive?

A. 270 B. 300 C. 200 D. 180 E. None of these.

27. Write an equation in slope-intercept form for the line that passes through the points (4,5)
and (20,25).

A. y = x + 1 B. y = x + 5 C. y = 45 x D. y = 4x E. None of these.

4
28. A right triangle has a base of 8 and a hypotenuse of 10. What is its area?

A. 40 B. 24 C. 48 D. 12 E. None of these.

29. Expand and simplify the expression: (a + b)3 + (a − b)3 .

A. 2a3 B. 2a3 − 2b3 C. 2a3 − 6a2 b + 6ab2 − 2b3

D. 2a3 − 6ab2 E. None of these.

30. What is the radius of the circle x2 + y 2 + 2x + 4y = 2?

√ √
A. 7 B. 5 C. 7 D. 2 E. 5

31. Which of the following points is on the line y = 5x + 4?

A. (24, 4) B. (4, 0) C. (14, 2) D. (2, −2/5) E. None of these.

32. The circle x2 + y 2 = 1 and the parabola y = x2 + k are tangent to each other at exactly
two points. What is the value of k?

A. 1 B. 4/5 C. -4/5 D. -5/4 E. None of these.

33. We define a two digit integer as “well-bahaved” if it is equal to four times the sum of the
two digits. What is the sum of all the two digit well-behaved integers?

A. 120 B. 84 C. 72 D. 60 E. None of these.

34. Notice that 2010 = 2 × 3 × 5 × 67 and 67 is a prime number, what is the number of ALL
positive integer factors of 2010?

A. 4 B. 8 C. 14 D. 16 E. None of these.

35. On a quiz with 4 multiple choice questions, each question has exactly five answers, where
one and only one of these are correct. A completely clueless student randomly chooses
one answer for each problem. What is the probability that this student will get at most
one of the correct answers?
 3  4  3  4
4 4 4 4 E. None of these.
A. B. C. 2 × D. 2 ×
5 5 5 5

5
36. If x + y + z = 0 and x + y + z 2 = 6, what is the largest possible value of x + y?

A. 3 B. 2 C. −2 D. −3 E. None of these.

37. If m and n are both positive integers and m > 1, such that m|(28n + 25) and m|(7n + 3).
What is m?

A. 9 B. 11 C. 13 D. 17 E. None of these.

38. The last digit of 132010 is

A. 1 B. 3 C. 7 D. 9 E. None of these.

√
39. In the regular hexagon ABCDEF , BF = 2 4 12, what is the area of the hexagon
ABCDEF ?
A F
A. 10

B. 12

B E C. 6
√
D. 6 12

C D E. None of these.

40. A bouncy ball is dropped to a flat surface from 100 meters above and each time it bounces
back up, it goes to half of its previous height, in other words, the first bounce sends it
back up to 50 meters above the surface and the next bounce sends it back up to 25
meters, etc. What is the total distance (in meters) traveled by the bouncy ball if it is
stopped exactly when it reaches the surface before the 10th bounce? (For the purpose of
this problem, ignore the size of the ball.)
6147 3068 1023 1023 E. None of these.
A. 100 × B. 100 × C. 200 × D. 400 ×
2048 1024 1024 1024

6
This page is left blank for your use.

7
 2010 Marywood Mathematics Contest
Level II

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 27, 2010

Directions:

1. This exam consists of 40 questions on 7 pages. Please check to make sure that you have
all the pages.

2. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

3. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

4. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

5. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
1. The largest whole number such that seven times the number is less than 100 is

A. 12 B. 13 C. 14 D. 15 E. 16

2. The degree of (x2 + 1)4 (x3 + 1)3 as a polynomial in x is

A. 5 B. 7 C. 12 D. 17 E. 72

2 5
3. (−1)5 + 12 =

A. -7 B. -2 C. 0 D. 1 E. 57
r
1 1
4. + =
9 16
1 1 2 5 7
A. B. C. D. E.
5 4 7 12 12
2
5. If the ratio of 2x − y to x + y is , what is the ratio of x to y?
3
1 4 C. 1 6 5
A. B. D. E.
5 5 5 4

6. In the following figure, CDE is an equilateral triangle and ABCD and DEF G are
squares. The measure of ∠GDA is
A B
E A. 90◦

B. 105◦
F
C. 120◦
D C
D. 135◦
G E. 150◦

√
7. A positive number x satisfies the inequality x < 2x if and only if
1 B. x > 2 C. x > 4 1 E. x < 4
A. x > D. x <
4 4

8. If a, b > 0 and the triangle in the first quadrant bounded by the coordinate axes and the
graph of ax + by = 6 has area 6, then ab =

A. 3 B. 6 C. 12 D. 108 E. 432

1
 9. A square is cut into three rectangles along two lines parallel to a side, as shown. If the
perimeter of each of the three rectangles is 24, then the area of the original square is

A. 24

B. 36

C. 64

D. 81

E. 96

10. If azb = ab + ba , find (2z3)z1.

A. 22 B. 48 C. 18 D. 126 E. 24

x−b
11. Let f (x) = for constants a and b. If f (2) = 0 and f (1) is undefined, what is
  x−a
1
f ?
2
A. 0 B. 1 C. 2 D. 3 E. 4

2
12. The length and width of rectangle AEF G are each of the corresponding parts of
3
rectangle ABCD. The area of ABCD is 72. The area of the shaded part is
A E B A. 24

B. 32

C. 36
G F
D. 40
D C E. 48

13. For any convex polygon, a diagonal is a line segment connecting two nonadjacent vertexes.
A certain convex polygon has 9 diagonals. How many sides does it have?

A. 3 B. 4 C. 5 D. 6 E. 7

14. Consider all integers from 1 to 2010 inclusive. What is the difference between the sum of
all the integers that are multiples of 3 and the sum of all the other integers in this range?

A. 2007 × 335 B. 2010 × 335 C. 2007 × 670 D. 2010 × 670 E. 0

2
15. The graph of the equation x2 + 2y 2 + 2x − 12y = −2 is

A. a line. B. an ellipse. C. a circle. D. a point. E. a hyperbola.

16. A right triangle has an area of 10 and a base of 4. What is the length of the hypotenuse?
√ √
A. 5 B. 41 C. 89/2 D. 3 E. None of these.

1
17. If sin(2α) = , what is sin4 (α) + cos4 (α)?
7
2 97 2305 4610 195
A. B. C. D. E.
98 98 2401 2041 196

18. In △ABC, AB = 5, BC = 7, AC = 9, and D is on AC with BD = 5. Find the ratio

AD : DC.
B A. 4 : 3

B. 7 : 5

C. 11 : 6

D. 13 : 5

A D C E. 19 : 8

19. Let f (x) = ax7 + bx3 + cx − 5, where a, b, and c are constants. If f (−7) = 7, then f (7) =

A. -17 B. -7 C. 14 D. 21 E. Not uniquely

determined

1 1 1
20. How many pairs (a, b) of non-zero real numbers satisfy the equation + =
a b a+b
A. 1 B. 2

C. one pair for each b 6= 0 D. two pairs for each b 6= 0

E. None of these.

21. In triangle ABC, AB = 9, BC = 10, and AC = 11. What is cos(∠ABC)?

2 1 2 3 1
A. B. C. D. E.
9 4 7 10 3

3
22. Solve for x: 2 log10 x = log10 54 − log10 6

A. 3 B. 9 C. 4 D. 24 E. 4.5

1
23. If f (x) = (x − 3) and g(x) = x3 , then f −1 (g −1 (1)) =
8
A. -823 B. 32 C. 12 D. 11 E. None of these.

24. Where do the lines y = 3x + 2 and y = −x + 6 intersect?

A. (0, 6) B. (−2, 8) C. (1, 5) D. (4, 2) E. None of these.

25. A square flag has a red cross of uniform width with a blue square in the center on a white
background as shown. (The cross is symmetric with respect to each of the diagonals of
the square.) If the entire cross (both the red arms and the blue center) takes up 36% of
the area of the flag, what percent of the area of the flag is blue?

A. .5
RED RED
B. 1
BLUE C. 2

RED RED D. 3

E. 6

26. A man walks x miles due west, turns √ 150◦ to his left and walks 3 miles in the new
direction. If he finishes at a point 3 from his starting points, then x is
√ √ 3
A. 3 B. 2 5 C. D. 3 E. Not uniquely
2 determined.

27. A lattice point is a point in the plane with integer coordinates. How many lattice points
are on the line segment whose endpoints are (3, 17) and (48, 281)? (Include both end-
points of the segment in your count.)

A. 2 B. 4 C. 6 D. 16 E. 46

28. For how many integers n between 1 and 100 does x2 + x − n factor into the product of
two linear factors with integer coefficients?

A. 0 B. 1 C. 2 D. 9 E. 10

4
29. Mr. and Mrs. Zhang just had their third baby girl. They want to name new baby Zhang
so that her monogram (first, middle, and last initials) will be in alphabetical order with
no letters repeated. How many such monograms are possible?

A. 276 B. 300 C. 552 D. 600 E. 15,600

30. Two strips of width 1 overlap at an angle of α as shown. The area of the overlap (shaded
region) is

A. sin α
1
1 B.
sin α
1
C.
1 − cos α
1 1
D.
α sin2 α
1
E.
(1 − cos α)2

31. The least whole number greater than 1 that is both a square and a cube is 64. What is
the least whole number greater than 1 that is a square, cube, and a fourth power?

A. 4096 B. 256 C. 243 D. 19683 E. None of these.

32. The equations L1 and L2 are y = mx and y = nx, respectively. Suppose L1 makes twice
as large of an angle with the horizontal (measured counterclockwise from the positive
x-axis) as does L2 , and that L1 has 4 times the slope of L2 . If L1 is not horizontal, then
mn is
√ √
2 2 C. 2 D. -2 E. Not uniquely
A. B. −
2 2 determined.

33. A particle move through the first quadrant as follows. During the first minute it moves
from the origin to (1, 0). Thereafter, it continues to follow the directions indicated in
the figure, going back and forth between the positive x and y axes, moving one unit of
distance parallel to an axis in each minute. At which point will the particle be after
exactly 2010 minutes? Note: This problem was discarded due to a mistake.

4 A. (35, 14)
B. (36, 24)
3
C. (37, 24)
2
D. (14, 35)
1 E. (24, 36)

O 1 2 3 5
34. A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic or wood),
3 sizes (small, medium, or large), 4 colors (blue, green, red, yellow), and 4 shapes (circle,
hexagon, square, triangle). If not two blocks are the same, how many blocks in the set
are different from the “plastic, medium, red circle” in exactly two ways? (For example
the “wood, medium, red square” is such a block.)

A. 29 B. 39 C. 48 D. 56 E. 62

35. In a certain cross-country meet between two teams of five runners each, a runner who
finishes in the nth position contributes n points to her team’s score. The team with the
lowest score wins. If there are no ties among the runners, how many different winning
scores are possible?

A. 10 B. 13 C. 26 D. 120 E. 126

36. The perimeter of an equilateral triangle exceeds the perimeter of a square by 2010 cm.
The length of each side of the triangle exceeds the length of each side of the square by
d cm. The square has perimeter greater than 0. How many positive integers are not
possible values for d?

A. 0 B. 16 C. 228 D. 670 E. infinitely many.

37. Find the sum of the roots of tan2 x − 9 tan x + 1 = 0 that are between x = 0 and x = 2π
radians.
π 3π
A. B. π C. D. 3π E. 4π
2 2
xy
38. If the operation ⊕ is defined for all positive x and y by x ⊕ y = ,which of the
x+y
following must be true for positive x, y, and z?

i. x ⊕ x = x/2
ii. x ⊕ y = y ⊕ x
iii. x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z

A. i. only B. i. and ii. only C. i. and iii. only

D. ii. and iii. only E. all three

6
39. The traffic on a certain east-west highway moves at a constant speed of 60 miles per
hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-
minute interval. Assume vehicles in the westbound lanes are equally spaced. Which of
the following is the closest to the number of westbound vehicles present in a 100-mile
section of highway?

A. 100 B. 120 C. 200 D. 240 E. 400

40. Suppose that 3 boys and 5 girls line up in a row. Let S be the number of places in the
row where a boy and a girl are standing next to each other. For example, for the row
BGGBGBGG we have S = 5. The average value of S (if all possible orders of these 8
people are considered) is closest to

A. 2 B. 3 C. 4 D. 5 E. 6

The rest of this page is left blank for your use.

7
This page is left blank for your use.

8
 2011 Marywood Mathematics Contest
Level I

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 19, 2011

Directions:

1. This exam consists of 40 questions on 6 pages. Please check to make sure that you have
all the pages.

2. No calculator or any other electronic device is allowed on this exam.

3. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

4. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

5. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

6. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
 −23 ∗ 10
 
· 37 − 20116 + 72 .


1. Evaluate 16 +
5
A. 0 B. 1 C. 2 D. 3 E. None of these.

2. Solve the equation 7x + 3(4x − 5) = 0 for x.

1 5 15 5 E. None of these.
A. B. C. D.
5 19 19 11

3. In which list are the three numbers arranged from smallest to largest?
7 12 7 12 12 7 7 12 E. None of these.
A. 2.32, , B. , , 2.32 C. , 2.32, D. , 2.32,
3 5 3 5 5 3 3 5

4. Which is the SMALLEST of these five numbers?

2 1 2 1 2 1 2 1 1 2
A. + B. − C. × D. ÷ E. ÷
3 6 3 6 3 6 3 6 6 3

5 7
5. Find a number which is one third of the way from to on the number line.
6 2
13 7 13 31 E. None of these.
A. B. C. D.
9 3 3 18

15
6. If x and y are real numbers then 32x16 y 15 =

A. 2x16/5 y 3 B. 4x16/5 y 3 C. 8x16/5 y 3 D. 16x16/5 y 3 E. None of these.

7. Which linear equation has x- and y-intercepts at (5, 0) and (0, −12)?

A. 5x − 12y + 144 = 0 B. 12x − 5y + 144 = 0 C. 5x + 12y + 144 = 0

D. 12x + 5y + 144 = 0 E. None of these.

8. How many real solutions does x2 − 5x + 4 = 0 have?

A. 0 B. 1 C. 2 D. 3 E. None of these.

9. The roots of x(x2 + 8x + 16)(4 − x) are

A. 0 B. 0, 4 C. 0, -4 D. 0, 4, -4 E. None of these.

1
10. The sum of all positive integer factors of 48 is

A. 75 B. 105 C. 108 D. 112 E. None of these.

11. A 27-year old mother has a 5-year old daughter. In how many years will the mother be
three times as old as her daughter?

A. 3 B. 4 C. 5 D. 6 E. None of these.

a·b+b
12. Suppose that the operation f is defined as afb = . What is the value of 3f(2f6)?
2
A. 12 B. 14 C. 15 D. 18 E. None of these.

2
13. What is the value of 23 ?

A. 512 B. 256 C. 128 D. 64 E. None of these.

14. A woodchuck could chuck as much wood as a woodchuck could chuck, if a woodchuck
could chuck wood. Assume that a woodchuck could chuck 3 cords of wood in 90 minutes.
How many cords of wood could 4 woodchucks chuck in 2 hours?

A. 12 B. 14 C. 16 D. 18 E. None of these.

15. Gillian has a collection of 50 songs that are each 3 minutes in length and 50 songs that
are each 5 minutes in length. What is the maximum number of songs from her collection
that she can play in 5 hours?

A. 70 B. 80 C. 90 D. 100 E. None of these.

16. Barry’s daily grades for one grading period are shown below:

94, 88, 87, 92, 78, 88, 93, 100, 92, 92, 90, 92, 85.

What is the mode of his grades?

A. 93 B. 92 C. 91 D. 90 E. None of these.

2
17. The minimum value of the parabola y = x2 − 6x + 8 occurs at x =

A. 2 B. 3 C. 4 D. 5 E. None of these.

18. If x − y = 3 and x2 − y 2 = 9, then xy =

A. 0 B. 1 C. 2 D. 3 E. None of these.

19. The prime factorization of 2000 can be written 2000 = 2x · 5y . The sum x + y equals

A. 6 B. 7 C. 8 D. 9 E. None of these.

20. If x + y = −23 and x − y = 7 then 2x − y equals

A. -31 B. -1 C. 1 D. 8 E. None of these.

21. To build a set of book shelves, a carpenter needs 30 identical boards, each 2 feet 7 inches
long. For this type of wood, the lumber yard only sells boards in 12-foot lengths.
How many 12-foot boards must the carpenter purchase?

A. 6 B. 7 C. 8 D. 9 E. None of these.

22. A math teacher buys three sport coats and six ties. He teaches Monday through Friday.
Starting on a Monday, he wears a different combination of new sport coat and new tie each
day that he teaches. On which day of the week does he wear his last new combination?

A. Monday B. Tuesday C. Wednesday D. Thursday E. None of these.

23. A Leap Year has 366 days and the year 2012 will be a Leap Year. Randy’s 28th birthday
on April 1, 2011 will be on a Friday. Randy’s 29th birthday will be on a

A. Sunday B. Monday C. Tuesday D. Wednesday E. None of these.

24. Two standard six-sided dice are rolled. The probability that the sum of the two dice is 9
or larger is
5 1 1 1 E. None of these.
A. B. C. D.
18 6 12 36

3
25. A square piece of paper is folded in half twice: from top to bottom, then from top to
bottom again. If the perimeter of the final rectangle is 10 cm, what was the perimeter of
the original square?

A. 14 B. 15 C. 16 D. 20 E. None of these.

26. Randomly select two different numbers from this set: {−5, −2, 4, 8}. The probability
that the product of the two selected number is positive is
1 1 1 2 E. None of these.
A. B. C. D.
6 3 2 3

27. What is the last digit of 20112011 ?

A. 0 B. 1 C. 2 D. 3 E. None of these.

28. If i2 = −1, then i2011 =

A. 1 B. -1 C. i D. −i E. None of these.

29. If each edge of a cube is increased by 30%, by what percentage does the surface area of
the cube increase?

A. 10% B. 20% C. 30% D. 40% E. None of these.

30. A farm consists of a right triangle and the three squares on the sides of the right triangle.
The length of the three sides of the right triangle are a, b, and c. The farmer decides to
keep the triangular piece of land for himself. He shares the rest of the farm equally, in
terms of area, between his two children.
What area of land does each child receive?
 a 2 2  
b c 2
A. + +
2 2 2
 2
a+b+c
B.
c a
2
(a + b + c)2
b C.
2
2
D. c
E. None of these.

4
31. In triangle ABC, AC = 51 and BC = 50. Point D on AB divides it into segments of
length AD = 1 and DB = 3. Which value best approximates the length ofDC?

A. 50 B. 50.25 C. 50.5 D. 50.75 E. 60

32. In a 6-horse race with no ties, Family finished fifth, 14 meters behind Doggie and 20
meters behind Boya. Alpha finished 8 meters ahead of Eppa and 10 meters behind
Captain. Which horse finished fourth?

A. Alpha B. Boya C. Captain D. Doggie E. None of these.

33. A cube measuring 1 cm on a side is sliced in half by a plane through vertices A, B, C,

and D. Two congruent prisms are formed. The surface area, in square centimeters, of
each prism is

A A.
1
2
B B. 3
C. 4
D D. 5
C E. None of these.

34. The four vertices of a square in a rectangular coordinate system are (9, 6), (5, 16), (a, b),
and (c, d). The vertices (9, 6) and (5, 16) lie on one diagonal of the square. Then the sum
a + b + c + d equals

A. 15 B. 20 C. 25 D. 30 E. None of these.

35. In a three-digit whole number N, the hundreds digit is the same as the units digit. When
N is divided by the sum of its digits, the quotient is 28. The tens digit of N is

A. 2 B. 3 C. 4 D. 5 E. None of these.

36. The natural numbers are written in sequence, without spaces:

1234567891011121314151617....

and so on. What is the 100th digit in this sequence?

A. 1 B. 3 C. 4 D. 6 E. None of these.

5
37. You have one sheet of very thin paper. One stack of 200 sheets is only one centimeter
tall. If you could fold the paper in half 50 times, how thick, in kilometers, would the
folded paper be? (Select the closest answer.)

A. 1 B. 5 C. 500 D. 50,000 E. 50,000,000

38. The first four figures below consist of 1, 5, 13, and 25 squares respectively. If the pattern
in the figures continues, how many squares will there be in figure 100?

A. 39801 B. 20201 C. 19801 D. 10401 E. None of these.

39. In rectangle ABCD, AD = 1, P is on AB, and the line segments DB and DP trisect
∠ADC. What is the perimeter of 4BDP ?
√ √
A P B A. 2 + 2 2 5 3
B. 2 +
3
√ √
4 3 3
C. 2 + D. 3 +
3 3
D C E. None of these.

40. In the figure shown, if the area of the letter “L” equals the area of the triangle, what is
the length x of the ends of the L?
√ √
x A.
6−2 6
B.
6−3 6
3 3
√ √
2 C.
6+2 6
D.
6+3 6
3 3
x
2 E. None of these.

6
This page is left blank for your use.

7
 2011 Marywood Mathematics Contest
Level II

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 19, 2011

Directions:

1. This exam consists of 40 questions on 6 pages. Please check to make sure that you have
all the pages.

2. No calculator or any other electronic device is allowed on this exam.

3. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

4. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

5. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

6. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
1. (−2)−2 + (−3)−3 =
23 23 31 31 E. None of these.
A. − B. C. − D.
108 108 108 108

2. For two consecutive days, Thomas ate 20% of the jellybeans that were in his jar at the
beginning of that day. There were 32 remaining at the end of the second day. How many
jellybeans were in the jar originally?

A. 40 B. 50 C. 55 D. 60 E. 75

3. Given that log10 2011 ≈ 3.303, how many digits does 201110 have (in base 10)?

A. 30 B. 33 C. 34 D. 40 E. None of these.

4. If a right triangle has one angle equal to 30◦ and the length of its hypotenuse is 8, what
is the length of the altitude that is perpendicular to the hypotenuse?

A. 2
8 B. 3
√
C. 2 3
√
D. 3 3
30◦ E. None of these.

5. Solve the equation log10 x + log10 (x − 2) = log10 3, to get x =

A. 3. B. 1. C. −1. D. 3 or −1. E. 3 or 1.

√
q p
6. 3 x 3 x 3 x =

A. x13/27 . B. x1/27 . C. x1/9 . D. x13/9 . E. None of these.

7. If cos θ = 0.6 and 0 < θ < π/2, what is tan θ?

A. 0.8 B. 0.75 C. 2/3 D. 4/3 E. None of these.

8. Find the sum of all integers from 1 to 2011: 1 + 2 + 3 + · · · + 2010 + 2011 =

A. 2011 × 1006 B. 2011 × 1005 C. 2012 × 1006 D. 2012 × 1005 E. None of these.

1
 9. If the two diagonals of a rectangle make an angle of 40◦ , what is the ratio between two
adjacent sides of the rectangle?

A. sin 20◦ : sin 70◦ B. sin 40◦ : sin 140◦ C. cos 20◦ : cos 70◦

D. cos 40◦ : cos 140◦ E. None of these.

10. The equation x2 − 2011x + 2010 = 0 has

A. no real solution. B. 1 real solution. C. 2 real solutions.

D. 3 real solutions. E. None of these.

11. Given two functions f (x) and g(x), the composite function (f ◦ g)(x) is defined as
1
(f ◦ g)(x) = f (g(x)). If f (x) = x2 + 5 and g(x) = , what is (f ◦ g)(x)?
x+1
1 x2 + 5 1
A. 2 B. C. 2
(x + 1) + 5 x+1 (x + 5) + 1
1 x+1
D. +5 E.
(x + 1)2 x2 + 5

12. There are 5 white beans, 9 black beans, and 7 red beans in a bag. If one randomly takes
out a certain number of beans without looking, what is the minimum number of beans
needed to guarantee that all three colors are represented among the beans taken out?

A. 13 B. 15 C. 16 D. 17 E. 18

13. If a fair coin is tossed 3 times, what is the probability that one will observe at least two
heads consecutively OR at least two tails consecutively?

A. 3/8 B. 1/2 C. 5/8 D. 3/4 E. None of these.

14. In the figure below, the smaller circle passes through the center of the larger circle and
its diameter is equal to the radius of the larger circle. The center of the smaller circle
C is on the diameter AB of the larger circle. AE is tangent to the smaller circle at D.
What is cos ∠BAE?
√
A. 2 2/3
D E B. 1/3
C. 8/9
B √
A O C D. 1/(2 2)
E. None of these.

2
 1
15. If cos x = , and 0 < x < π/2, what is sin(2x)?
4
√ √ √ √ √
15 15 15 15 15
A. B. C. − D. − E.
16 8 16 8 4

16. If a circle centered in the first quadrant passes through the points (0, 1) and (0, 5) on the
y-axis, and it is tangent to the x-axis, where is the center?

A. (3, 3)
√
B. (3, 5)
(0, 5) √
C. ( 5, 3)
√
D. ( 8, 3)
E. None of these.
(0, 1)

17. The area enclosed by the graph of y = 3|x| − 5 and the x-axis is
25 25 25 25 25
A. B. C. D. E.
6 9 18 12 3

18. Find the intersection point (x0 , y0 ) of the lines y = −2x + 1 and x + 3y = −7, then
determine the value of x0 + y0 .

A. −5 B. −1 C. 5 D. 1 E. None of these.

19. A circle of radius 1 is centered at (3, 4). What is the distance from the origin (0, 0) to
the point on the circle closest to the origin?

A. 3 B. 4 C. 5 D. 6 E. 7

20. For what value of k > 0 will the circle x2 + y 2 = 3k and the line y = x + k be tangent to
each other?

A. 8 B. 6 C. 4 D. 2 E. None of these.

3
21. If a circle and a regular hexagon (six sided shape with equal side length) have equal area,
and the circle radius is 3, what is the side length of the hexagon?
√ √ √ √
27π √
r
A. 2 3π 27 3 C. 2π · 4 3 E. None of these.
B. π D. · 43
2 2

22. Find the remainder of x2011 − x2010 + 1 divided by x − 2.

A. 22010 B. 22010 + 1 C. 22011 D. 22011 + 1 E. None of these.

23. There are two math teachers and 3 physics teachers in a meeting sitting at a round table.
If the two math teachers cannot sit next to each other, how many distinct ways can the
seats be arranged, NOT counting rotations of the same seating.

A. 12 B. 24 C. 96 D. 120 E. None of these.

24. If a parabola representing the quadratic equation y = x2 + ax + b has y-intercept equal

to 12, and one of the x-intercepts is 3, what is the value of a + b?

A. 5 B. −7 C. 4 D. 7 E. None of these.

25. Find the radius of the circle inscribed in the triangle with sides 5, 12, and 13.

A. 1 B. 1.5 C. 2 D. 2.5 E. 3

26. Suppose that a function f (x) satisfies 3f (x) + 2f (1 − x) = 2x + 9 for every real number
x. What is the value of f (1)?

A. 0 B. 1 C. 2 D. 3 E. 4

27. Along the edges of a 3 × 3 square, there are 12 lattice points arranged as in the figure
below. How many triangles can be formed with these lattice points?

A. 48

B. 64

C. 204

D. 220

E. 256

4
28. Let r and s be the two roots of the equation x2 + 5x + 2 = 0. Find r3 + s3 .

A. −95 B. 95 C. 115 D. −115 E. None of these.

√ 1 1
29. Given 3 = a + √ where a > 1, what is a − ?
a a
√ √
A. 5 B. 6 C. 3 5 D. 7 E. 5 2

√
30. Let i be −1, what is i + i2 + i3 + i4 + · · · + i2010 + i2011 ?

A. 1 + i B. 1 − i C. −1 + i D. −1 − i E. None of these.

31. Consider the periodic continued fraction

1
x= 1 .
1+ 2+ 1
1+ 1
1
2+ 1+···

The value of x is equal to

√ √ √ √
A. −1 + 3. B. −1 − 3. C. 1 + 3. D. 1 − 3. E. None of these.

32. The stairs leading up to the main entrance of the Marywood University Liberal Arts
Center (the building you are in right now) have 9 steps. If one is only allowed to go up
1 step or 2 steps at a time, how many different ways are there to go up the stairs?

A. 34 B. 54 C. 89 D. 35 E. 55

√
1 3 √
33. Let z = + i, where i = −1. Find z 2011 .
2 2
√ √ √ √
1 3 1 3 1 3 1 3 E. −1
A. + i B. − + i C. − − i D. − i
2 2 2 2 2 2 2 2

34. Let x be the repeating decimal number 3.19 = 3.191919 · · · , which can also be represented
a
as a reduced fraction where a and b are relatively prime positive integers. What is a+b?
b
A. 418. B. 415. C. 319. D. 316. E. None of these.

5
35. From a group of two math teachers and three physics teachers, a committee of three is
to be formed. If the selection process is completely random, what is the probability that
exactly 2 physics teachers and 1 math teacher are selected to be on the committee?

A. 0.3 B. 0.4 C. 0.5 D. 0.6 E. 0.7

36. Let u, v, w, x, y, and z be the degree measures of the six angles of a hexagon. Suppose
that u < v < w < x < y < z and u, v, w, x, y, and z form an arithmetic sequence. Find
the value of w + x.

A. 60 B. 72 C. 90 D. 108 E. 240

x
37. How many pairs of positive integers (x, y) with y < x ≤ 100 are there such that both
y
x+1
and are integers?
y+1
A. 86 B. 85 C. 84 D. 83 E. 82

38. If x, y, and z satisfy

x y z
= = = 3.
y−6 z−8 x − 10
What is the value of x + y + z?

A. 24 B. 30 C. 32 D. 36 E. 40

39. How many real numbers x satisfy the following inequality?

|x4 − 4x2 + 3| ≥ |x4 − 4x2 + 5|

A. Infinitely B. 0 C. 1 D. 2 E. 4
many.

40. How many three-digit integers (numbers between 100 and 999) are there such that the
three digits are in strictly increasing order from left to right?
Example: 137 has the digits in strictly increasing order, but 215 or 115 does not have
this property.

A. 28 B. 85 C. 56 D. 84 E. None of these.

6
This page is left blank for your use.

7
 2012 Marywood Mathematics Contest
Level I

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 24, 2012

Directions:

1. This exam consists of 40 questions on 5 pages. Please check to make sure that you have
all the pages.

2. No calculator or any other computing device is allowed on this exam.

3. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

4. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

5. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

6. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
 1. Write fifteen-and-a-half billion in scientific notation a × 10n . What is the value of n?

A. 8 B. 9 C. 10 D. 11 E. 12

2. The positive difference between 1/2 and its reciprocal is

A. 3/2 B. 1/2 C. 5/2 D. 1 E. None of these.

3. 12 + 34 =

A. 14 + 32 B. 13 + 42 C. 12 + 43 D. 13 + 24 E. None of these.

4. The product of 2012 distinct integers is an even number, at most how many of those
2012 integers can be odd?

A. 0 B. 1 C. 2011 D. 2012 E. None of these.

5. (a + b + c)2 =

A. a2 + b2 + c2 B. a2 + b2 + c2 + 2ab + 2ac C. a2 + b2 + c2 + 2ab + 2bc

D. a2 + b2 + c2 + 2ab + 2bc + 2ac E. None of these.

22
6. 22 =

A. 16 B. 64 C. 128 D. 256 E. None of these.

7. The area of a circle with diameter 6 is

A. 6π B. 12π C. 18π D. 36π E. None of these.

8. 11 × 22 × 33 × 44 × 55 × 66 × 77 × 88 × 99 = 9!×

A. 119 B. 11 × 9 C. 1110 D. 11 × 10 E. None of these.

9. If 3x + 24 = 7x + 2012, then x =

A. 497 B. −497 C. 506 D. −506 E. None of these.

10. In 4ABC, ∠A measures 40◦ , and ∠B measures 6 times as ∠C. What is m∠B − m∠C?

A. 20◦ B. 100◦ C. 120◦ D. 140◦ E. None of these.

1
 √ √ √
11. If 36 + x = 100, then x =

A. 2 B. 4 C. 8 D. 64 E. None of these.

12. log3 81 =

A. 4 B. 27 C. 9 D. 2 E. None of these.

13. The slope of the line through the points (3, 24) and (20, 12) is
12 17 12 17 E. None of these.
A. B. C. − D. −
17 12 17 12
p√
14. 16 × 16 × 16 × 16 =

A. 2 B. 4 C. 8 D. 16 E. None of these.

15. 57x = 253x+100 , then x =

A. 25 B. 50 C. 100 D. 200 E. None of these.

16. If two different students are randomly selected from a class of 12 boys and 13 girls, what
is the probability that both students are boys?

A. 22% B. 23% C. 24% D. 26% E. None of these.

17. How many odd numbers between 1 and 1,000,000 (inclusive) are perfect cubes?

A. 49 B. 50 C. 51 D. 99 E. 100

18. If a rectangle has a perimeter of 60 inches, and its length is 5 times the width, then the
area of the rectangle is

A. 125 in2 B. 225 in2 C. 250 in2 D. 500 in2 E. None of these.

19. The radius of the circle x2 + y 2 + 8x = 0 is

A. 2 B. 4 C. 8 D. 16 E. None of these.

2
20. The two lines 3x − y = 11 and x + 2y = −1 intersect at the point (x0 , y0 ). What is the
product of x0 and y0 ?

A. 6 B. -6 C. 1 D. -1 E. None of these.

21. 22012 + 22012 + 32012 + 32012 + 32012 =

A. 42012 + 92012 B. 24024 + 36036 C. 42013 + 92013 D. 22013 + 32013 E. None of these.

a 1
22. The fraction is equal to , and a + b = 27. What is the value of a × b?
b 8
A. 24 B. 8 C. 324 D. 72 E. None of these.

23. Which of the following is NOT the sum of three consecutive integers?

A. 444 B. 12 C. 48 D. 76 E. None of these.

24. If a building has 6 entrances, in how many different ways can a person enter through one
entrance and exit through a different entrance?

A. 36 B. 30 C. 18 D. 15 E. None of these.

25. In the figure below, AB = 5, CD = 9, and the height of the trapezoid AF = BE = 4.

What is the area of the shaded region in the trapezoid?
A B
A. 8
B. 18
C. 36
D. 16
D F E C
E. None of these.

26. Alina took 10 quizzes and had an average of 94% for all 10. If her average on the first 6
quizzes was 92%, what was her average on the last 4 quizzes?

A. 95% B. 96% C. 97% D. 98% E. None of these.

27. What is the degree measure of the angle formed by the hour hand and the minute hand
of a clock at 9:30?

A. 60◦ B. 90◦ C. 105◦ D. 75◦ E. None of these.

3
28. If a fair coin is tossed four times, what is the probability that no two consecutive tosses
have the same results?
1 1 1 1 E. None of these.
A. B. C. D.
16 8 4 2

29. Notice that 324 = 22 × 34 . How many positive divisors does 324 have?

A. 8 B. 10 C. 12 D. 15 E. None of these.

30. The sum of the two roots of the equation x2 − 3x − 28 = 0 is

A. 3 B. −3 C. 11 D. −11 E. None of these.

31. A circle and a square have the same area, what is the ratio between the circumference of
the circle and the perimeter of the square?
√ √ √
A. 1 : 1 B. π : 2 C. π : 2 D. 1 : π E. 2 : π

32. How many rectangles are there in the 2 × 4 grid below?

A. 12
B. 17
C. 28
D. 30
E. None of these.

33. Two standard six-sided dice are rolled. What is the probability that the two numbers
rolled will differ by exactly 2?
4 8 2 1 E. None of these.
A. B. C. D.
11 11 9 9

34. 10 ÷ 0.2 =

A. 5 B. 20 C. 45 D. 50 E. None of these.

35. What is the last digit of the number 72012 ?

A. 1 B. 3 C. 7 D. 9 E. None of these.

4
36. During March Madness, the NCAA men’s basketball tournament features 64 teams in a
single elimination bracket. After the first round, the winner of each game moves on to
the second round, where 32 teams compete, and the winner of each second round game
moves on to the third round (sweet 16), and so on. Two friends are trying to predict
the outcomes of all the games in the tournament. For each first round game predicted
correctly, one gets 1 point. For each second round game predicted correctly, one gets
2 points. For each third round game predicted correctly, one gets 4 points, and so on
(with the point value per correctly predicted game doubled at the successive rounds). If
a person magically predicts ALL of the outcomes correctly, how many points would this
person get?

A. 80 B. 160 C. 192 D. 224 E. None of these.

37. In the figure below, how many paths are there from A to Z, if the paths must trace the
line segments downward and no retracing is allowed?
A
A. 20

B. 35

C. 55

D. 70

Z E. None of these.

38. A circular cone has the same height as a circular cylinder, but the base circumference
of the cone is twice of the base circumference of the cylinder. What is the ratio of the
volume of the cone to the volume of the cylinder?

A. 2 : 1 B. 2 : 3 C. 1 : 2 D. 3 : 2 E. None of these.

39. A positive integer will be called an awesome-five number if it does not contain the digit
zero and all the digits sum up to 5. For example, 1121 is an awesome-five number, but
2012 and 324 are not. How many awesome-five numbers are there?

A. 14 B. 15 C. 16 D. 17 E. None of these.

40. What is the sum of the remainders when 100, 101, 102, . . ., 998, 999, and 1000 are each
divided by 9?

A. 3600 B. 3601 C. 4500 D. 4501 E. None of these.

5
This page is left blank for your use.

6
 2012 Marywood Mathematics Contest
Level II

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 24, 2012

Directions:

1. This exam consists of 40 questions on 6 pages. Please check to make sure that you have
all the pages.

2. No calculator or any other electronic device is allowed on this exam.

3. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

4. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

5. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

6. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
1. (−3)−2 + (−2)−3 =
1 1 17 17 E. None of these.
A. B. − C. D. −
72 72 72 72

2. On day one of a long journey, Corey drove 20% of the way to his destination, got tired,
pulled over, and slept. The second day he drove 30% of the remaining way to his desti-
nation, got tired, pulled over, and slept. If there were 56 miles left after the second day,
how many miles did Corey drive the first day?

A. 10 B. 15 C. 20 D. 25 E. None of these.

3. Solve the equation log10 (x + 2) + log10 (x − 3) = log10 6, to get x =

A. −3. B. 3. C. −4. D. 4. E. None of these.

4. Find the sum of all integers from 1 to 2012: 1 + 2 + 3 + · · · + 2010 + 2011 + 2012 =

A. 2013 × 1006 B. 2013 × 1005 C. 2012 × 1006 D. 2012 × 1005 E. None of these.

5. If sin θ = −0.8 and −π/2 < θ < 0, what is tan θ?

A. 3/4 B. 4/3 C. −3/4 D. −4/3 E. None of these.

6. A sleepy student wakes up in the morning and needs some socks. She reaches into the
drawer to grab a pair, but since the room is dark, she cannot see their colors. She knows
that she has 2 pairs of green socks, 3 pairs of purple socks, 4 pairs of aquamarine socks,
and 5 pairs of pink socks in the drawer. What is the minimum number of socks she needs
to take out of the drawer in order to guarantee that she has at least two socks of the
same color?

A. 2 B. 3 C. 4 D. 5 E. None of these.

7. Find the radius of a circle circumscribed about a triangle with sides 3, 4, and 5.

A. 1 B. 1.5 C. 2 D. 2.5 E. None of these.

8. How many nonreal complex solutions does the equation x2 + 2012x + 2013 = 0 have?

A. 0 B. 1 C. 2 D. 3 E. None of these.

1
 1
9. If sin x = , and 0 < x < π/2, what is cos(2x)?
3
√ √
1 8 8 7 E. None of these.
A. B. − C. D.
2 3 3 9

a·b−a
10. The binary function > is defined by a > b = . What is (2 > 3) > 2?
a·b−b
A. -3/2 B. -2/3 C. 2 D. 1/2 E. None of these.

11. A palindrome is a number which is same whether it is written forwards or backwards.

For example, the number 13931 is a palindrome. How many positive integers less than
1000 are palindromes?

A. 117 B. 108 C. 99 D. 90 E. None of these.

12. The sum of the third and fourth terms in a sequence of consecutive integers is 53. The
sum of the first five terms of the sequence is

A. 130 B. 102 C. 99 D. 81 E. None of these.

13. The function f (x) = |4 cos(2012x − π) − 2| has a maximum value of

A. 2 B. 3 C. 4 D. 5 E. None of these.

14. Which of the following numbers is least?

A. 6100 B. 5200 C. 4300 D. 3400 E. 2500

15. Jillian drives to school at an average speed of 75 miles per hour. At what average speed
would she have to travel on the return trip in order to average 60 miles per hour for the
round trip? (So her parents don’t bust her for speeding.)

A. 45 mph B. 50 mph C. 55 mph D. 60 mph E. None of these.

16. Find the intersection point (x0 , y0 ) of the lines y = −3x + 1 and 2x + 4y = −1. The value
of x0 + y0 is

A. −2 B. −1 C. 2 D. 1 E. None of these.

2
17. A cube measuring 100 units on each side is painted only on the outside and cut into unit
cubes. The number of cubes with paint only on two sides is

A. 1000 B. 1125 C. 1176 D. 980 E. None of these.

√
q p
18. x 3 x 4 x =

A. x13/24 . B. x15/24 . C. x16/24 . D. x17/24 . E. None of these.

19. Rogelio can mow a yard in 30 minutes. It takes Erin only 20 minutes to mow the same
yard. How many minutes would it take them to mow if they worked together using the
two mowers?

A. 12 B. 50 C. 13 D. 14 E. None of these.

20. In the configuration below consisting of 25 one-by-one squares, how many total squares
with horizontal and vertical sides can be formed using the points as vertices?

A. 53

B. 54

C. 55

D. 56

E. None of these.

21. In an urn, 5 red marbles and 9 blue marbles are mixed together. If two marbles are drawn
at random, what is the probability that one is red and one is blue?

A. 1/2 B. 45/91 C. 1/14 D. 4/45 E. None of these.

22. An 8 foot tall sasquatch is standing 20 feet away from a light pole which is 24 feet tall.
How long is the beast’s shadow?

A. 8 B. 11 C. 14 D. 17 E. None of these.

23. A lattice point in the plane is a point both of whose coordinates are integers. How many
lattice points (including the endpoints) are there on the line segment joining the points
(2, 0) and (16, 203)?

A. 15 B. 8 C. 9 D. 14 E. None of these.

3
24. Consider the continued fraction
1
x= 1 .
1+ 1
1+ 1+···

The value of x is equal to

√ √ √ √
1+ 5 1− 5 −1+ 5 −1− 5
A. 2
B. 2
C. 2
D. 2
E. None of these.

25. The four angles of a quadrilateral inscribed in a circle are α, β, γ, and δ as shown. Which
of the following is necessarily true?

A. α + β + γ + δ = 180◦
α
B. α + β = 180◦
C. β + δ = 90◦
D. α + γ = β + δ
β
E. None of these.
δ
γ

26. The largest constant C such that sin x ≥ Cx for all x in [0, π/2] is

A. 0 B. 1/2 C. 2/π D. π/2 E. None of these.

27. Circle C is tangent to line l. Two circles C1 and C2 of equal radii are each tangent to one
another, to C, and to l. If the radius of C is 2, then the radius of C1 is

A. 6
B. 8
C1 C2
C. 10
C D. 12
l E. None of these.

28. The unit digit (base 10) of 22012 is

A. 1 B. 2 C. 4 D. 8 E. None of these.

4
29. For a certain integer n, 5n + 16 and 8n + 29 have a common factor larger than one. That
common factor is

A. 19 B. 17 C. 13 D. 11 E. None of these.

30. For what values of a does the system of equations

x2 = y 2
(x − a)2 + y 2 = 1

have exactly 3 solutions?

A. a ≥ 0 B. −1 ≤ a ≤ 1 C. a = −1, 0, 1 D. a = −1, 1 E. None of these.

31. Given a square whose sides have length 2a, find the area of the region bounded by the 4
semi-circles which are in the interior of the square and have the four sides of the square
as diameters.

A. (2π − 4)a2

πa2
B.
8
(8 − π)a2
C.
4
D. 2a2

E. None of these.

32. The sum of two integers is S. Two digits in one of the integers are interchanged and a
new sum T is produced. The difference S − T is necessarily divisible by

A. 9 B. 7 C. 5 D. 10 E. None of these.

1 1 1
33. Let X = 2012 + , Y = 2012 + , and Z = 2012 +
2012 1 1
2012 + 2012 +
2012 1
2012 +
2012
The numbers X, Y , and Z arranged in increasing order are

A. X, Y , Z B. Z, X, Y C. Y , Z, X D. Z, Y , X E. None of these.

5
34. Let a and b be the two roots of the equation x2 − x − 1 = 0. What is a2 + b2 ?

A. 2 B. 4 C. 6 D. 8 E. None of these.

35. If a fair four sided die is tossed 3 times (with possible outcomes of 1, 2, 3, and 4), what
is the probability that one will observe at least two 1’s consecutively OR at least two 3’s
consecutively?

A. 1/16 B. 1/8 C. 3/16 D. 1/4 E. None of these.

36. A certain function f satisfies f (x) = 2f (6 − x) − x for all real numbers x. The value of
f (1) is

A. -9 B. 1 C. 2 D. 3 E. None of these.

37. There are 120 five-digit numbers that can be formed by permuting 1, 2, 3, 4, and 5, such
as
12345, 12354, 21435, . . . , 54321.
The sum of all these numbers is

A. 2,876,540 B. 3,999,960 C. 4,969,960 D. 5,600,610 E. 6,975,640

38. In quadrilateral ABCD, AB = 2, BC = CD = 4, DA = 5, and the opposite angles A

and C are congruent. The length of the diagonal BD is
2√
A A. 30
D 3
B. 5
B √
C. 2 6
√
D. 6 2

C E. None of these.

sin 10◦ + sin 40◦

39. If tan x = , and x is between 0◦ and 90◦ , then x is
cos 10◦ + cos 40◦
A. 20◦ B. 30◦ C. 22.5◦ D. 25◦ E. 50◦

40. When x100 − 2x99 + 4 is divided by x2 − 3x + 2, the remainder is

A. x + 1 B. 2x + 1 C. x − 1 D. x + 2 E. 3x − 2

6
This page is left mostly blank for your use.

7
 2013 Marywood Mathematics Contest
Level I

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 16, 2013

Directions:

1. This exam consists of 40 questions on 6 pages. Please check to make sure that you have
all the pages.

2. No calculator or any other computing device is allowed on this exam.

3. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

4. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

5. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

6. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
 1. What is the average amount of time you have per problem to complete this exam?

A. 30 seconds B. 1 minute C. 90 seconds D. 2 minutes E. 150 seconds

2. Solve the equation 3x + 5 = 2(x + 7) for x.

A. 1 B. 3 C. 5 D. 7 E. None of these.

3. If the radius of a circle is a rational number, its area is given by a(n) number.

A. rational B. irrational C. integral D. whole E. None of these.

4. The average of 4 and its reciprocal is

A. 4.25 B. 2.125 C. 3.75 D. 0.25 E. None of these.

22
5. Order the numbers 3.1, , and π from largest to smallest.
7
22 22 22 22 E. None of these.
A. 3.1, π, B. , 3.1, π, C. π, 3.1, D. π, , 3.1
7 7 7 7
1
6. In 4ABC, m∠B = 3 · m∠A and m∠C = · m∠B. Then m∠A =
6
A. 10◦ B. 20◦ C. 30◦ D. 40◦ E. None of these.

7. If (x, y) is the intersection of the two lines 3x − 2y = −2 and −2x + 3y = 8, then x + y =

A. 6 B. 7 C. 8 D. 9 E. None of these.

3
8. 32 =

A. 6561 B. 729 C. 512 D. 216 E. None of these.

 −1/2
4 1
9. If 25 · 2 = 5a · 3b , then a + b =
27
A. -2 B. -1 C. 1 D. 2 E. None of these.

10. The distance between the points (2, 7) and (5, 3) is

A. 3 B. 4 C. 5 D. 6 E. None of these.

1
11. The maximum value of y = x2 − 2x − 15 is

A. −16 B. 1 C. −3 D. 5 E. None of these.

12. The coefficient of a2 b2 in the expansion of (a + b)4 is

A. 3 B. 4 C. 5 D. 6 E. None of these.

13. The new binary operator ? is defined by a ? b = a · b − a − b. What is (1 ? 2) ? 3?

A. -7 B. -5 C. -3 D. -1 E. None of these.

14. The sum of the roots of the quadratic 2x2 − 6x + 1 is

A. 1 B. 2 C. 3 D. 4 E. None of these.

15. Which of the following relationships must hold in the following figure?

δ γ β

A. α + β = δ B. α + β < δ C. α + β > δ D. δ = β E. None of these.

16. You are playing Dungeons and Dragons with your friend. Her orc will kill your troll unless
you roll a higher number than she does. Since your friend’s orc is more experienced than
your measly troll, she can roll an eight sided die (d8) while you are only allowed to roll
a six sided die (d6). Which of the following fractions best represents the chance that
you will roll a higher number than your friend thus allowing your troll the opportunity
to live and fight again?
1 1 1 1 3
A. B. C. D. E.
5 4 3 2 4

17. Successive discounts of 10% and 20% are equivalent to a single discount of

A. 30% B. 15% C. 72% D. 28% E. None of these.

18. If the radius of a circle is increased by 100%, the area of the circle will increase by

A. 100% B. 200% C. 300% D. 400% E. None of these.

2
19. As the number of sides of a polygon increases from 3 to n, the sum of the exterior angles
formed by extending each side in succession

A. remains constant
B. increases
C. decreases
D. cannot be predicted
E. None of these.

20. A six place number if created by repeating any three digit number; for example, 256,256
or 678,678, etc. Any number of this form is always exactly divisible by

A. 7 only B. 11 only C. 13 only D. 101 E. 1001

21. The formula which expresses the relationship between x and y as shown in the accompa-
nying table is

x 0 1 2 3 4
y 100 90 70 40 0
A. y = 100 − 10x B. y = 100 − 5x2 C. y = 100 − 5x − 5x2

D. y = 20 − x − x2 E. None of these.

22. A polynomial of degree 6 has 1 + 5i and 2 + 3i as the only roots whose imaginary part is
positive. How many real roots does the polynomial have?

A. 4 B. 3 C. 2 D. 1 E. 0

23. Cool Clint drives 120 miles to take a math test at Marywood on a Saturday. Due to
construction on I-81, his average speed on the way to Scranton is only 30 miles per hour.
However, on the way home things have picked up and he is able to average 40 miles per
hour. The average speed for his round trip is closest to

A. 34 mph B. 35 mph C. 36 mph D. 37 mph E. 38 mph

3
24. What is the last digit of 20132013 ?

A. 1 B. 3 C. 7 D. 9 E. None of these.

25. The number of circular pipes with an inside diameter of 1 inch which will carry the same
amount of water as a pipe with an inside diameter of 6 inches is

A. 6π B. 6 C. 12 D. 36π E. None of these.

26. Two high school classes took the same test. One class of 20 students made an average of
80%; the other class of 30 students made an average of 70%. The average grade for all
students in both classes is

A. 75% B. 74% C. 72% D. 77% E. None of these

27. When the circumference of a balloon is increased from 20 inches to 25 inches, the radius
is increased by
5 5
A. 2π
inches B. π
inches C. 2.5 inches D. 5 inches E. None of these

28. In the rectangle ABCD, the point X is located on BC as shown in the figure. If CD = 8,
AX = 17, and DX = 10, what is the area of 4ADC?

A D

8 17 10 8

B X C

A. 108 B. 92 C. 85 D. 84 E. None of these.

29. On a spelling exam, taken by 7 students, 2 scored an A, 3 scored a B, and 2 scored a C.

If 3 of these 7 students are chosen at random, what is the probability that none of them
scored an A?
1 2 3 4 E. None of these.
A. B. C. D.
7 7 7 7

4
 √ √
3− 2
30. After rationalizing the numerator of √ , the denominator in simplest form is
3
√ √ √ √ √ √ √
A. 3( 3 + 2) B. 3( 3 − 2) C. 3 − 6
√
D. 3 + 6 E. None of these.
 
 a b 
31. If the expression   has the value ab − cd for all values of a, b, c, and d, then the
  c d 
 2x 1 
equation   = 3 is true for how many values of x?
x x 
A. 0 B. 1 C. 2 D. ∞ E. None of these.

32. Given x > y and z 6= 0, which of the inequalities below is not always correct?

A. x + z > y + z B. x − z > y − z C. xz 2 > yz 2

x y
D. > 2 E. xz > yz
z 2 z

33. How many triangles are in the following figure?

A. 10 B. 16

C. 17 D. 27

E. None of these.

34. You are given a cylinder of radius 5 and height 10, a sphere of radius 5, a right circular
cone of radius 5 and height 10, and a cube of side length 10 as shown in the figure. Which
object has the smallest surface area to volume ratio?

10 5
10 10
5 5

A. cylinder B. sphere C. cone D. cube E. my head

5
35. The front of a box has area 12 square inches, the side has area 8 square inches, and the
bottom has area 6 square inches. What is the volume of the box (in cubic inches)?

A. 576 B. 109 C. 24 D. 9 E. None of these.

36. A tired student wakes up and grabs two socks out of her drawer without looking. If she
has 3 identical black socks and 3 identical green socks, what is the probability that she
will choose a matching pair?

A. 0% B. 10% C. 20% D. 30% E. None of these.

37. A number when divided by 10 leaves a remainder of 9, when divided by 9 leaves a

remainder of 8, by 8 leaves a remainder of 7, and so on, down to where, when divided by
2, it leaves a remainder of 1. A possible value for the number is

A. 59 B. 419 C. 1259 D. 2519 E. None of these.

38. A total of 28 handshakes was exchanged at the conclusion of a party. Assuming that each
participant was equally polite toward all the others, the number of people present was

A. 7 B. 8 C. 14 D. 28 E. None of these.

39. A square is cut into four equal congruent squares and the upper left square is shaded
in. The lower right square is then split into four more congruent squares and the upper
left square is shaded in. This process is continued forever as demonstrated in the figure.
What proportion of the square will eventually be shaded in?

1 1 2 3 E. None of these.
A. B. C. D.
2 3 3 4
40. A group of high school boys and girls are taking a math exam. After an individual
finishes, they are free to leave. At first 15 girls leave. At this point there are two boys
for each girl left in the room. After this, 45 boys leave. Now there are 5 girls for every
boy left in the room. How many girls were in the room to start with?

A. 40 B. 43 C. 29 D. 50 E. None of these.

6
Except for this notice and the page number, this page has been left blank for
your use.

7
 2013 Marywood Mathematics Contest
Level II

Sponsored by

SEMI-GROUP
The Student Mathematics Club of

Marywood University

March 16, 2013

Directions:

1. This exam consists of 40 questions on 6 pages. Please check to make sure that you have
all the pages.

2. No calculator or any other electronic device is allowed on this exam.

3. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on
any one problem. If a question seems to be too difficult, make your best possible guess.
Your score will be the number of correct responses.

4. On the scantron form provided for you, darken in the space corresponding to the correct
answer. Please mark all answers carefully and erase completely when changing an answer.
Mark only one answer for each question. Only those answers on the answer sheet will
be counted.

5. There is a sheet of blank paper on the last page which you can tear off and use as scratch
paper. You may also use the back of the pages.

6. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer

any questions pertaining to specific problem content.

Please do NOT open the test until you are told to do so.
1. 20130 + 2013−1 =

A. −2013 B. −1/2013 C. 1/2013 D. 2012/2013 E. 2014/2013

2. (x − 1)2 =

A. x2 − 1 B. x2 + 1 C. x2 − 2x − 1 D. x2 − 2x + 1 E. None of these.

3. Thomas made 12 shots in a basketball game, which is approximately 52% of his attempted
shots. How many shots did he attempt?

A. 6 B. 7 C. 24 D. 23 E. None of these.

4. When a pair of fair dice are rolled, what is the probability that the sum of the two
numbers represented by the dice is 10?
1 1 1 1 E. None of these.
A. B. C. D.
9 10 11 12

5. 32013 + 32013 =

A. 34026 B. 32014 C. 2 × 32013 D. 62013 E. None of these.

6. Notice that 2013 = 3 × 11 × 61, how many distinct positive factors does 2013 have?

A. 3 B. 6 C. 7 D. 8 E. 9

7. If f (x) = x2013 + x + 1, what is f (−1)?

A. 1 B. −1 C. 2012 D. 2013 E. 2014

8. The operation  is defined as a  b = a2 + ab + b2 . Find 3  2.

A. 20 B. 19 C. 13 D. 12 E. None of these.

9. If one leg of a right trangle is twice the length of the other leg, what is the ratio between
the hypotenuse and the shorter leg?
√ √ √
A. 3 : 1 B. 3 : 2
x √ √ √
C. 5:1 D. 5: 2

2x E. 5 : 1

1
10. Given two points A(4, 12) and B(12, 20), find the point Q on AB such that |AQ| = 3|BQ|.

A. (8, 16) B. (10, 18) C. (6, 14) D. (18, 10) E. None of these.

11. The sum of the two roots of the equation x2 + 5x − 1 = 0 is

A. −5 B. −1 C. 5 D. 1 E. None of these.

12. If all of the following have the same area, which one has the largest perimeter/circumference?

A. An equilateral triangle B. A square C. A regular pentagon

D. A regular hexagon E. A circle

13. The units digit of 20132012 is

A. 1 B. 3 C. 7 D. 9 E. None of these.

14. Professor Johnson lives 10 miles from the university where he works. His morning com-
mute to work takes 30 minutes, while his evening commute home only takes 20 minutes.
What is Professor Johnson’s combined average speed for his commute each day?

A. 12 mph B. 24 mph C. 25 mph D. 26 mph E. None of these.

15. The point P (−3, 4) is on the circle x2 + y 2 = 25. If a tangent line of the circle is drawn
through P , where would it intersect with the x-axis?
         
25 25 25 25 25
A. − , 0 B. ,0 C. − , 0 D. ,0 E. 0,
3 3 4 4 4

16. How many positive integers less than 2013 would make 1n + 2n + 3n divisible by 4?

A. 0 B. 1005 C. 1006 D. 2010 E. 2011

17. The edge of a cube is of length 2 in. What is the distance from the center point on one
face to a vertex on the opposite face?
√ √ √ √
A. 8 in B. 7 in C. 6 in D. 5 in E. None of these.

2
18. A rectangle has a perimeter of 40 cm. If the length and width are each increased by 2
cm, by how much will the total area increase?

A. 40 cm2 B. 44 cm2 C. 80 cm2 D. 96 cm2

E. It depends on the exact dimension of the rectangle.

1 1
19. If + = 6, what is the denominator of x when it is written in the simplest fraction
x 2x
form?

A. 2 B. 3 C. 4 D. 18 E. None of these.

20. A regular polygon with n vertices has more than 15 diagonals. What is the smallest
possible value of n? (A diagonal of a polygon is defined as the line segment between any
two non-adjacent vertices.)

A. 9 B. 8 C. 7 D. 6 E. None of these.

21. In how many ways can the number 10 be written as the sum of three distinct positive
integers? Note that the order of the three integers is insignificant. For example, 1+2+7
and 2+7+1 are considered the same.

A. 4 B. 5 C. 6 D. 7 E. 8

22. The pages in a book are numbered from 1 to 314. How many pages have the digit 3 in
its page number?

A. 68 B. 69 C. 71 D. 72 E. None of these.

1
23. If sin θ =, and 0◦ < θ < 90◦ , what is tan θ?
10
1 1 10 √
A. √ B. √ C. √ D. 99 E. None of these.
99 9 99

24. If x and y are positive real numbers such that xy = 12, what is the minimum possible
value of x2 + y 2 ?

A. 40 B. 25 C. 24 D. 20 E. 12

25. If f (x) = x − 3, g(x) = x2 , then f ◦ g(x) = f (g(x)) =

A. x2 − 3 B. (x − 3)2 C. x2 − 9 D. x − 9 E. None of these.

3
 1 + 2 + 3 + · · · + 2012
26. =
1 + 2 + 3 + · · · + 2012 + 2013
1 2012 2012 2013 E. None of these.
A. B. C. D.
2013 2013 2014 2014

27. The four corners of a square are cut and a regular octagon is formed. If the resulting
octagon has side length equal to 1 cm, what is the length of a side in the original square?
√ √
1 2 2
A. B. 1 +
2 √ √ 2
C. 1 + 2 D. 2

E. 3

28. How many digits are in the number 45 · 510 ?

A. 8 B. 9 C. 10 D. 11 E. 12

29. What time was it 2013 minutes after the beginning of January 1, 2013?

A. January 1 at 9:33PM B. January 1 at 11:53PM C. January 2 at 3:13AM

D. January 2 at 9:33AM E. January 2 at 6:03PM

30. Allie, Bobby, Cathy and Dan are seated at random around a square table with one person
at each side. What is the probability that Allie and Dan are seated across from each other
on opposite sides of the table?
1 1 2 1 3
A. B. C. D. E.
2 3 3 4 4

31. In how many ways can 2015 be written as the sum of two positive prime numbers?

A. 0 B. 1 C. 2 D. 3 E. 4

4
32. A circle is inscribed in a square and circumscribed about another square. Which of the
following best approximates to the ratio of the area between the circle and the smaller
square to the area between the two squares?

A. 1/2 B. 1
C. 3/2 D. 2
E. 5/2

33. How many triangles are there in the figure below?

A. 25
B. 30
C. 35
D. 40
E. None of these.

1 1 1 1
34. + + + ··· + =
1·2 2·3 3·4 2012 · 2013
2012 2014 2013 D. 1 E. None of these.
A. B. C.
2013 2013 2012

35. Bag A contains 10 chips labeled 1, 3, 5, · · · , 19, and bag B contains 10 chips labeled
2, 4, 6, · · · , 20. If one chip is selected from each bag, how many different values are
possible as the sum of the two numbers on the two chips selected?

A. 18 B. 19 C. 20 D. 39 E. 40

36. If an equilateral triangle and a regular hexagon have the same area, what is the ratio
between their perimeters?
√ √ √
A. 1 : 1 B. 3 : 6 C. 6 : 1 D. 6 : 1 E. 3 : 6

5
 
20x + 13y = 20
37. If x and y satisfy the system of equations , what is the value of
20x − 12y = 15
x + y?

A. 1.07 B. 0.67 C. 0.2 D. 0.93 E. None of these.

√ √
qp q p
38. If M = 3, then M M M =

A. 3 B. 27 C. 81 D. 729 E. None of these.

39. The three sides of 4ABC are all integers in length, and |AB| = 7, |AC| = 12. If the
maximum possible value of |BC| is x, and the minimum possible value of |BC| is y, what
is the sum x + y?

A. 18 B. 19 C. 24 D. 25 E. 26

40. If a point is only allowed to move to the right or up from one lattice point to another.
How many different paths are there from (0, 0) to (5, 5)? Note: A lattice point is one
with integer coordinates.

A. 5
B. 10
C. 126
D. 252
E. None of these.

An example path.

6
This page is left mostly blank for your use.