1. What is
8 − 6 ÷ 3 + 2 × (−1)?
CSU FRESNO MATHEMATICS FIELD DAY
MAD HATTER MARATHON 9-10
PART I
April
17th ,
2010
2. How many zeros does (1010 )2 have?
(a)
(b)
(c)
(d)
12
13
20
21
(a) −1
1
3
(b) 8
(c) 4
(d) 3
3. A painting is 10 feet by 12 feet, and the artist plans to frame the
painting with a picture frame and a matte (i.e., a border between the
painting and the inner edge of the frame). If he wants the matte to
measure 3 inches from the edge of the painting to the edge of the
frame, what will the perimeter of the inner edge of the frame be?
(a) 56 feet
(b) 46 feet
(c) 23 feet
1
(d) 22 feet
2
4. Simplify
(a)
!
z −2 y 3
22 34
(c)
.
5. An isoscoles triangle has equal sides of length 5. Determine the
length of the third side.
18z
(a)
(b)
(c)
(d)
3
y2
3
(b)
"−1/2
y2
18z
z
4
3
6
It cannot be determined from the information given.
3
18y 2
3
(d)
18y 2
z
6. Find the fourth term of (x − 2y )12 .
(a) 840x 10 y 2
(b) −840x 10 y 2
(c) 1760x 9 y 3
(d) −1760x 9 y 3
7. If ab = 2, a + b = 2, and a2 − b 2 = 8, find (a + b)2 .
(a)
(b)
(c)
(d)
2
4
16
None of the above.
8. Find the sum
1 1
+ + 1 + 2 + · · · + 1024.
4 2
(a) 8191
(b) 8190
9. Pam is 10 years older than Jake, who is 4 years older than Tom,
whose age is one half the difference between Pam’s age and twice
Jake’s age. How old is Pam?
(a)
(b)
(c)
(d)
8191
4
4095
(d)
2
(c)
16 years old
10 years old
6 years old
2 years old
10. Write in simplest form
x3 − x2 + x − 1
.
x −1
(a) (x + 1)2
2
(b) x + 1
(c) (x − 1)2
(d) x 2 − 1
11. Find the solution(s) of the inequality
(x + 2)2 ≥ 4.
(a)
(b)
(c)
(d)
x ≥0
x ≥ 0 or x ≥ −4
x ≥ 0 or x ≤ −4
It is true for all values of x.
12. Three separate awards are to be presented to selected students from a
class of 20. How many different outcomes are possible if a student
can receive any number of awards?
(a)
(b)
(c)
(d)
8,000
1,140
120
60
13. Which of the following is a true statement about irrational numbers?
(a)
(b)
(c)
(d)
The
The
The
The
square root of an irrational number is irrational.
sum of two irrational numbers is irrational.
quotient of two irrational numbers is irrational.
product of two irrational numbers is irrational.
14. Determine the fraction denoted by 0.5421.
15. What is the radius of the largest circle that can be inscribed in a right
triangle with legs of length 6 and 8?
5367
9990
5367
(b)
9900
5421
(c)
9999
5421
(d)
10000
(a)
(a)
(b)
(c)
(d)
5
4
3
2
16. An insulated cup is created by taking two cylinders of different radii
and heights and filling in the interior with an insulating material, then
sealing. If the outer cylinder has height 9 cm and radius 4 cm, and
the inner cylinder has height 8 cm and radius 3 cm, determine the
volume of insulation.
(a)
(b)
(c)
(d)
72π cubic centimeters
36π cubic centimeters
9π cubic centimeters
8π cubic centimeters
18. In a cup, there are 4 quarters, 5 dimes, 6 nickels, and 10 pennies. If
one coin is selected at random, what is the probability that the coin
has at least one letter “n” in its name?
6
25
1
(b)
2
9
(c)
16
16
(d)
25
(a)
17. The perimeter of rhombus RSTU is 52 and the diagonal RT = 24.
What is the area of the rhombus?
(a)
(b)
(c)
(d)
312
240
120
78
19. The values of 10 houses on a certain street are given below:
Value per House
$150,000
$200,000
$250,000
$400,000
Number of Houses
1
4
4
1
Find the average value of a house on this street.
(a)
(b)
(c)
(d)
$250,000
$235,000
$150,000
$100,000
20. Jaime has $0.40 stamps and $0.16 stamps. If she needs to mail a
package to her friend, the postage will cost $5.76. Just for fun, she
decides to use twice as many $0.40 stamps as $0.16 stamps. How
many of each type of stamp does she need?
(a)
(b)
(c)
(d)
Twelve $0.40 stamps and six $0.16 stamps.
Six $0.40 stamps and twelve $0.16 stamps
Thirteen $0.40 stamps and four $0.16 stamps
Four $0.40 stamps and eight $0.16 stamps
21. The electronics store, in its attempt to maximize its profit, charges
10% over the standard price on a video recorder on a regular basis.
Then, the store puts the video recorder on sale for 25% off. What is
the net discount on the standard price?
(a)
(b)
(c)
(d)
15%
17.5%
20%
25%
23. Solve the equation
|x + 1| ≤ 3x.
22. A substance is 99% water. Some water evaporates, leaving a
substance that is 98% water. How much of the water evaporated?
(a)
(b)
(c)
(d)
49.5%
50%
50.5%
55%
1
2
(b) x ≥ −1
(a) x ≥
(c) x ≥ −
(d) x ≥
1
4
1
or x ≤ −1
2
24. In a geometric progression, the sum of the first 5 terms is 11 and the
11
sum of the next 5 terms is
. What is the common ratio?
32
1
11
1
(b)
32
5
(c)
32
1
(d)
2
25. One of the sides of a rectangle is 3cm shorter than the other one.
Find the sides of the rectangle if we know that, if we increase every
side by 1cm, the area of the rectangle will be increased with 18 cm2 .
(a)
26. Compute f(g(4)) if f(4) = -4, g(4) = -2, and f(-2) = -1.
(a)
(b)
(c)
(d)
8
4
-2
-1
(a)
(b)
(c)
(d)
18 cm and 15 cm
10 cm and 7 cm
8 cm and 5 cm
7 cm and 4 cm
27. Solve for x:
log4 x + log4 (x + 3) = 1.
(a)
(b)
(c)
(d)
x
x
x
x
= −4
=1
= −1
= 1 or x = −4
28. Find the difference between the least common multiple and the
greatest common divisor of 315 and 36.
(a)
(b)
(c)
(d)
29. A boat is able to make 16 knots traveling downstream but only 10
knots traveling upstream. What is the speed of the current?
(a) 13 knots
(b) 6 knots
(c) 3 knots
13
(d)
knots
2
1260
1251
18
9
30. The value of
x + x2 + x3 + x4 + x5 + x6 + x7
x −3 + x −4 + x −5 + x −6 + x −7 + x −8 + x −9
31. Given a line segment with endpoints (-12, 16) and (-3, 4), determine
the coordinates of a point on the line whose distance from the left
endpoint is two-thirds the length of the line segment.
is
(a)
(b)
(c)
(d)
x 20
x 10
x9
x −9
(a) (2,
√ $
# 9) √
(b) −3 + 41, 4 + 41
(c) (-6, 8)
!
"
20
(d) 5, −
3
32. If a chain letter requires you to send copies to three friends, what is
the total number of letters sent after five mailings?
33. Find a function g (x) such that for f (x) = 2x − 5, f (g (x)) = x.
(a) g (x) = 2x + 5
(a)
(b)
(c)
(d)
15
243
363
729
34. If all people eat the same amount of pizza, and a pizza 12 inches in
diameter serves two people, how many inches in diameter should each
of two pizzas be in order to serve three people? (Pizzas are circular
and are eaten entirely.)
(a)
(b)
(c)
(d)
√
6√3 inches
3√6 inches
9 3 inches
9 inches
5
2
x −5
(c) g (x) =
2
x +5
(d) g (x) =
2
(b) g (x) = x +
35. What is the sum of the real solutions of
x 6 − 5x 5 + 6x 4 − x 2 + 5x − 6 = 0?
(a)
(b)
(c)
(d)
5
7
9
11
36. Put the following objects in order of increasing volume:
I a rectangular pyramid with base 3 cm by 2 cm and height 4 cm
II a cylinder with radius 2 cm and height 4 cm
III a cone with radius 2 cm and height 6 cm
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
I, II, III
I, III, II
III, II, I
III, I, II
38. Find all real roots of
4
2
r − 13r + 36.
(a)
(b)
(c)
(d)
4 and 9
-2 and 2
-3 and 3
both (b) and (c)
37. Given that 80% of all California drivers wear seatbelts, if 4 drivers
were pulled over, what is the probability that all 4 would be wearing
their seat belts?
0.80
(0.80)2
(0.80)3
(0.80)4
39. A quiz consists of true and false questions. The ratio of the number
of true questions to the number of false questions is 4:3. About what
percent of the questions are false?
(a)
(b)
(c)
(d)
34%
43%
57%
67%
40. An office requires P sheets of paper each month for each employee. If
there are E employees in the office, for how many months will T
sheets of paper last?
P
TE
PE
(b)
T
T
(c)
PE
TE
(d)
P
(a)
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
C
D
B
A
B
C
A
D
B
D
B
B
C
B
C
C
D
Solutions
1 C
2 D
3 B
4 A
5 D
6 D
7 D
8 C
9 A
10 B
11 C
12 A
13 A
14 B
15 D
16 A
34
A
35
A
36
B
37
D
38
D
39
B
40
C
CSU FRESNO MATHEMATICS FIELD DAY
MAD HATTER MARATHON 9-10
PART II
April 17th , 2010
2. The four numbers a, b, c, and d, where a < b < c < d, can be paired
in six different ways. If each pair has a different sum, and if the four
smallest sums are 1, 2, 3, and 4, what are all possible values of d?
(a) 1
7
2
(c) 4
(d) Both (b) and (c) are possible values.
(b)
1. Today is Saturday. What day of the week will it be in exactly 2010
days from today?
(a)
(b)
(c)
(d)
Friday
Saturday
Sunday
Monday
3. An odd integer between 600 and 800 is divisible by 7 and also
divisible by 9. What is the sum of its digits?
(a)
(b)
(c)
(d)
7
12
18
21
4. At night a man who is 6 feet tall stands 5 feet away from a lamppost.
The lampposts lightbulb is 16 feet above the ground. How long is the
man’s shadow?
15
feet
8
(b) 3 feet
5. Let !ABC be a right triangle with hypotenuse AB and with the
measure of ∠BAC equal to 32◦ . A square with side AB is placed so
that the interior of the square does not overlap the interior of !ABC .
Let P be the center of the square. What is the measure of ∠PCB?
(a)
10
feet
3
(d) 4 feet
(c)
6. Given that 29a031 × 342 = 100900b02 where a and b denote two
missing digits, what is the value of a + b?
(a)
(b)
(c)
(d)
8
9
10
11
(a)
(b)
(c)
(d)
32◦
45◦
58◦
60◦
7. Two points A and B lie on a sphere of radius
√ 12. The length of the
straight line segment joining A and B is 12 3. What is the length of
the shortest path from A to B if every point of the path must lie on
the sphere?
(a)
(b)
(c)
(d)
6π
8π
9π
12π
8. For what values of a does the system of equations
x2 = y2
(x − a)2 + y 2 = 1
have exactly three solutions?
(a)
(b)
(c)
(d)
for
for
for
for
a = ±1
−2 ≤ a ≤ 2
a ∈ {−1, 0, 1}
all a ≥ 0
10. Assume that b and c are integers greater than 1. In base b, c 2 is
written as 10. Then b 2 written in base c is
(a)
(b)
(c)
(d)
100
101
1010
10000
9. Trains leave from Philadelphia for Harrisburg every hour on the hour.
The trip takes three hours. Each train waits at the Harrisburg depot
one half hour and then returns to Philadelphia. Train A leaves
Philadelphia at 9 a.m. The number of trains going the other way that
it will pass on its return trip is
(a)
(b)
(c)
(d)
3
4
5
6
11. Sam and Susie are brother and sister. Sam has twice as many sisters
as brothers. Susie has twice as many brothers as sisters. How many
sisters does Sam have?
(a)
(b)
(c)
(d)
2
3
4
5
12. Points A and B are in the first quadrant and O is the origin. If the
slope of OA is 1, the slope of OB is 7, and the length of OA is equal
to the length of OB, then the slope of AB is
13. If the system of equations
2x − 3y = −4
3x − y = 1
x − ky = 5
is to have at least one solution, then the value of k must be
2
3
3
(b) −
4
6
(c) −
7
1
(d) −
2
(a) −
(a)
(b)
(c)
(d)
-2
-1
1
2
15. For all real numbers x, the function f (x) satisfies
14. An 8 inch chord is twice as far from the center of a circle as a 10 inch
chord. What is the circumference of the circle?
(a)
(b)
(c)
(d)
√
8√7π
6√7π
4√7π
2 7π
inches
inches
inches
inches
2f (x) + f (1 − x) = x 2 .
Find f (5).
(a) 4
34
3
(c) 8
(b)
(d)
19
3
16. Both solution X and solution Y contain alcohol and water. In
solution X , the ratio of alcohol to water is 3:2. When equal amounts
of solution X and solution Y are mixed, the ratio of alcohol to water
is 3:4. What is the ratio of alcohol to water in solution Y ?
(a)
(b)
(c)
(d)
9:26
10:25
10:24
none of the above
18. The smallest positive integer n for which the decimal expansion of n!
ends in 3 zeroes is
(a)
(b)
(c)
(d)
10
12
14
15
17. Suppose that 1 and 2 are roots of x 3 + ax 2 + bx + c = 0 and that
a + b = −15. Then a =
(a)
(b)
(c)
(d)
−19
−10
−5
4
19. A contest among n ≥ 2 players is held over a period of 4 days. On
each day each player receives a score of 1, 2, . . . , n points with no two
players getting the same score on a given day. At the end of the
contest it is discovered that every player received the same total of 26
points. How many players participated?
(a)
(b)
(c)
(d)
12
11
10
9
20. Given that
510 ?
(a)
(b)
(c)
(d)
1025
= 1.0009765625, what is the sum of the digits of
1024
36
40
41
50
22. Parallelogram ABCD is such that AB = 17 cm. There are points E
on AB and F on CD such that EF = 10 cm and EF is perpendicular
to AB. Find the area of ABCD
(a)
(b)
(c)
(d)
85 cm2
167 cm2
170 cm2
340 cm2
21. What is the coefficient of x 3 in the expansion of
!
(a)
(b)
(c)
(d)
"6
1 + x + x2 + x3 + x4 + x5 ?
40
48
56
62
2
3
23. In a given city, of the adult men are married to of the adult
3
7
women. The number of married men and women are equal, and the
adult population is over 1800. What is the smallest possible number
of adult residents in the city?
(a)
(b)
(c)
(d)
1809
1817
1821
1825
24. Consider a convex body having 13 vertices. For example, a cube is
convex body having 8 vertices. How many distincts triangles can be
constructed by connecting three different vertices of this body?
(a)
(b)
(c)
(d)
283
286
289
292
26. A final exam was taken by 32 students. The mean score of those who
passed was 62, the mean score of those who failed was 54, and the
mean score of all the students was 60. How many students did not
pass the test?
(a)
(b)
(c)
(d)
24
18
14
8
25. A wierd number is a number that is the product of two consecutive
primes, such as 7x11=77 . What is the least common multiple of the
2 smallest wierd numbers?
(a)
(b)
(c)
(d)
10
20
30
60
27. How many digits are in the number 588 · 291 ?
(a)
(b)
(c)
(d)
86
87
88
89
28. Let T be a point outside a circle. From T we draw two secant lines.
The first secant line intersects the circle in two points V and R such
that V is between R and T . The second secant line intersects the
circle in two points S and U such that S is between U and T .
Suppose that the measure of ∠RTU is 38◦ , the measure of ∠TRS is
31◦ , RT = UT and VU and RS intersect at Q . What is the measure
of ∠RQV ?
(a)
(b)
(c)
(d)
80◦
79◦
78◦
77◦
30. A two-inch elastic band is fastened to the wall at one end, and there is
a bug at the other end. Every minute (beginning at time 0), the band
is instantaneously and uniformly stretched by 1 inch, and then the bug
walks 1 inch towards the fastened end. The bug will reach the wall in
(a)
(b)
(c)
(d)
under 3 minutes
in under 6 minutes, but over 3 minutes
in under 60 minutes, but over 6 minutes
never
29. There are 100 people in a room. 95 of them speak Russian, 80 speak
Mandarin, and 90 speak Spanish. At least how many speak all 3
languages?
(a)
(b)
(c)
(d)
65
35
70
30
31. In order to compute the area of a square, you measure the length of a
side. However, your measurement might have been off by up to 10%.
What is the most your area calculation might be off?
(a)
(b)
(c)
(d)
10%
19%
21%
100%
32. A cylinder with radius r and height h has volume 1 and total surface
1 1
area 12. Compute + .
r
h
1
12
1
(b)
6
(c) 6
(d) 12
33. Define a reverse prime to be a positive integer N such that when the
digits of N are read in reverse order, the resulting number is a prime.
For example, the numbers 5, 16, and 110 are all reverse primes.
Compute the largest two-digit integer N such that the numbers
N, 4 · N, and 5 · N are all reverse primes.
(a)
34. Jill rides her bike around a course in the shape of an equilateral
triangle. Her speed is 10 miles per hour on the first side of the course,
15 miles per hour on the second side of the course, and 20 miles per
hours on the third, and final, side of the course. Jill’s average speed
during her ride is
(a)
(b)
(c)
(d)
less than 13 miles per hour
at least 13 but less than 14 miles per hour
at least 14 but less than 15 miles per hour
at least 15 miles per hour
(a)
(b)
(c)
(d)
79
91
95
97
35. The arithmetic mean (i.e the average) of N real numbers is N. The
arithmetic mean of a subset of M of the given numbers is M, where
M < N. What is the arithmetic mean of the remaining N − M
numbers?
(a)
(b)
(c)
(d)
M
N
N −M
N +M
36. If a is not zero and the three roots of ax 3 + bx 2 + (a − 4)x + 2 = 0
are equal integers, with r the common value of these integers, what is
the value of r ?
(a)
(b)
(c)
(d)
1
2
3
4
2
38. Suppose that of 10 bananas are worth as much as 8 oranges. How
3
1
many oranges are worth as much as of 5 bananas?
2
(a) 2
5
2
(c) 3
(b)
(d) 72
37. In a collection of red, blue, and green marbles, there are 25% more
red marbles than blue marbles, and there are 60% more green marbles
than red marbles. Suppose that there are r red marbles. What is the
total number of marbles in the collection?
(a)
(b)
(c)
(d)
2.85r
3r
3.4r
3.85r
39. Points A and C lie on a circle centered at O, each of BA and BC are
tangent to the circle, and !ABC is equilateral. The circle intersects
BD
BO at D. What is
?
BO
√
2
3
1
(b)
2
√
3
(c)
3
√
2
(d)
2
(a)
Solutions
16
C
D
C
B
B
D
B
A
D
D
A
D
A
C
B
A
34
B
35
D
36
A
37
C
38
C
39
B
40
C
1
40. What number is
1
1
3
of the way from to ?
3
4
4
2
3
4
5
1
2
1
(b)
3
5
(c)
12
2
(d)
3
(a)
6
7
8
9
10
11
12
13
14
15
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
D
D
A
B
C
C
B
B
C
D
D
A
A
B
C
C
A