Download Heracles lies on Monday, Tuesday, and Wednesday. Theseus lies

1. Heracles lies on Monday, Tuesday, and Wednesday. Theseus lies on Thursday, Friday, and
Saturday. At all other times they both tell the truth. “Yesterday was one of my lying days,” says
Heracles. “Yesterday was one of my lying days too,” says Theseus. Which day of the week is it
in the context of the problem?
A. Monday
B. Tuesday
C. Wednesday
D. Thursday
E. None of the Above
2. Which number is the next number in this sequence: 2, 28, 126, 344, 730, _____?
A. 1332
B. 860
C. 5110
D. 898
E. None of the Above
3. The driver of a car glanced at the odometer and saw that it read 15,951 miles. He said to
himself; “That’s interesting. The mileage is a palindrome: it reads the same backward as
forward. It will be a long time before that happens again.” Just two hours later, however, the
mileage shown on the odometer was a new palindrome. How fast was the car going in those two
hours in miles per hour?
A. 25 mph
D. 105 mph
B. 51 mph
E. None of the Above
C. 55 mph
4. Given the function g ( x)  x  1 to the parent function f ( x)  x , which of the following is
NOT true about the graph of g ( x)  x  1 ?
A. It is a function.
B. It is a one-to-one function.
C. It is reflected over the y-axis.
D. It is vertically shifted up one unit.
E. None of the Above
5. The points (4, 1) and (-2,3) are reflected over the line y = x. Find the number of square units in
the area of the quadrilateral whose vertices are the points and their images.
A. 12
D. 18
B. 14
E. None of the Above
C. 16
***Consider the odd function f ( x) and the even function g ( x ) , both with a domain of all real
numbers. Use the following chart to answer questions 6 and 7:
x
1
2
3
4
5
6
-5
-2
1
2
4
8
f (x)
2
4
6
8
-2
-4
g( x )
))).
6. Calculate the value of: f(1f(f(1
A. -5
B. -4
C. -2
D. -1
E. None of the Above
(4g
(g
(
2
))).
7. Calculate the value of g
A. 8
B. 4
C. 2
D. -2
E. None of the Above

1
ta
n
x1
?
8. What are the asymptotes to the graph of y2
A. y  
B. y  

D. y  2
E. None of the Above
2

4
y



C.
1
 1 2
9. What is the reciprocal of the coefficient of the fourth term of 1 x plus the coefficient of
 2 
the third term of 23x ?
3
2021
16
215
B.
128
2001
C. 
16
1005
8
E. None of the Above
A. 
D. 
10. Harry Potter is flying between Hogwarts and Hogsmeade. On the way to Hogsmeade, there is a
head wind that decreases his speed by 20 mph. Flying the other way, the wind at his back
2
increases his speed by the same amount. If it takes him 5 hours to get to Hogsmeade and 1 3
hours to get back, how many miles is it between Hogwarts and Hogsmeade?
A. 40
B. 60
C. 80
D. 100
E. None of the Above
11. An ellipse centered at the origin with major axis on the line y  x has focus (3, 3) and
2
eccentricity . Which of the following is a co-vertex of the ellipse?
3
A. (-3, 3)
 31
031
0

,


B. 
2 
 2

3 5 3 5
,

C. 
 2
2



j
12.
j 2
 31
531
5

,

D. 
 2
2 


E. None of the Above
8

1
2
A. 4
B. 6
C.
13
2
D. 8
E. None of the Above
13. If x

1

x

2
y

2
y

2
y

.
.
.and x  1, what is the value of 4y ?
2
2
x

x1

x4
x

A. 
2
2
x

x1

x3
x

B. 
2
2
x

2
x

1

2
x

3
x

C. 
2
2
x

2
x

1

2
x

4
x
 E. None of the Above
D. 
14. Of the 26 letters in the alphabet, what is the probability that one selected at random is not
printed on this page (side) of the test? Consider upper- and lowercase equivalent.
A. 0
B.
1
26
C.
1
13
D.
3
26
E. None of the Above
15. How many of the following graphs bound an area of 16 ? Assume all lengths are in units, and
areas in units squared.
x

1
6

1
6
4

y

I. The graph of 
in the Cartesian Plane.

4
c
o
s
t
,y

1
64
s
i
n
tin the Cartesian Plane.
II. The graph of x
III. A circle with circumference 8 .
IV. An isosceles trapezoid of height 4 and median 4.
V. The graph of r 8cos in the Polar Coordinate Plane.
x y
 1.
VI. The graph bounded by the x- and y-axes and the graph of
4 8
2
A. 2
B. 3
 
2
C. 4
D. 5
E. None of the Above
 
v

w
u

u

u
u

w
16. u


2
A. u vuu
C. uv  u
2


2
B. u vuu
D. uv  u
2
E. None of the Above
17. A lattice point is a point ( x, y ) where x and y are both integers. If there are L lattice points that lie
in the solution set of the system below, then find the sum of the digits of L.
2
2

1
6
x

9
y

6
4
x

1
8
y

7
1

 1
y
x

21

 2
A. 4
B. 6
C. 8
D. 10
E. None of the Above
18. A regular hexagon’s apothem measures
5 . The hexagon is the base of a right prism. The prism’s
height is 2 6 . The volume of the prism is x y in simplest radical form. Find the sum of the digits
of the product ( xy ) .
A. 3
B. 6
C. 9
D. 11
E. None of the Above
19. Rahn, Jorge, Thierry, Will, and Caitlin will enter the MAO Hall of Fame one at a time. Jorge must
not enter first, and Thierry must be one of the last three to enter. In how many different orders can
these five MAO alumni enter?
A. 36
B. 54
C. 60
D. 72
E. None of the Above
20. A one-to-one function is graphed in the Cartesian plane. Its domain is all real numbers on the
closed interval [2,5] , and its range is all real numbers in the closed interval [3,10] .
The graph is reflected about the line y  x , and then reflected again about the x-axis.
Let g be the function defined by the resultant graph. Which of the following statements are true?
I. The range of g includes 1 .
II. The domain of g includes 3.
III. The range of g includes 3.
A. I and II only
I.
B. I and III only C. II and III only
D. I, II, and III
E. None of the Above
21. A line perpendicular to the line 3x4y12 passes through the points (0, 2k 1) and (3  k, 0) .
P
If k can be expressed as Q , where P and Q are relatively prime integers, then evaluate P  Q .
A. 1
B. 6
C. 17
D. 18
E. None of the Above
22. The graph of a conic section is centered at the point (1,1) and has a vertex at the point (4,1) .
The conic section has eccentricity = 2. Which of the following is true of the graph?
A. Its conjugate axis has length 10.
B. Its conjugate axis has length 6 3 .
C. Its transverse axis has length 10.
D. Its transverse axis has length 6 3 .
E. None of the Above
0
23. Let x  22i , y  ln() , and z (xy) . Which of the following inequalities is true?
A. x  y  z
B. x  z  y
C. y  z  x
D. z  y  x
E. None of the Above
24. Two cars drive toward each other at 20 mph and 30 mph respectively. When their front bumpers are
100 miles apart (time=0), a fly flies from the bumper of one car to the bumper of another, and back,
continuously, at a constant rate. If the fly flies at 40 miles per hour, how far has the fly flown from time
0 to the time the cars meet?
A. 80 mi
B.
2 mi
C.
1
mi
2
D.
1
mi E. None of the Above
40
25. Three numbers are chosen at random and without replacement from the set {1, 2, 2, 2, 3, 3, 4, 5, 6}.
What is the probability that at least two of the numbers are equal?
A.
11
84
B.
2
9
C.
13
42
D.
29
84
E. None of the Above
26. Assuming the expression converges, find the largest value of x such that x5000, and
x
x
x
x

.
.
.is rational.
A. 4890
B. 4900
C. 4970
D. 4998
E. None of the Above
()

a
xb

x

c
xd
27. A cubic function with integral coefficients has equation fx
, for
3
2
a  0 . If the graph of f has a relative (local) maximum at (3, k) for k  1 and relative (local)
minimum at (1, 1) then which function below has exactly two distinct roots?
A. f (x)k1
B. f (x) 1
C. f (x1)
D. f (x k) E. None of the Above
28. For positive even integers n, and i  1 , which is NOT a possible value of
A. 64
B. 32i
C. 4
D. 8i
1  i 
E. None of the Above
n
?
29. A two-digit, base-ten number is three times the sum of its digits. What is the tens digit of this
number?
A. 9
B. 7
C. 3
D. 2
E. None of the Above
30. Janine will give Bill $3 on every even day (Sunday=1, Monday=2, …) and Bill will give Janine $2
on every odd day. If they both begin with $12 at the beginning of Sunday (before money has changed
hands), on what day of the week will Bill first have the same amount that Janine has, after Sunday?
A. Wednesday B. Thursday C. Friday
D. Saturday
E. None of the Above