Sophomore Olympiad
2013
1
1.
The distance between the lines
2 y

1
2 x
 
3 and 3 y

3 x

18 is:
A) 2 3
B) 6 2
C) 12
D) 16
E) None of these
2.
If the sides of a triangle are 4, 6, and 8 units, then the triangle is:
A) Acute
B) Obtuse
C) Right
D) No triangle can be formed
E) None of these
1 a 2 b 3
2

3
4
1 a 3 b 4
1
2
3.
Where defined, equals:
 a
1
12


1
2
A) 0
B) 4 a
C) b 3 a
1
D) a
3
E) None of these

4.
The prices (in dollars) for a sample of personal computers are 550, 700, 420, 580, 450, 700, 390 and
350. If a computer with a price of $2500 was added to the list, then the measure (mean, median, mode) affected the most would be:
A) Mean
B) Median
C) Mode
D) All would be affected the same
E) None of these
5.
When a tortoise crosses a highway, his speed is 2 feet per hour faster than normal. If this tortoise can cross a highway 24 feet in width in 24 minutes less time than he can travel that same distance off the highway, then his speed crossing the highway, in feet per hour, is:
A) 10
B) 12
C) 14
D) 16
E) None of these
6.
An insurance company conducted a survey and 2700 people responded to the question: “How much would you trust your doctor to put your health above cost?” The following relative frequency bar graph represents the results. The number of people who said that they completely trusted or mostly trusted their doctor is:
A) Less than 500
B) Less than one-fourth of the respondents
C) Less than one-third of the respondents
D) More than half of the respondents
E) None of these

7.
If a haircut at Joe’s Barber Shop costs $50 dollars less than a haircut at Renee’s French Salon, and you can get five haircuts at Joe’s for less than the cost of one haircut at Renee’s, then the price range in dollars, using interval notation, for a haircut at Joe’s is:
A)
B)
C)
D)



0 , 12 .
5

0 ,
0 ,
12 .
5
62 .
5



0 , 62 .
5

E) None of these
8.
The median price of a new home in Prescott Valley in 2004 was $88,000 and in 2007 it was $92,800. If the median price can be modeled by a linear function of the year, then the estimated median price for a new home in 2013 is:
A) $73,600
B) $102,400
C) $104,000
D) $108,800
E) None of these
9.
A function expressing the volume V of a cube as a function of the length of the diagonal d of a face of the cube is:
A) V ( d )
 d
3
4
2
3
B) V ( d )
 d 3
3
C) V ( s )
 d
3
2
4
D) V
 d
3
3
9
E) None of these
10.
If a

2 7 , b

7
5

3
A)
B)
C)
D) b b c a b




 a c a


 c a b c
E) None of these
2
, and c

72 are written from smallest to largest, then the order is:

Use the following for questions 11 and 12:
Sheila charges $49 to give a walking tour of Honolulu to one person. In order to increase her business, she advertised that for each additional person she would lower the price per person for everyone by $1, up to 49 people.
11.
Sheila’s revenue as a function of the number of people, n , on the tour is:
A) R ( n )
  n
2 
49 n
B)
C)
D)
R ( n )
R ( n )
R ( n )


 n
2 
49 n n ( 50
49 ( 49

 n ) n )
E) None of these
12.
The maximum revenue for her tour is:
A) $ 425
B) $ 625
C) $ 650
D) $ 2401
E) None of these
13.
A jar contains nine marbles numbered 1 through 9. Two marbles are randomly selected one at a time without replacement. The probability that the sum of the numbers selected is less than five is:
A)
B)
1
36
4
81
C)
5
72
7
D)
72
E) None of these

14.
The solution of 3

1
3

1
1
3

1 x
is:
A)

1
3
1
B)
3
8
C)
3
D) 24
E) None of these
15.
Where defined, 1

1

1
1
1

1
1

1 x
A) 1
equals:
B)
1 x
5 x

C)
D)
3 x
5 x


2 x
3
2
3

1
E) None of these
16.
If ab

0 , a
 b and a
 b
2
 b
2 
A)
 a
 b
 a

B)
C) b
 a a

D) a
 b b b

2 a
2
E) None of these
17.
If x and y satisfy x
3  y
3 x
 x
 b a

2 , then x equals:

341 and x
 y

11 , then xy equals:
A) 10
B) 18
C) 24
D) 30
E) None of these

18.
For the given histogram the boxplot that represents the same data set is:
A) (1)
B) (2)
C) (3)
D) (4)
E) None of these
19.
The ratio of the angles of a triangle is 1:2:3. If the circle circumscribing the triangle has a radius of
50 cm, then the perimeter of the triangle, in centimeters, is:
A)
B)
C)
D)
25


3

3

25 6

3
50


1

3

50 2

3


E) None of these
20.
For all real numbers, x

5 , the range of f ( x )
 x

5 x

5
is:
A)
B)
C)
D)
 
 


1
1 , 1


0 ,


E) None of these

21.
Using interval notation, the solution set for
A)
B)
C)
D)





4 ,

2




2
4

,
,
,






E) None of these x
2 
4 x

4
 x

6 is:
22.
The number of digits in the product of the first 50 positive-integer powers of 10 is:
A) 1275
B) 1276
C) 2550
D) 2551
E) None of these
23.
If a 500 meter train needs 1 minute to go through a 500 meter tunnel, then the speed of the train in kilometers per hour is:
A) 30
B) 60
C) 100
D) 120
E) None of these
24.
A right circular cone shaped cup is filled with juice. If you drink half of the juice, then the ratio of the depth of the juice that is left in the cup to the height of the cup is:
1
A)
3
B)
1
2
1
C)
2
D)
3
1
2
E) None of these

25.
Where defined, d d
2
3 

64
16
 d d
2
2 
8 d

4 d

16

16
equals:
A) d

4
B)
C)
D)
( d

4 )
2 d
2 ( d


4
4 ) d

4

2 ( d

4 )
2
E) None of these
26.
The area of an equilateral triangle, in square centimeters, with a height of 9 cm is:
A)
B)
27
2
3
81
2
C) 27 3
D) 81
E) None of these
27.
If the ratio of x

2 y to 5 y
 x is 3 to 5 , then the ratio of x to y is:
A) 10 to 7
B) 5 to 8
C) 8 to 5
D) 7 to 10
E) None of these
2
2
28.
If
2
0


2

1
2

2

2 x

2
3
A)

4
B)

3
 
2

1 , then x equals:
C)
4
3
D) 8
E) None of these

29.
Using interval notation, the solution set for 25

7 x

6

18 is:
A)
B)



C)




,
7 ,



,

5
5

7

 

5 ,


D) The empty set
E) None of these
30.
Where defined, x
2
8 x
 y
2
 y
5
 x
equals:
A)
B)
C)
D)
( x
13 x
 y

)( x
5 y
 y )
( x
3 x

5 y
 y )( x
 y )
( x
13 x
 y

)( x
5 y
 y )
( x
3 x
 y

5 y
)( x
 y )
E) None of these
31.
The record R for a certain school for the 400 meter run t years after 1960 is given by
R ( t )

73 .
1

0 .
43 t . The domain of this function is:
A)
B)
C)
D)
 t
 t
 t
 t
|
| t is
| t

| t

0
 t a real number
0

1960


170


E) None of these
18
 x

7

7
 x

10 is: 32.
The sum of the solution(s) of
A)

4
B) 0
C) 3
D) 7
E) None of these

33.
Where defined, 8 x
7 y
9
A) 16 x
8 y
10 xy equals:
B) 16 x
7 y
9
C) x
11 y
13
8 xy
D) xy 8 x
3 y
5
E) None of these
34.
The minimum value of f ( x )

2 x

3

4 is:
A)

8
B)

4
C) 
3
D) 2
E) None of these
35.
A jet flying the 3000 miles from LA to NYC has a 50-mph tailwind. The flight’s point of no return is the point at which the flight time required to return to LA is the same as the time required to continue to
NYC. If the plane’s speed in still air is 750 mph, then the distance, in miles, from LA to the point of no return is:
A) 1300
B) 1400
C) 1500
D) 1800
E) None of these
36.
If f ( x )

16

15 x
and g ( x )

2
2
, x

0 , then the sum of the solution(s) of x f ( x )
 g ( x ) is:
A)

16
15
B) 
4
15
C)
4
15
D)
16
15
E) None of these

37.
If 2 ax 2 
4 ax
 a

1 value of a ( a

0 ) is:

0 has two rational roots and one root is three times the second root, then the
A) 1
B) 2
C) 3
D) 4
E) None of these
38.
Where defined, x
2 x
2


4 x
4

4

A)

1
B) 0 x
2 x
2

5 x
 x


6
6
equals:
C) 1
D) x x


2
2
E) None of these
39.
The number of integral ordered pair solutions to the inequality x
2  y
2 
9 is:
A) 4
B) 20
C) 25
D) 29
E) None of these
40.
If f ( x )


 x
2 x

2

3
1 for x odd for x even
, then f
 f ( 5 )

is:
A)
B) 1
C) 3

1
2
D) 17
E) None of these

41.
Working together at a constant rate each, it takes Jane and Mike 2 hours 55 minutes to paint a room.
Alone, Jane would require 2 more hours than Mike to paint the same room (both painting at a constant rate). The number of hours it would take Mike to paint this room by himself is:
A) 1
B) 4
C) 5
D) 7
E) None of these
42.
If the points (0, 0), (5, 1), (8, 8), (3, 5), (

2 , 10) and (0, 0) are connected sequentially, then the area of the resulting polygonal region is:
A) 36
B) 44
C) 48
D) 52
E) None of these
43.
The value of log
3


4 27
3

 is:
A)

4
B)

1
4
C) 
1
8
D) Does not exist as a real number
E) None of these
44.
A circle with radius 1 is tangent to both sides of a 60 degree angle. A second circle, larger than the first, is tangent to the first circle and to both sides of the angle. The radius of the second circle is:
A) 3
B) 2 3
C) 4
D) 3 2
E) None of these

45.
If the slope of a line perpendicular to the line x
4
 common factors other than

1 , then pq is: y
5

2 is written as p q where p and q have no
A)

20
B)

1
C) 
16
25
D)
25
16
E) None of these
46.
If f ( x )

1
1
 x
for x
 
1 and f ( a )

3 , then f ( 1
 a ) is:
A) 
5
2
B)
C)

2
3
1
8
D)
3
8
E) None of these
47.
If cos x

4
5
and sin x

0 , then tan x equals:
A)
B)

5
3

4
3
C)
D)

3
4
4
5
E) None of these

48.
All the students in Ms Smith’s class are in math club, science club , and/or chess club. Fourteen students are in the math club, 12 are in the science club, and 11 are in the chess club. Two students are in all three clubs, and 12 students are in exactly two clubs. The number of students in Ms Smith’s class is:
A) 19
B) 21
C) 37
D) 39
E) None of these
49.
If 2 sin x

5 cos x , then 58 sin x cos x equals:
A) 10
B) 15
C) 20
D) 25
E) None of these
50.
Joe walks to work and back home along the same path. He walks on level ground at 4 mph, uphill at
3 1
3
mph, and downhill at 5 mph. To get to work, he walks first on level ground and then up a hill. If his total time for the round trip is 2 1
2
hours, then the number of miles he travels to work is:
A) 5
B) 5 1
2
C) 6
D) 6 2
3
E) None of these