Sophomore Olympiad 2006
1. If n is a real number, then 2 n

1
A) 1

2 n equals:
B) 2
C)
D) n
2
2 n

1
E) none of these
2. The 599 th
odd number is:
A) 599
B) 1197
C) 1199
D) 1201
E) none of these
3. The number 3
100
is:
A) even
B) odd
C) prime
D) multiple of 6
E) none of these
4. The domain of the function f ( x )
 x
2 x

1
is
A)
B)
C)
(
(


1 ,
,

0 )
1 ]


( 1 ,
[ 1 ,


)
)
[

1 , 0 )

[ 1 ,

)
D) [

1 , 0 )

( 0 ,

)
E) none of these
5. The range of the function
A) (

,

)
B) [ 0 ,

)
C) [ 4 ,

)
D) [ 5 ,

) f ( x )

4 x
4 
4 x
2 
5 is:
E) none of these

6. The radius of the circle that passes through the points (0,0), (1,0), and (0,1) is equal to:
1
A)
4
1
B)
2
2
C)
2
D) a circle that passes through the given points does not exist
E) none of these
7. The inverse, g ( x ) , of the function f ( x )

7 x
4

1
is:
A) g ( x )

7 x
4

1
B) g ( x )

4
7 x

1
C) g ( x )

7
4 x

1
4
D) g ( x )

4
7 x

E) none of these
1
7
8. If the graph of f ( t )
 ce kt passes through the points (0,1) and (1,1), then f ( 2 ) equals:
A) 1
B) 2
C) e
D)
2 e
E) none of these
9. The sum of the values of m for which the two lines of equations
 mx
 y

4

0 are perpendicular is: mx
 y

1

0 and
A) -1
B) 0
C) 1
D) 2
E) none of these


2 x

1
10. If f
3
2 x

A)
18
18 x

B)
1
C)
1
9 x
2

3

6 x for all x , then
D) impossible to determine
E) none of these f
 
equals:
11. Three coins are drawn at once from a jar containing two half dollars, four quarters, two dimes, and two nickels. The probability of drawing sixty cents is:
3
A)
120
14
B)
120
26
C)
120
28
D)
120
E) none of these
12. The sum of all solutions to the equation log( x

5 )
 log( x

4 )

1 is:
A) -1
B) 1
C) 5
D) the equation has no solutions
E) none of these
13. The sum of the solutions of the equation x
2  x

3
 x

1 is:
A)

2

B) 0
C) 2
D) 1

3
3
E) none of these

14. Let ABC be a right triangle with m

A =
 one has side length AB, square two has side length AC, and square three has side length BC, then:
A) area of square three – area of square two < area of square one
B) area of square three – area of square one > area of square two
C) area of square three – area of square two = area of square one
D) the relation between the areas depends on the lengths of the sides of the triangle
E) none of these
15. The solution set for 1

2 x

1

4 is:
B)
C)
D)
1
 x

5
A)

0

2
2
3

3 x

 x
 x
2
1


0
5
2 or
E) none of these
1
 x

5
2
16. The product of the values of the parameter m for which the line x
 y
 m

0 is tangent to the circle of center (0,0) and radius 2 is:
A) 8
B)

4
C) 2 2

D) m
2
2
4
E) none of these
17. If A
 h
2
( b
1
 b
2
) and h

0 , then b
2
is equal to:
A)
2 A h
 b
1
B)
C)
D)
A
 h b
1 b
1
2
2 A
 hb
1
2 A
 h
E) none of these

18. The solution set for the inequality x x

1

1
A) [

3 ,

1 )
B)
C) [

3 ,

)
D)
(

,

3 ]

[

1 ,

)
(

,

3 ]

(

1 ,

)

2 is:
E) none of these
19. Let m be an arbitrary real number. The solution set of the quadratic inequality
 x
A)
2
(

2

, mx

)

2 m
2 
0 is:
B) (
 m
2
, m
2
)
C) the empty set
D) (
 m

3 ,
 m

3 )
E) none of these
20. The solution set for the inequality x
2  x

1
 x

0 is:
A) the empty set
B) [

1 , 1 ]
C) (

,

)
D) [

1 ,

)
E) none of these
21. The sum of the solutions of the equation 3 x
A) -3
B) 0
C) 1
D) 3
E) none of these

4

3
 x

1 
0 is:

1

2
 x
2

1 is: 22. The solution set for the inequality
 x
2
A) (

, 0 ]
B) [ 0 , 1 ]
C) the empty set
D) [ 0 ,

)
E) none of these

23. Let ABC be a triangle with D a point on the side AB such that m

ABC = m

ACD.
If AD = 2 cm and BD = 8 cm, then AC in cm is:
A) 2 2
B) 4
C) 2 5
D) 6
E) none of these
24. Given three arbitrary points that are not on the same line, then:
A) there is not necessarily a circle that passes through all three points
B) there is more than one circle that passes through all three points
C) there is only one circle that passes through all three points
D) there is only one circle that passes through any two points
E) none of these
25. If 1, -1 , and 1+ i are zeroes of a real polynomial P( x ), then the most accurate conclusion about the degree of the polynomial is:
A) degree = 3
B) degree

3
C) degree = 4
D) degree

4
E) none of these
26. Wal-Mart is having a “25% off the price marked” on items that are already marked
50% off. The total discount on the original price of these items is:
A) 62.5%
B) 65.5%
C) 70%
D) 75%
E) none of these
27. If a polynomial p ( x ) changes its signs five times over

 
,


, then the number of real zeros of p ( x ) is:
A) 4
B) 5
C) 6
D) not enough information to determine
E) none of these

28. Consider a semicircle with diameter AB

4 cm and center O . The line through O perpendicular to AB meets the semicircle at C , and M is the midpoint of OC . The tangent at B to the semicircle meets line AM at D . If line OD crosses the semicircle at E , then the area of the region EDB is:
A)
4
  cm
2
B)
8
2
  cm 2
C
C)
8
4
  cm
2
M
D)
16
2

4
 cm 2
A O
E) none of these
E
B
D
29. Let x denote the gross monthly sales of a small company. If the manager receives a fixed salary of $4000 a month plus a commission rate of 5% on the monthly sales that exceed $50,000, then a function f ( x ) that represents the managers monthly salary is:
A)
B)
C) f ( x )
 f ( x )
 f ( x )

4000
4000

0 .
05 x x x


50 , 000
50 , 000
4000
4000

0 .
05 x x


50 , 000
50 , 000
4000
1500
4000


0 .
05 x
0 .
05 ( x x x


50 , 000
50 , 000

50 , 000 ) D) f ( x )

E) none of these
30. A real polynomial p ( x ) has exactly 4 real zeros and an absolute minimum. A possible equation of p ( x ) is:
A)
B)
C) p ( x ) p ( x )


 x
4 x
5
 ax
 ax
4
3  bx
 bx
3
2  cx
 cx
2
 d
 dx
 e p ( x ) p ( x )


 x
5 x
6
 ax
4
 ax
5
 bx
 bx
4
3

 cx
2 cx
3
 dx

 dx
2 e
 ex

D)
E) none of these f

31. The number of irreducible factors contained in the factorization over the real numbers of x
12 
1 is:
A) 3
B) 4
C) 5
D) 6
E) none of these
32. Two circles of centers A and B and radii 1 cm and 2 cm , respectively, are externally tangent. Let CD be a line segment that is tangent to one circle at C and the other at
D . The area of the region ABCD is:
A)
4

35 cm
2
4
B) 2
2
3 cm
C) 2
2
4 cm
D)
2
6 cm
E) none of these
33. If f ( x )
  x
2 
1 , then f ( x
 h )
 f ( x )
where h

0 is equal to: h
2 x
2 
2 xh
 h
2
A)
B) h

2 x
 h
C) f
D)

2 x
 h
E) none of these
34. The number of distinct 8-letter permutations of the letters in the word PARALLEL is:
A)
8 !
12
B)
C)
8 !
6
8 !

D) !
2 !

3 !
E) none of these

35. The statement “If some Vulcans are Cardasians and some Cardasians are from the delta quadrant, then some Vulcans must be from the delta quadrant” is:
A) true
B) false
C) is such that the validity cannot be determined
D) neither true nor false
E) none of these
36. Let ABC be a triangle with such that BD

DE

EC m

B
 m

C

, then the triangle
30 o . If
ADE
D and
must be:
E are two points on BC
A) isosceles but not equilateral
B) equilateral
C)
D)
45 o
30 o


45 o
60 o


90 o
90 o
E) none of these
triangle
triangle
37. Assume that your grade on algebra tests is inversely proportional to how many hours you sleep the night before the test. Assume also that your grade is 50 % when you sleep 6 hours the night before the test. If you get 12 hours of sleep on the night before the test, then your grade is:
A) 10 %
B) 20 %
C) 25 %
D) 100 %
E) none of these
38. If the area of the circular base of a right circular cone is quadrupled and the resulting cone is similar to the original cone, then the volume of the larger cone is:
A) 2 times the volume of the original cone
B) 4 times the volume of the original cone
C) 8 times the volume of the original cone
D) 16 times the volume of the original cone
E) none of these
39. If it takes ten workers to paint 60 houses in 120 days, then the time it takes five workers to paint 30 houses is:
A) 15 days
B) 30 days
C) 60 days
D) 120 days
E) none of these

40. Let a

0 . If log 2
 a
0 .
301 and log 3
 a
0 .
477 , then log a
 
is:
A) 1 .
723
B) 3 .
857
C)
D)
1
1 .
.
857
857 a

2
2 a
E) none of these
41. Missy is the 50 th best and the 50 in Missy’s class is: th worst student in her class. The number of students
A) 50
B) 99
C) 100
D) 101
E) none of these
42. Given that i
 
1 is a root for the polynomial p ( x )
 x
4 
3 x
3 
3 x
2 
3 x

2 , then the sum of all real roots of p ( x ) is:
A) 0
B) 3
C) 4
D) 5
E) none of these
43. If ABCD is a quadrilateral such that its diagonals are angle bisectors and perpendicular to each other, then the quadrilateral ABCD is:
A) a rhombus
B) a rectangle
C) a square
D) all of the above
E) none of these
44. The remainder of
A)

3
B)

2
C)

1
( 2 x
100
D) 4
E) none of these

3 x
49 
1 )

( x

1 ) is:

45. Let A be the center of two concentric circles with radii of 1 cm and 2 cm, and let the measure of angle BAC be 45 o
. The area of the region between the two circles in the interior of the angle is:
3

A) cm
2
3
8

B) cm
2
C)
D)
3
4

3 cm
2

2
2
E) none of these
46. Let S be the set
A)
B)
C)
D)


   x , y
 
  
|
|
|
|

 y y x x

 x
 y



 y
 x

 x
 x

| y
2  x
2

. Then S equals:
E) none of these
47. The expression


4 x

2
12 x

4 y
2 y

1


3
equals: x
6 y
9
A)
3 x
2 y
3
B)
3 x
6 y
9
C)
27 x
8 y
6
D)
9
E) none of these
48. A breakfast cereal company produces a brand of cereal with a stated net weight of
18 oz .
Only boxes with a net weight within 0 .
02 oz .
of the stated amount are acceptable. Among the following, the box that is of acceptable weight is:
A) a box with a weight of 17 .
00 oz .
B) a box with a weight of 17 .
03 oz .
C) a box with a weight of 18 .
01 oz .
D) a box with a weight of 18 .
03 oz .
E) none of these

49. The graph of y
 x is moved to the right 3 units and lifted up 2 units. The equation of the new graph is:
A) y
 x

3
B)
C) y
 x

2

3 y
 x

3

2
D) y
 x

2

3
E) none of these
50. The average on a math test was 60 . Feeling bad, the teacher raised the grade of every student by 5 points. The new average and standard deviation result in the following:
A) they are the same as the original ones
B) they are 5 higher than the original ones
C) the average is the same, but the standard deviation is 5 higher.
D) the average is 5 higher, and the standard deviation remains the same
E) none of these