Download 2007 Exam

Junior Olympiad
1. If f x   2 x 3  Ax 2  Bx  5 , f 2  3 , and f  2  37 , then the value of
A  B is
A) -14
B) -1
C)
1
2
D) 20
E) none of these
 4  x if x  2
2. If f x   
, then the value of  f  1  f 4 is
 x  2 if 2  x
A) -6
B) 0
C) 6
D) 10
E) none of these
3. The coordinates of the point on the line 3x  2 y  0 that is equidistant from 0,0
and  2,3 is
3

A)  ,1
2

 3
B) 1, 
 2
3 
C)  ,1
2 
3

D)   1, 
2

E) none of these
4. If the point 2,5 is at a distance of 65 from the midpoint of the segment
joining 4,2 and x,4 , then the value of x is
A)  2
B) 2
C)  2
D)  2
E) none of these
5. Using interval notation, the solution set for the inequality
A)  6,4
x
 3 is
x4
B)  ,4  6, 
C)  6,4
D)  ,4  6, 
E) none of these
6. Using interval notation, the solution set for 6  3x  7  10 is
 11
A) 1, 
 3
B)  , 
17 

C)   , 
3

11 
D)  ,1   ,  
3

E) none of these
7. Given the arithmetic sequence 5, 16, 27, 38, 49, …, the 138th number in the
sequence is
A) 1372
B) 1378
C) 1512
D) 1518
E) none of these
8. A student tosses a 6-sided die twice. The probability that the sum is odd and no
more than 5 is
1
A)
36
B)
1
9
C)
1
6
D)
1
2
E) none of these
9. A student draws two cards without replacement from a regular fair deck. The
probability that the second is black, given that the first is a spade is
1
A)
25
B)
25
52
C)
25
51
D)
1
2
E) none of these
10. If n A  54 , n A  B  22 , n A  B  85 , n A  B  C   4 , n A C   15 ,
nB C   16 , nC   44 , and nB'  63 , then nB  C  is
A) 39
B) 81
C) 108
D) 118
E) none of these
11. If n is any real number, then 2 n  2 n is equal to
A) 2 n 1
B) 4 n
C) 2 2 n
D) 4 2 n
E) none of these
12. The number of times the digit nine occurs in the numbers from 1 to 100 is
A) 10
B) 12
C) 19
D) 20
E) none of these
13. The equation of a line is given by x  y  5 . The acute angle that the line makes
with the x -axis is
2
A)
2
B)

4
C)

3
D)

2
E) none of these
14.
2 4 3 is equal to
A)
8
6
B)
6
6
C)
5
6
D)
4
12
E) none of these
15. If n is any real number, then the number below that is not a perfect square is
A) 2 2 n
B) 9 n
C) 5 6 n  2
D) 312 n
E) none of these
16. The number of real solutions of x 3  3 x  1 is
A) none
B) one
C) two
D) three
E) none of these
17. If h  0 and f x   x  1 , then
A)
2h
h
B)
 2x  h  2
h
C)  1
D) 1
E) none of these
f x  h   f x 
is equal to
h
x2
2
18. Where defined, the expression x  4
is equal to
2x  2
1
x
x4
A)
4
B)
C)
D)
x
 2 x  4 x
x  4x  2
2
x2  2
8
x  4x
x4
E) none of these
19. The graph that represents a population of a city that increases at a decreasing rate
is
C)
A)
D)
B)
E) none of these
20. Let M be the midpoint of AB in a triangle ABC with mA  25 . If
CM  MB , then mB is equal to
A) 45
B) 55
C) 65
D) 75
E) none of these
21. United States imports increased by 20% and exports decreased by 10% during a
certain year. The ratio of imports to exports at the beginning of the year was x.
The ratio of imports to exports at the end of the year is
11
x
A)
8
B)
4
x
3
C)
3
x
2
D) 2x
E) none of these
22. Let ABC be an isosceles triangle with base BC . Let BM and CN be the
perpendiculars from B and C to AC and AB , respectively. If AB  5 cm and
AM  4 cm , then NC is
A) 3 cm
B)
10 cm
C) 4 cm
D) 5 cm
E) none of these
23. Let ABC be an isosceles triangle with base BC . If BH is the perpendicular
from B to AC , then CBH is congruent to
A)  A
B)
A
2
C)
A
3
D) not enough information to tell
E) none of these
24. Let f be a function over the negative real numbers. If f
then f x  is
1
A)
x
1
x   1
x
for all x  0 ,
B) x
C)  x
D) 
1
x
E) none of these
25. If you can move your finger up, down, left, or right only, then the number of 4digit telephone number extensions that start with 2 that you can dial is
A) 1800
B) 2000
C) 2800
D) 10000
E) none of these
26. The solution set for the inequalities 4  x 2  9 is
A) 2  x  3
B)  3  x  3
C)  3  x  2 2  x  3
D)  3  x  2 x  3
E) none of these
27. The number of different vertical arrangements of 8 flags (4 whites, 2 blues and 2
reds) is
A) 16
B) 20
C) 70
D) 420
E) none of these
28. If x  1 and y  2 , then
A) xy  2
B)
x 1

y 2
C) y  x  1
D) x  y  1
E) none of these
29. If x  0 , then 413 log2
A) 4
x
equals
3
x2
B) 12 x
C) 6x 2
D) 4x 3
E) none of these
30. If k is a real number, then the set of values of k for which the graph of x 2  kx
does not intersect the graph of y  1 is
A)  2  k  2
B) k  2 k  2
C) k  1
D) the empty set
E) none of these
31. Let C be a semi-circle centered at the origin O and diameter AB  4 cm. Let P
be a point in the second quadrant on C . The arc AP , for which the area of
OPB is 3 cm 2 , has length (in cm)

A)
4
B)

3
C)
2
3
D)
5
6
E) none of these
32. The price P (in dollars) and the quantity x sold of a certain product obey the
demand equation Px  5x  160 , 0  x  32 . The price that maximizes the
company’s revenue is
A) $16
B) $24
C) $19.2
D) $32
E) none of these
33. The faculty at a college consists of 80 full-time teachers and 52 part-time
teachers. Of the 80 full-time teachers, 40 are female. Of the 52-part time teachers,
26 are female. The probability that a randomly selected teacher is male or works
part time is
13
A)
66
B)
1
2
C)
23
33
D)
59
66
E) none of these
34. The expression
A)
B)
C)
49  x 
1
2 2
 3x 49  x
49  x 2
2
2


1
2
equals
3x 2
49  x 
3
2 2
49  2 x 2
49  x 
3
2 2
49  x 2
49  x 
3
2 2
3x 2
D)
49  x 2
E) none of these
35. If the probability of having a girl or a boy is the same, then the probability of a 4child family having 4 girls is
1
A)
32
B)
1
16
C)
1
8
D)
1
4
E) none of these
3
7
36. The expression 13 x  9 x
5
8
 

A) x 7 13 x 7  9 x 



5
7
equals
3
8
8


B) x 7 13 x 7  9 x 9 


8


C) x 13 x 7  9 



5
7
8



D) x 13  9 x 9 


1
3
E) none of these
37. Missy can wallpaper 2 rooms in a new house in 10 hours. Together with her
trainee they can wallpaper the 2 rooms in 5 hours. By herself, the trainee can do
the job in
A) 10 hours
B) 12 hours
C) 22 hours
D) 24 hours
E) none of these
38. The domain of the real-valued function g x  
A) 49, 
B)  7,7
C)  ,7  7, 
D) all real numbers except 7 and -7
E) none of these
x
x 2  49
is
39. The sum of the solution(s) to the equation log 4 x  4  log 4 x  2  2 is
A)  6
B)  2
C) 4
D) 5
E) none of these
40. In a survey of 462 computer buyers, 256 put price as a main consideration, 211
put performance as a main consideration, and 56 listed both price and
performance. The number of computer buyers who listed performance only as a
main consideration is
A) 51
B) 107
C) 155
D) 211
E) none of these
41. The owners of a candy store want to sell, for $6 per pound, a mixture of
chocolate-covered raisins, which sells for $3 per pound, and chocolate-covered
macadamia nuts, which sells for $8 per pound. They have a 40-pound barrel of
raisins. The number of pounds of the nuts they should mix with the barrel of
raisins so that they hit their target value of $6 per pound for the mixture is
A) 52
B) 56
C) 60
D) 64
E) none of these
42. If f  x  
f
1
x3
and  x   2
, then g x is
x
x x
g
A)
x 1
x3
B)
x3
x 1
C)
x 1
x3
D)
x3
x 1
E) none of these
43. If f  x  
1
, then f  f  f x is
x
A)
1
x
B)
1
x3
C) x
D)
1
x2
E) none of these
44. If x100  x  1 is divided by x  1 , then the remainder is
A)  1
B) 1
C) 2
D) 3
E) none of these
45. The value of m that makes the two lines of equations 3x  y  2 and mx  2 y  1
perpendicular to each other is
A) 
2
3
B) 
1
3
C)
1
3
D)
2
3
E) none of these
46. The vertex of the parabola that passes through the points 0,1, 1,0, 2,3 is
 3 1
A)  , 
 4 8
 3 
B)   ,10 
 2 
C) 1,1
 1
D) 1, 
 2
E) none of these
47. A value of m that makes the graph of f x   x 4  m 2  1 cross the
x-axis is
A)  4
B)  1
C) 0
D) does not exist
E) none of these
48. A value of m that makes the graph of f x   x 3  x  m have no x-intercepts is
A) 0
B) 1
C) 100
D) does not exist
E) none of these
49. Among the following, the one that is a true identity is
x4  x  2
A)
B) ln x  y   ln x  ln y
C) e x  y  e x  e y
D)
1
1 1
 
x y x y
E) none of these
50. If f 2x  1  x  4 for all x, then f x  is
A) 5
B)
x 1
2
C)
x2
2
D)
x9
2
E) none of these