Download 2007 Exam

Senior Olympiad
2007
 x 2  1 for x  4
1. If f ( x)  
, then f  f  2 is:
6
x

7
for
x

4

A)
B)
C)
D)
E)
–19
5
23
362
none of these
2. If f ( x)  2 x  1  1 for all real numbers x, then the range of f is:
A)
B)
C)
D)
E)
 ,  1
 ,  1
 1, 
 , 
none of these
3. The vertex of the parabola defined by y  2 x 2  x  1 is:
9
 1
A)   ,  
8
 4
9
1
B)  ,  
8
4
15 
 1
C)   ,  
 4 16 
17 
 1
D)   ,  
 4 16 
E) none of these
4. If x 4  x 3  x 2  1 is divided by x 2  1 , then the remainder is:
A)
B)
C)
D)
E)
1
 x 1
x 1
x2  x 1
none of these
5. The unit’s digit of 1!  2!  3!  4!  5!  6!  7!  8!  9!  10!
is:
1
A)
B)
C)
D)
E)
2
3
4
5
6
7
8
9
10
0
1
5
7
none of these
6. The area of the triangle bounded by the graphs of 4 x  3 y  0, 3x  4 y  25 and
y  0 is:
A)
B)
C)
D)
E)
25
3
75
8
25
2
50
3
none of these
7. If A  2007 2007  2007 2007 and B  2007 2007  2007 2007 , then A 2  B 2 is:
A)
B)
C)
D)
E)
1
0
1
2
none of these
8. The coordinates of the vertices of a rectangle are (1, 1), (5, 1), (1, 7), and (5, 7). If a
point P is randomly selected within the interior of the rectangle, then the probability
that the x-coordinate of P is greater than the y-coordinate of P is:
A)
B)
C)
D)
E)
1
4
1
3
1
2
1
none of these
9. If the sum of two numbers is 50 and the product of the same two numbers is 25, then
the sum of the reciprocals of these two numbers is:
A)
B)
C)
D)
E)
3
50
2
75
1250
none of these
10. The area of the polygon bounded by the graphs of 3x  4 y  6, 4 x  3 y  18, x = 0,
and y = 0 is:
A)
B)
C)
D)
E)
3
4
5
6
none of these
11. If A  B  C , C  D  E , E  A  F , B  D  F  100 , and A  8 , then F is:
A)
B)
C)
D)
E)
46
48
56
58
none of these
2
1
12. The sum of the solutions of x 3  6 x 3  7 is:
A)
B)
C)
D)
E)
6
6
342
343
none of these
13. The negation of “If x  0 , then x 2  0 ” is:
A)
B)
C)
D)
E)
If x  0 , then x 2  0 .
If x 2  0 , then x  0 .
x  0 or x 2  0 .
x  0 and x 2  0 .
none of these
14. The set of all (x, y) such that 2 x 2  y 2  4 x  4 y  1 is:
A)
B)
C)
D)
E)
the interior of a circle
the exterior of a circle
the interior of an ellipse
the exterior of an ellipse
none of these
15. The conic section defined by x 2  2 x  4 y 2  1 is:
A)
B)
C)
D)
E)
a circle
a parabola
a hyperbola
an ellipse
none of these
16. Let i   1 . If 1 3i is a zero of f ( x)  x 4  3x 3  6 x 2  2 x  60 , then the sum of
all the real zeros of f (x) is:
A)
B)
C)
D)
E)
 60
3
1
3
none of these
17. The number of solutions of ln x  2  ln 2x  3  2 ln x is:
A)
B)
C)
D)
E)
0
1
2
3
none of these
18. If there are 100, 1000, and 10,000 elements in sets A, B, and C, respectively; two
elements common to each pair of sets; and one element common to all three sets, then
the number of elements in A  B  C is:
A)
B)
C)
D)
E)
1
11,093
11,107
11,109
none of these
19. If 400 books are placed onto thirteen shelves, then at least one shelf must have at
least:
A)
B)
C)
D)
E)
31 books
32 books
33 books
34 books
none of these
20. The number of subsets of {a, b, c, d, e} that do not contain the letter a is:
A)
B)
C)
D)
E)
15
16
31
32
none of these
21. A fair coin is flipped eight times where each flip comes up either ‘heads’ or ‘tails’.
The number of possible outcomes that contain exactly three heads is:
A)
B)
C)
D)
E)
5
8
56
336
none of these
22. The number of permutations of the letters A, B, C, D, E, F, and G that contain the
strings ABC and DE is:
A)
B)
C)
D)
E)
24
35
840
5040
none of these
23. Where defined, the expression cot 2   sin 2   csc 2  equals:
A)
B)
C)
D)
E)
sin 2 
 sin 2 
cos 2 
 cos 2 
none of these
24. The sum, in degrees, of the solutions of 2 sin 2   3 cos  in the interval
0    360 is:
A)
B)
C)
D)
E)
60
120
240
360
none of these
25. The area of a parallelogram with an interior angle of measure 150 and two sides of
length 8 and 26 is:
A)
B)
C)
D)
E)
52
104
104 3
208
none of these
26. If the points  3, 22 , 1, 6 , and 0, 1 lie on the graph of y  ax 2  bx  c , then
a  b  c equals:
A)
B)
C)
D)
E)
2
6
27
29
none of these
27. The sum of the solutions of x  1  5x  1  6  0 equals:
4
A)
B)
C)
D)
E)
7
5
0
2 2  3 2
none of these


2
x  y  1
28. If k is a constant such that 
has no solution (x, y), then k is:
kx  2 y  2
A)
B)
C)
D)
E)
1
0
1
2
none of these
29. The domain of definition of the real-valued function defined by f ( x) 
A)
B)
C)
D)
E)
 ,  4  4, 
 ,  4  4, 
 4, 4  4, 
 4, 4
x 2  16
is:
x4
none of these
16
cm arc on a circle of radius 16 cm,
3
then the smaller angle, in degrees, between the tangent lines is:
30. If tangent lines are drawn from each end of a
A)
B)
C)
D)
E)
30
60
90
120
none of these




31. The sum of all solutions of sin  x    sin  x    1 in the interval 0    2 is:
3
3


A)
B)
C)
D)
E)

4

3

2
7
6
none of these
32. The sum of the solutions of sin   3 cos   1 in the interval 0    2 is:
A)
B)
C)
D)
E)

6

2
5
3
7
3
none of these
33. If i   1 , then 1  i  is:
12
A)
B)
C)
D)
E)
 64
 2
1
64i
none of these
2
2 

 i sin
34. If i   1 , then the sum of the square roots of 4 cos
 is:
3
3 

A) 2  2 3i
B)
C)
D)
E)
 2  2 3i
0
2
none of these
35. The number of nonempty bit strings of length five or less is:
A)
B)
C)
D)
E)
32
62
64
128
none of these
36. The number of functions from {1, 2, 3, 4} into {a, b, c, d} that do not map 1 onto a is:
A)
B)
C)
D)
E)
18
256
456
512
none of these
37. The number of functions from {a, b, c, d} into {e, f, g, h} that are not one-to-one is:
A)
B)
C)
D)
E)
24
64
232
256
none of these
 x2 y2

1

4
9
38. The number of solutions of 
is:
2
2
 x  4  y  1
 4
16
A)
B)
C)
D)
E)
0
1
2
4
none of these
39. The sum of the x and y - coordinates of the point on the graph of
x  12   y  22  1 that is closest to the graph of x  7 2   y  62  4 is:
A)
B)
C)
D)
E)
2
5
6
51
5
10
none of these
9
40. The number of real numbers x for which
x
i
 1 is:
i 1
A)
B)
C)
D)
E)
0
1
2
3
none of these
19
41. The value of
2
i
is:
i 1
A)
B)
C)
D)
E)
2 20  1
1  2 18
1  2 19
1  2 20
none of these
42. If f (3x  1)  4 x  2 for all x, then f (x) is:
A) x  3
4x  3
B)
3
C) 7 x  3
4x  3
D)
10
E) none of these
43. The sum of all the coefficients in the expansion of x  2 y 
100
A)
B)
C)
D)
E)
– 2000
– 200
200
2000
none of these
is:
 x  1 for x  0
44. If f ( x)  
, then f  f (x) is:
2 x  1 for x  0
 x  2 for x  1

A) 2 x  3 for  1  x  0
4 x  3 for x  0

 x 2  2 x  1 for x  0
B)  2
4 x  4 x  1 for x  0
 x  2 for x  0
C) 
4 x  3 for x  0
2 x  1 for x  0

D) 4 x  2 for 0  x  1
4 x  1 for x  1

E) none of these
45. If tan x 
3
and sin x  0 , then cos x equals:
4
4
5
3

5
3
5
4
5
none of these
A) 
B)
C)
D)
E)
46. If f ( x)  x 2 and h  0 , then
A)
B)
C)
D)
E)
h
2a  h 2
f
2a  h
none of these
f ( a  h)  f ( a )
is equal to:
h
2
47. If the x-intercepts of the graph of y  x  (a  b) x  ab are  5, 0 and 2, 0 ,
then a  b is:
A)
B)
C)
D)
E)
7
3
3
7
none of these
48. The expression
A)
B)
C)
D)
E)
123456790
123456789
2
 123456789 123456791
equals:
0
1
123456789
987654321
none of these
49. If x and y are positive integers such that y is the hundred’s digit of the six-digit integer
2
492y04 and 3x  690  492 y 04 , then x + y is:
A)
B)
C)
D)
E)
6
12
14
15
none of these
50. A simple polygonal region is constructed by sequentially connecting the points
0,  2, 4,  3 , 6, 3 , 1, 1 ,  2, 2,  4,  3 , and 0,  2 . The area of the
region is:
A) 24
67
B)
2
C) 48
D) 67
E) none of these