Senior Olympiad 2006
1. Where defined, equals:
A)
B)
C)
D)
E) none of these
2. Where defined, the solution set of is:
A)
B)
C)
D)
E) none of these
3. The value of is:
A)
B)
C)
D)
E) none of these
4. Using interval notation, the solution set for is:
A)
B)
C)
D)
E) none of these
5. If , then, for , is:
A)
B)
C)
D) 0
E) none of these
6. A small sphere is contained inside a larger sphere. If the surface area of the small sphere is
and the surface area of the larger sphere is , then the volume in cubic centimeters of the region insidethe larger sphere but outside the small sphere is:
A)
B)
C)
D)
E) none of these
7. The parabola passes through the points and (1, 5). The value of b is:
A)
B) 0
C) 2
D) 4
E) none of these
8. If x + 2 is a factor of , then k is:
A)
B)
C) 2
D) 56
E) none of these
9. The area of the equilateral triangle that is circumscribed about a circle whose diameter has
length 4 is:
A) 6
B) 12
C)18
D)24
E) none of these
10. The domain of the real-valued function defined by is:
A)
B)
C)
D)
E) none of these
11. If , then the sum of the solutions of cos x + sin (2x) = 0 is:
A)
B)
C)
D)
E) none of these
12. As a wheel with a diameter of length 24 inches rotates, a point on the rim of the wheel travels a
linear distance of 240 inches each second. The number of revolutions the wheel makes each
second is:
A)
B)
C)
D)
E) none of these
13. If and , then equals:
A)
B)
C)
D)
E) none of these
14. On a recent trip, a man noted that his car averaged 26 mpg without using the air conditioning
and 23 mpg while using the air conditioning. If 30 gallons of gas were used to travel 720 miles,
then the number of miles the air conditioning was used is:
A) 10
B) 20
C) 260
D) 460
E) none of these
15. Given , the sum of the x- and y-intercepts of the graph of f is:
A)
B) 0
C) 1
D) 2
E) none of these
16. If for all real numbers x, then the range of f is:
A)
B)
C)
D)
E) none of these
17. In a triangle ABC, the midpoints of , , and are D, E, and F, respectively.
If the lengths of BD, BF, and CE are 4, 5, and 6, respectively, then the area of triangle DEF is:
A)
B)
C)
D) 60
E) none of these
18. If for , then the value of k such that the inverse of f will be is:
A)
B)
C) 2
D) 3
E) none of these
19. The slant (or oblique) asymptote for the graph of is:
A)
B) y = x + 1
C) y = x
D) y = x + 3
E) none of these
20. The sum of the solutions to for is:
A)
B)
C) 2
D)
E) none of these
21. The graph of is:
A) a circle
B) an ellipse
C) a hyperbola
D) a parabola
E) none of these
22. The value of is:
A)
B)
C)
D)
E) none of these
23. A regular 12-gon is inscribed in a circle of radius 12. The area of the 12-gon is:
A) 36
B)
C) 432
D)
E) none of these
24. If the zeros of the graph of are , 1 and 5, then is:
A)
B)
C)
D) 11
E) none of these
25. In triangle ABC the lengths of and are 2 and 3, respectively. If the measure of
angle ABC is , then the length of is:
A)
B)
C)
D)
E) none of these
26. The center of a circle lies on the graph of where k > 0. If the circle is tangent to
the y-axis, then the length of the segment of the x-axis that is intercepted by this circle is:
A)
B) k
C) 2k
D)
E) none of these
27. There are 1000 students at a certain high school. Eighty percent of the student body is involved in
some type of athletics. Ten percent of the student body is not involved in athletics or math club.
The number of students that are not involved in athletics but are involved in math club is:
A) 0
B) 100
C) 200
D) 300
E) none of these
28. Ally Gator and Ryan Ocerus play for different teams in the Animal Baseball League. In games
played on natural grass, Ally hits safely in 30% of her at-bats and Ryan hits safely in 28% of his
at-bats. In games played on AstroTurf, Ally hits safely in 35% of her at-bats and Ryan hits safely
in 33% of his at-bats. Ally ended the season with 220 at-bats on grass and 80 at-bats on
AstroTurf, while Ryan ended the season with 100 at-bats on grass and 200 at-bats on AstroTurf.
If we subtract the percentage of hits for Ryan from the percentage of hits for Ally, the result is:
A) –2%
B) 0%
C) 0.2%
D) 2%
E) none of these
29. A teacher decides to divide 10 students into two groups of 5 each. The number of possible
distinctdivisions is:
A) 126
B) 252
C) 504
D) 30240
E) none of these
30. Suppose that a person chooses a letter at random from the word RESERVE and then chooses a
letter at random from the word VERTICAL. The probability that the same letter was chosen from
each word is:
A) 0
B)
C)
D)
E) none of these
31. Suppose that Urn A contains 3 white balls and 5 red balls, and Urn B contains 4 white balls and
6 red balls. A six sided die is rolled, and if a 1 or a 2 is rolled, a ball is selected at random from
Urn A; otherwise, a ball is selected at random from Urn B. If we know that a white ball was
selected, thenthe probability that it came from Urn A is:
A)
B)
C)
D)
E) none of these
32. If and , then the sum of all of the values of and that satisfy both
and is:
A)
B)
C)
D)
E) none of these
33. If and , then is:
A) 0
B)
C)
D) 1
E) none of these
34. If and one of the three solutions is 2, then the sum of the unspecified
coefficient k and the other two solutions is:
A)
B) 0
C) 1
D) 20
E) none of these
35. If and , then the sum of the solutions is:
A)
B)
C)
D)
E) none of these
36. The negation of the conditional statement “If , then ” is:
A) If , then
B) If , then
C) and
D) and
E) none of these
37. The sum of the values of k such that the vector is a unit vector is:
A)
B)
C) 3
D) 17
E) none of these
38. The distance between the two parallel lines given by and is:
A)
B)
C) 2
D) 12
E) none of these
39. Consider the set . The
number of elements in is:
A)3
B) 7
C) 10
D) 21
E) none of these
40. If , then x is:
A) 2
B) 3
C) 4
D) 5
E) none of these
41. If a wheel with a radius of 14 inches makes revolutions per mile and a wheel with a radius
of 15 inches makes revolutions per mile, then is:
A)
B)
C)
D)
E) none of these
42. If , , and x is in the fourth quadrant, then is:
A)
B)
C)
D)
E) none of these
43. The number of points of intersection of the graphs of and
is:
A) 0
B) 1
C) 2
D) 3
E) none of these
44. If , then is:
A)
B)
C) 23
D) 90
E) none of these
45. The value of k such that the point lies on the line containing the points and
is:
A)
B)
C) 1
D) 2
E) none of these
46. An equation for a line perpendicular to is:
A)
B)
C)
D)
E) none of these
47. The value of a such that the graph of contains the point is:
A)
B)
C) 8
D) 16
E) none of these
48. The manager of a Starbucks store decides to experiment with a new blend of coffee. She will mix
some B grade Colombian coffee that sells for $5 per pound with some A grade Arabica coffee
that sells for $10 per pound to get 100 pounds of the new blend. If the selling price of the new
blend is to be $7, and there is to be no difference in revenue from selling the new blend versus
selling the other types, then the number of pounds of grade B Columbian coffee required is:
A)30
B)40
C)50
D)60
E)none of these
49. Mary is four fifths as old as Jane. Ten years ago, Mary was three fifths as old as Jane was then.
The sum of the ages in years of Mary and Jane in 20 years is:
A)58
B)64
C)76
D)85
E)none of these
50. The base of a regular pyramid is a regular hexagon whose perimeter is 36 centimeters. If the
volume of this regular pyramid is 216 cubic centimeters, then the surface area of the pyramid,
in square centimeters, is:
A)
B)
C)
D)
E) none of these