Sophomore Olympiad 2005

Sophomore Olympiad 2005

1.Where defined, is equal to:

A)

B)

C)

D)

E) none of these

2. If x ≠ 0, then is equal to:

A)

B)

C)

D)

E) none of these

3.If u 2, then (u – 2)-4 (u – 2)-3is equal to:

A)

B) (u-2– 4u + 4)-7

C)

D) (u2 + 4u – 4)7

E) none of these

4.If r 1, S L, and S = , then r is equal to:

A)

B)

C)

D)

E) none of these

5.There are 80 students in a class. If the 75th percentile on a test in that class was 80, then:

A) there were 75 students that scored 80

B) there were 5 students that scored 80 or above

C) there were 60 students that scored 80 or above

D) there were 20 students that scored 80 or above

E) none of these

6.A group of 114 people was polled about a contest between Freddy and Jason. If the result of the poll was two to one in favor of Jason, then the number of people in favor of Jason in that group was:

A) 2

B) 38

C) 57

D) 78

E) none of these

7.The diagram below is the side view of a staircase that is 4 steps high. This staircase is made by stacking 10 blocks as shown. The number of blocks that would be used to make a similar staircase 12 steps high is:

A) 24

B) 76

C) 77

D) 78

E) none of these

8. The non-terminating decimal number is equal to:

A)

B)

C)

D)

E) none of these

9. The simplified form of is:

A)

B)

C)

D) 36

E) none of these

10. Let ABCD be a rectangle with sides of measures 3 and 4 meters. The diagonals intersect at a point O. Let M be a point on andlet N be the point where intersects. If the length of is 2 meters, then the length in meters of is:

A)

B)

C)

D)

E) none of these

11.Let ABC be a right triangle. LetM be the midpoint of the hypotenuse. Ifthe measure of angle BAM = 20, then the measure in degrees of angle AMC is:

A) 20

B) 30

C) 40

D) 90

E) none of these

12.The sum of all solutions to |x – 10| = x2 – 10x is:

A) 0

B) 9

C) 10

D) 20

E) none of these

13.A group of farmers agree to share equally in the cost of a $48000 piece of machinery. If they could find two more farmers to join the group, then each person’s share of the cost would decrease by $4000. The number of farmers that are presently in the group is:

A) 3

B) 4

C) 5

D) 6

E) none of these

14.In October, a company’s total profit was 12% more than it was in September. If the total profit for the two months was $689000, then the difference between the October profit and the September profit is:

A) $12000

B) $39000

C) $68700

D) $73821

E) none of these

15.Two integers from 1 through 60 are chosen at random. The probability that the same integer is chosen twice is:

A)

B)

C)

D)

E) none of these

16.The solution set ofx2 + 4x < 2x is:

A) (-2, 2)

B) (-  , -2)  (2, )

C) (-2, 0)

D) (- , )

E) none of these

17.The solution set of  is:

A) (-, 3]

B) (-1, 3]

C) [-1, 1]

D) (-1, 1]

E) none of these

18.A small college needs four faculty members: a mathematician, two chemists, and an engineer. If there are eleven applicants for the positions, two mathematicians, six chemists, and three engineers, then the number of ways these positions can be filled is:

A) 36

B) 90

C) 180

D) 330

E) none of these

19. If the length of a rectangular field is increased by 5 yards and its width is increased by 10 yards, then its area is increased by 450 square yards. If the length the field is increased by 5 yards and its width diminished by 10 yards, then its area is diminished by 350 square yards. The length of the field in yards is:

A) 10

B) 35

C) 40

D) 45

E) none of these

20. Two numbers are in the ratio 3:4 and the ration of their sum to the sum of their squares as 7:50. The numbers are:

A) 3 and 4

B) 6 and 8

C) 7 and 25

D) 14 and 100

E) none of these

21. If Harry gave Sally $100, Harry would have half as much money as Sally. If Sally gave Harry $100, then Sally would have one third as much as Harry. The original amount in dollars Sally had is:

A) 20

B) 40

C) 120

D) 220

E) none of these

22. If , then y equals:

A)

B)

C)

D) 2

E) none of these

23. In a box that is 3 inches deep and 6 inches wide, a wire 1 foot long can be stretched to reach from one corner to the diagonally opposite corner. The length of the box in inches is:

A) 3

B) 9

C) 3

D) 3

E) none of these

24. A square pyramid has a base of 4 feet on a side and a slant height of 18 feet. The surface area of the pyramid in square feet is:

A) 144

B) 160

C) 292

D) 304

E) none of these

25. Two right triangles have bases of 15 inches and 21 inches and hypotenuses of 25 inches and 35 inches respectively. The ratio of the area of the smaller triangle to the area of the larger triangle is:

A)

B)

C)

D)

E) none of these

26. Seven lines, no three of which lie in the same plane, pass through the same point. The number of planes determined by the lines is:

A) 7

B) 21

C) 35

D) 5040

E) none of these

27. If the circumference of the base of a right circular cylinder is 8π units and the total surface area is 96πsquare units, then the volume in cubic units is:

A) 160π

B) 256π

C) 512π

D) 768π

E) none of these

28. Where defined, is equal to:

A)

B)

C)

D)

E) none of these

29. The domain of the real-valued function is:

A) [-2, 2]

B) (-∞, 2]

C) [2, ∞)

D) (-∞, -2] [2, ∞)

E) none of these

30. If and , then is equal to:

A)

B)

C)

D)

E) none of these

31. Iffor all real values, then the range is:

A) (-∞, ∞)

B) [-1, ∞)

C) [2, ∞)

D) [1, ∞)

E) none of these

32. Let be an acute triangle with BC = 2cm. Let and be the two heights from C and B, respectively. If BD = 1cm, then the measure in degrees of angle DEAis:

A) 30

B) 45

C) 60

D) 75

E) none of these

33. A rectangle ABCD is divided into three congruent squares as shown. Given that triangles AHG and CHA are similar, then the sum of the measures in degrees of angles DCA and DGA is:

A) 20

B) 30

C) 45

D) 60

E) none of these

34. Let where k is a positive integer. Let (M, 0) and (N, 0) be the

x-intercepts of, when they exist. The sum of the values of k for which M and N are integers is:

A) 6

B) 7

C) 10

D) 14

E) none of these

35. Where defined, is equal to:

A)

B)

C)

D)

E) none of these

36. The solution to is:

A)

B)

C)

D)

E) none of these

37. The solution to is:

A)

B)

C)

D)

E) none of these

38. If and , then is:

A) -8

B) 0

C) 1

D) 5

E) none of these

39. If and , then is:

A) -98

B) 53

C) 74

D) 98

E) none of these

40. The solution to is:

A) 0

B) 1

C)

D)

E) none of these

41. The sum in degrees of the measures of the angles of a triangle and a pentagon is:

A) 180

B) 360

C) 540

D) 720

E) none of these

42. The sum of the solution(s) of is:

A) 2

B) 4

C) 9

D) 13

E) none of these

43. Tommy, weighing 90 pounds, sat down 3 feet from the center of a seesaw. Susan, weighing 72 pounds, balanced Tommy by sitting on the other end of the seesaw. The distance in feet that Susan sat from the center is:

A) 2.3

B) 2.4

C) 3.5

D) 3.75

E) none of these

44. Given the rhombus ABCD, the one statement that is not always true is:

A)

B)

C)

D)

E) none of these

45. Given a right triangle ABCinscribed in a circle with right angle at C and AB = 10, the area of the circle is:

A) 10π

B) 20π

C) 25π

D) 100π

E) none of these

46. Onelinear factor of is:

A)

B)

C)

D)

E) none of these

47. An equation of the line passing through (-4, 2) and perpendicular to the line y – 4x = 5 is:

A) y = -4x + 2

B) y = .25x + 2

C) y = .25x +5

D) y = 4x + 5

E) none of these

48. Where defined, is equal to:

A)

B)

C)

D)

E) none of these

49. The remainder when is divided by 7 is:

A) 1

B) 3

C) 4

D) 6

E) none of these

50. The vertex of the graph of is:

A) (3, 22)

B) (3, 4)

C) (6, -5)

D) (-6, -77)

E) none of these