1. an Equation of the Line That Passes Through the Points and Is

Junior Olympiad

1. An equation of the line that passes through the points and is:

A) 

B) 

C) 

D) 

E) None of these

2. Using interval notation, the solution of is:

A) 

B) 

C) 

D) 

E)  None of these

3. If x and y satisfy and , then equals:

A) 

B) 

C)  3

D)  7

E)  None of these

4. If , then equals:

A)

B)

C)

D) 1

E) None of these

5. If 121! is written as a product of primes, then the total number of times 11 would appear as a factor is:

A) 2

B) 11

C) 12

D) 22

E) None of these

6. The value for c in the equation such that the product of the roots is is:

A) 

B) 

C)  4

D)  8

E)  None of these

7. If , then is:

A) 

B) 

C) 

D)  undefined

E)  None of these

8. If the points , , and lie on the graph of , then equals:

A) 

B) 

C)  5

D)  189

E)  None of these

9. Given , the sum of and is:

A) 

B)  29

C)  31

D) 

E)  None of these

10. If the sum of the lengths of the diagonals of a rhombus is 10 cm and the positive difference of their lengths is 2 cm, then the area of the rhombus is:

A) 3 cm2

B) 12 cm2

C) 24 cm2

D) 48 cm2

E) None of these

11. The sum of the solution(s) of equals:

A) 

B) 

C)  5

D)  13

E)  None of these

12. If and , then is equal to:

A) 

B) 

C) 

D) 

E)  None of these

13. If the circle circumscribing an equilateral triangle has a radius of length 10 cm, then the area, in square centimeters, of the equilateral triangle is:

A) 50

B)

C)

D)

E) None of these

14. The distance from the point (2, 4) to the line is:

A) 

B)  6

C) 

D)  7

E)  None of these

15. The number of solutions of is:

A)  0

B)  1

C)  2

D)  3

E)  None of these

16. If the Imagine Cookies store (where ) sells cookies for $(1 – i) each, then a box of 3 + 3i cookies would cost:

A) $(3 – 3i)

B) $(6 + 6i)

C) $ 6.00

D) $ 9.00

E) None of these

17. If x, y, and z are positive integers, then the number of ordered triples (x, y, z) that are solutions to the equation is:

A) 21

B) 27

C) 31

D) 120

E) None of these

18. If the three lines ,, and are concurrent, then m is equal to:

A)

B)

C)

D)

E) None of these

19. On a certain test, the average score for the boys in the class is 83 while the average score for the girls in the class is 71. If the average score of all the students in the class is 80, then the percentage of the class that is boys is:

A) 25%

B) 33%

C) 66%

D) 75%

E) None of these

20. A box contains 64 same-size gumballs. Each gumball is one of 8 colors and there are 8 gumballs of each color. If the gumballs are thoroughly mixed and you randomly choose two of them without replacement, then the probability that these two balls will have the same color is:

A)

B)

C)

D)

E) None of these

21. If a rectangle has a width of 15 cm and a length of 20 cm, then the length, x, of the segment from a vertex perpendicular to the diagonal shown is:

A) 10 cm

B) 12 cm

C) 15 cm

D) 24 cm

E) None of these

22. If is factored into two quadratic expressions with integer coefficients, then the sum of all the coefficients of the quadratics is:

A) 7

B) 13

C) 18

D) 19

E) None of these

23. In a group of 50 girls, each girl is either blonde or brunette and each has either blue or brown eyes. If 14 girls are blue-eyed blondes, 31 are brunettes, and 18 have brown eyes, then the number of brown-eyed blondes is:

A) 5

B) 13

C) 18

D) 19

E) None of these

24. A square centered at the origin has its vertices on the x-axis and y-axis. If the graph of the function passes through three of the square’s vertices, then the value of a is:

A)

B)

C) 2

D) 4

E) None of these

25. If the product of three positive, consecutive odd integers is 39 times their sum, then the sum of the consecutive odd integers is:

A) 33

B) 39

C) 1287

D) 2145

E) None of these

26. A semicircle is placed on one side of equilateral triangle as shown. A point halfway along the arc of the semicircle is labeled D. If BD = 1, then the length of each side of the equilateral triangle is:

A)

B)

C) 1

D)

E) None of these

27. In the figure to the right, if is a right angle, then the area of is:

A) 7 in2

B) 9 in2

C) 15 in2

D) 24 in2

E) None of these

28. Suppose that . If for all real numbers x, then a + b equals:

A) 2

B) 3

C) 5

D) 6

E) None of these

29. Let A represent the area of a square that circumscribes a circle and B represent the area of a square inscribed in the same circle. The ratio A/B is:

A)

B)

C)

D) 2

E) None of these

30. If three fair, 6-sided dice of different colors are tossed simultaneously, then the probability that the numbers shown on two of the dice add up to the number shown on the remaining die is:

A)

B)

C)

D)

E) None of these

31. In the figure shown, and are both right angles. If and , then the value of y is:

A) 6

B) 9

C) 20

D) 32

E) None of these

32. If the ordered pair , satisfies the system of equations , then the number of different pairs that satisfies the system is:

A) 1

B) 2

C) 3

D) 4

E) None of these

33. Using interval notation, the solution of is:

A) 

B) 

C) 

D) 

E)  None of these

34. The minute hand of a clock is 8 inches long. In 20 minutes the distance the tip of the minute hand moves is:

A)  inches

B)  inches

C)  inches

D)  inches

E)  None of these

35. Using interval notation, the domain of the real-valued function is:

A)

B)

C)

D)

E) None of these

36. Using interval notation, the range of the real-valued function is:

A)

B)

C)

D)

E) None of these

37. Where defined, the sum of the solution(s) of is:

A) 0

B) 2

C) 4

D) 6

E) None of these

38. Where defined, the sum of the solution(s) of is:

A)

B)

C)

D)

E) None of these

39. Using interval notation, the solution set for is:

A)

B)

C)

D)

E) None of these

40. If and , then equals:

A)

B)

C)

D)

E) None of these

41. If , then x equals:

A)

B)

C)

D)

E) None of these

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

A)

B) 0

C) 2

D) 6

E) None of these

43. The sum, in radians, of the solutions of in the interval is:

A) 

B) 

C) 

D) 

E)  None of these

44. Using interval notation, the range of the real-valued function is:

A)

B)

C)

D)

E) None of these

45. Where defined, if , then the inverse of f, , is equal to:

A) 

B) 

C) 

D) 

E)  None of these

46. The value for c in such that has a double zero is:

A) 0

B) 1

C) 2

D) 3

E) None of these

47. Square EFGH is inside square ABCD so that each side of EFGH can be extended to pass through a vertex of ABCD. Square ABCD has side length , E is between B and H, and BE = 1. The area of the inner square EFGH is:

A)

B) 25

C) 36

D) 49

E) None of these

48. Let A, M, and C each be a single-digit number (0, 1, 2, …, 9) with . The product AMC is:

A) 6

B) 8

C) 10

D) 12

E) None of these

49. A farmer had to sell a piece of his land each time one of his three children went to college. He had to sell of his land to send the oldest to college. However, land prices went up by the time the middle child went to college and he only had to sell of the remaining land. Finally when the youngest child went off to college he had to sell of what remained. This left 120 acres for the farmer. The total number of acres the farmer sold to send his kids

to college is:

A) 150

B) 180

C) 270

D) 320

E) None of these

50. Three circles of radius s are drawn in the first quadrant of the xy-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the x-axis, and the third is tangent to the first circle and the y-axis. A circle of radius r > s is tangent to both axes and to the second and third circles, but does not intersect the first circle. The ratio r/s is:

A) 1

B) 3

C) 6

D) 9

E) None of these