6) Let U = 2 * 2004^2005, V = 2004 ^ 2005, W = 2003 * 2004^2004, X = 2 * 2004^2004, Y = 2004^2004, and Z = 2004^2003. Which of the following is the largest?

U - V

V - W

W - X

X - Y

Y - Z

Easy.

U-V = 2004^2005

V-W = 2003 * 2004^2004

W-X = 2001 * 2004^2004

X-Y = 2004^2004

Y-Z = 2004^2004 - 2004^2003

They're already in size order. U-V is the largest.

Triangles EAB and CBA share base AB. If <EAB and <ABC are right angles, AB = 4,, BC = 6, AE = 8, and line AC and line BE intersect at D. What is the difference between the areas of triangle ADE and triangle BDC?

2

4

5

8

9

ADE Area = 1/2 (6*4) + 1/2 (8*4) + 1/2 (28*20) - 1/2(22*20)

BDC Area = 1/2 (28*20) - 1/2(22*20)

Difference in area = 1/2 (6*4) + 1/2 (8*4) = 12 + 16 = 28.

Hmmm. I've crapped up somewhere there. :unsure:

14) A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?

1

4

36

49

81

Oooh, this one uses algebra. Fun.

9 9

B B+2

C C+20

(C+20)/(B+2) = (B+2)/9

C-B = B-9

B+B = C+9

B = (C+9)/2

(C+20)/(((C+9)/2)+2) = (((C+9)/2)+2)/9

9*((C+20)/(((C+9)/2)+2)) = (((C+9)/2)+2)

(9C+180) = ((C/2)+6.5)*((C/2)+6.5)

9C+180 = (1/4)C² + 42.25 + 6.5C

(1/4)C² -2.5C - 137.75 = 0

Use Quadratic Formula

C=-19

B = (9-19)/2

B = -5

Arithmetic Progression: 9, -5, -19. (Change: -14)

Geometric Progression: 9, -3, 1. (Multiplier: -1/3)

Answer: 1

21) If the sum of (cos(x))^(2n), with lowerlimit n = 0 and upper limit infinity, is equal to 5, what is the value of cos(2x)?

1/5

2/5

rt(5)/5

3/5

4/5

Hmmm, can you re-check the wording in that question? I'm getting odd answers. :/

22) Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?

3 + rt(30) / 2

3 + rt(69) / 3

3 + rt(123) / 4

52 / 9

3 + 2 * rt(2)

Blah, that's just trig - the three spheres form a 2x2x2 equilateral triangle, with the large one forming it into a pyramid. I can't be arsed to calculate the exact value, it's about five.

24) A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover line AB. What is the area of S?

2pi + rt(3)

8pi / 3

3pi - rt(3)/2

10pi / 3 - rt(3)

4pi - 2 * rt(3)

Is that the center of the disk is on the line, or just some part of the disk is on the line? It's too vague.