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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

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8) 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

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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

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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

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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)

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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)

omg wtf these problems i could not do at all. #22 and #24 are ESPECIALLY bitchy because I have absolutely not even the SLIGHTEST clue how to even BEGIN.

i have no idea what all these terms are in my language and i do not feel the urge to translate ehm :huh:
(great excuse for me not to look into them :P )

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.




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:




Oooh, this one uses algebra. Fun. :D

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



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


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. :P


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.
 

For #8, its best to draw a picture.

I worded every problem exactly as it was worded. Maybe I'll upload the pics instead.

I did draw a picture. :miffed:

This stuff is giving me horrible flashbacks to what we were doing a few weeks ago. The first problem is easiest. Its alot of raising to the power of, which isn't all that hard. :D


Now, the one that SS did with Aritmetic, we did stuff like that last month. We didn't do geometric, so, just fall back on SS.



And the problem with (cos(2x), is, really with me, finding out what x is. You find out what x is, and its all solved. Though, there are two of the x. I can't remeber if you go 2 * x in that situation, or you take two of x's value as the answer. I think its 2 * x. And whatever x is.

Scanner b0rked

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page 6 is the smaller, external link

(http://home.ripway.com/2004-1/58049/amc/2004-6.jpg)