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 Algebra Marathon

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skittles
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PostSubject: An Algebra Problem   Wed Aug 13, 2008 6:47 pm

Let a and b be the roots of the equation x^2-mx+2=0. Suppose that a+(1/b) and b+(1/a) are the roots of the equation x^2-px+q=0. What is q?
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jellymuffins
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PostSubject: Re: Algebra Marathon   Thu Aug 14, 2008 11:01 am

Well from vieta

q = (a + 1/b)(b + 1/a) = ab + 1 + 1 + 1/ab

And again by vieta we know that ab = 2 from x^2 - mx + 2 = 0

2 + 1 + 1 + 1/2 = 4 1/2

Nice problem! =)


Edit: Try to find p. =)
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AIME15
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PostSubject: Algebra Marathon   Wed Aug 20, 2008 1:54 pm

Would anyone like to have an algebra marathon?

If so: Normal marathon rules. Remember to POST A PROBLEM AFTER YOU SOLVE ONE.

First problem: Johnny's Beard is very long, in fact, it's 3 feet long. It costs 6 dollars per inch cut, and Johnny wants his beard cut in 13 days. His beard grows one inch every day. How much will it cost him to cut his beard to just 3 inches long?
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johnnybeard
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PostSubject: Re: Algebra Marathon   Thu Aug 21, 2008 10:05 am

All right man, that's not nice.

Anyway:

In 13 days, MY beard will be 49 inches long. 49-3=46. Multiply by 6 to get 276 dollars. I would never pay that much for a beard cut.
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isabella2296
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PostSubject: Re: Algebra Marathon   Thu Aug 21, 2008 1:05 pm

AIME15 wrote:


If so: Normal marathon rules. Remember to POST A PROBLEM AFTER YOU SOLVE ONE.

Please read directions, Johnny.
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AIME15
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PostSubject: Re: Algebra Marathon   Thu Aug 21, 2008 1:57 pm

Yes johnny, it's a very important part of life, reading directions. Anyway:

When Maya divides her favorite number by 11, she gets a remainder of 4.What will the remainder be if she multiplies her favorite number by 4 and then divides by 11?

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herefishyfishy1
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PostSubject: Re: Algebra Marathon   Thu Aug 21, 2008 3:04 pm

Well, if we call the number a, then a=4 (mod 11). If we multiply both sides by 4, we get 4a=16 (mod 11). 16=5 (mod 11). Therefore, 4a=5 (mod 11). And therefore, the answer is 5.

What is the product of the roots of 4x7+7x6-2x5+x3-8x2-8x+2?
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AIME15
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PostSubject: Re: Algebra Marathon   Thu Aug 21, 2008 3:29 pm

Using vieta: it's -2/4=-1/2, right?

New Problem: How many integer divisors does 5625 have, given 5625=3^2*5^4?

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isabella2296
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PostSubject: Re: Algebra Marathon   Fri Aug 22, 2008 8:43 am

3*5 = 15.

rendeer can run 15 mph. He ran 24000 miles. How many days was he running?
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Dojo
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PostSubject: Re: Algebra Marathon   Fri Aug 22, 2008 9:52 am

24000/15=1600
1600/24=66 2/3

(Heres a sad question I'm gonna make up)
Currently, Joe is half the age of sally.
In 24 years, Sally will be 1 and 1/3 Joe's age.
How old is Joe?

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FantasyLover
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PostSubject: Re: Algebra Marathon   Mon Aug 25, 2008 11:42 am

J=1/2 S
S+24=4/3 (J+24)

S+24=4/3J+32

2J+24=4/3J+32, 2/3J=8, J=12

answer : 12



Fantasy runs 12 miles in 3 hours. How many miles would he run in 4 hours?
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isabella2296
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PostSubject: Re: Algebra Marathon   Tue Aug 26, 2008 11:39 am

-sigh-

Algebra doesn't mean amazingly easy peasy.

12/3 = x/4

x = 16

Fantasy buys 60 apples for 5 dollars, and then sells them for an average of 10 cents per apple. How many apples does he have to buy and sell to get a total profit of at least 200 dollars?
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Dojo
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PostSubject: Re: Algebra Marathon   Tue Aug 26, 2008 11:23 pm



Is profit/apple.



Is how many apples she needs. Nervous


Find the polynomial that has the roots that are recriprocals of the roots of:

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ternary0210
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PostSubject: Re: Algebra Marathon   Mon Sep 08, 2008 4:55 pm

x^2+6x-9

Find (sqrt(12+sqrt(12+sqrt(12...
Replace 12 with...
a)2
b)6
c)20
d)30
I love you Dojo
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