Algebra Marathon

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?

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. =)

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?

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.

AIME15 wrote:

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

Please read directions, Johnny.

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?

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?

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

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

3*5 = 15.

can run 15 mph. He ran 24000 miles. How many days was he running?

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?

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?

-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?

Is profit/apple.



Is how many apples she needs.


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

x^2+6x-9

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