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MATH EXAM FINISHED!!!!

Problem of the week

Upon browsing through a mensa book, I found this little gem. Of course, the author being no mathmatician offered no proof for the solution. The problem is outlined for you mathmaticians below.

An elipse is drawn out on a euclidian surface. Determine a method for finding the largest inscribable triange within the elipse and support your technique with proof. You may not assume that the largest incribable triangle in a circle is equilateral (but you may offer a proof for it to support further conclusions).

Last week

Two circles A and B are described by the equations x^2 + y^2 = 1 and (x-10)^2 + y^2 = 16 respectively. Point P on circle A and point Q on circle B define a line that is tangent to both circles. Determine the length(s) of the line segment PQ.

2 weeks ago

is the point C = 0.25 part of the mandelbrot set? full solution required

Solution

The definition of the set is as follows

Z(n+1) = Z(n) + C

the point C is in the set if this series never diverges (in other words, "...if the series of Zs should always stay, close to Z and never trend away, that point is in the mandelbrot set").

our task is to determine if C is indeed in the mandelbrot set,

C = 0.25

we will call the infinite sum of this series S, so

S = ( ( ( (C)^2 + C )^2 + C ) ^ 2 + C ) ^2 ...

so then S^2 + C = ( ( ( (C)^2 + C )^2 + C ) ^ 2 + C ) ^2 ...

but S^2 + C = S

substituteing values in the quadratic formula, we find that S = 0.5. Indeed it seems that the sequence does converge on this value so we can accept it as valid. I don't feel this techniqe works for all values of C, but it seems to work for positve real numbers. Now, try this week's problem.

for more fun with the set, try this link
http://www.h-schmidt.net/MandelApplet/mandelapplet.html