Recreational Problems (SM level)

[Collin Li]

My eyes hurt from reading that fluro-green page.

[/0]

Lol when I saw this it said the topic had been read 1729 times

[humph]

mmmm, number theory. quite possibly the best area of maths.

[/0]

Just read about Lagrange Multipliers, gotta say they're awesome. They can basically kill off any inequality problem easily, and seem to be the most powerful tool for optimization, you gotta read about them if you haven't seen them:

http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx

[kido_1]

This number theory stuff, is pretty thought provoking.

[Ahmad]

They can basically kill off any inequality problem easily

I wouldn't say easily, maybe systematically is a better word. Here is how lagrange multipliers usually look when proving inequalities:

http://documents.wolfram.com/mathematica/Demos/Notebooks/InequalityProof.html

[humph]

This number theory stuff, is pretty thought provoking.

number theory = win. srsly. read a book on elementary number theory if you're interested. it's (supposedly) 3rd year stuff, but really doesn't need any more maths than you've learnt in primary school.

analytic number theory, on the other hand, is much much more difficult. i'm interested in doing that later on, but you need a fair grasp of complex analysis & integral transforms and the like...

[Mao]

18. (n-radicals).

n^th crack at this:

this time i'm not going to find all the answers, i'll just show one of the answer(s)
















and a wild dab why why there are no other solutions:

the same expression with n+1 radicals:



(n+1) radicals

that means the original expression can be rewritten as
which has 2n radicals
or similarly for kn radicals, where k is a natural number.

this means, if given an expression in this form with x radicals, we can break this down into x.1 radicals (in the form of k.n), which means, the result each radical equals the next and so forth.....

^^^really badly worded...

but yeah

[Ahmad]

Well done Mao, your answer is correct, although your solution is somewhat difficult to understand.

Here is one way to think about why the solution x=3 works.

If , then





(Upon replacing by .







We can continue in this fashion for n-radicals. And thus, we have showed that x=3 is a solution.

It is difficult to understand your solution for why those are the only two solutions, can you explain further?

[Mao]

right, lets say we have a 5-radicals expression:



this expression can be developed from

has 1 radical

then the 2 radicals expression (applied to both sides) will be



hence with this identity we can substitute for continuously to make our n-radical expression one by one

but also note that , so everyone of these radicals equal one another, which is great.

in reality, i should have identified this first and solved , but instead i identified that two consecutive expressions of n and n-1 radicals are equal. I then expanded the n-radicals expression in terms of n-1 radicals, and solved it the hard and painful way

yeah

[Mao]

OKAY a solution that is more acceptable for ahmad's pragmatic butt

n radicals

as otherwise the innermost radical would be not yield a real result.

let

If we let

then an n-radical expression can be expressed as:







since

repeating this will arrive at a point which:







If we let

then an n-radical expression can be expressed as:







since

repeating this will arrive at a point which:





which is clearly false.


if we let

by the same token we will arrive at:



which is also clearly false.


therefore the only solutions to the n-radicals expression are


*much better*

[Mao]

solve this:





(ps do NOT assume that there are infinite terms, as that is clearly false

[Ahmad]

IIRC this was posed before in the Methods subforum.

[Ahmad]

You can use a bit of common sense, or you might have learnt about direction cosines.

http://en.wikipedia.org/wiki/Unit_vector ("direction cosines")

In any case, I'm afraid my rough calculations do not agree with yours.

[gfb]

Ok. Nevermind, I'll just delete my post .