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Perhaps easy $Let the function$ \ f(n) \ $be the digit sum of the integer$ \ n \ $so$ \ f(21) = 3 \ $etc.$ \\ $Let the $ \ k \ $time nested function$ \ f_k(n) = f(f(\dots(f(n)\dots) \ $where the function is nested$ \ k \ $times$ $Show that$ \ 1 \leq f_k(n) \leq 9 \ $for all$ \ n \ $and...

Interpret 'all the numbers from 1 to 1 000 000' as every combination of digits in the 6 digit places Each digit, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can occupy it Pick a random spot for these digits, it is found that if we fix a digit, say 1 xxx x2x Where the x's are variable, there are 10^5...

Imagine writing all these numbers on top of each other, in columns, the 'units' column (the far right), every 10'th number is the same, so you have continuing cycles of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 So in this column, the sum is \frac{1000000}{10} ( 1 + 2 + \dots + 9) The next column, the...

Re: HSC 2014 4U Marathon - Advanced Level Not all those solutions are valid since they give a non-zero Re(z)! For instance, try your first one, z = cis \frac{\pi}{3(n-1)} (when k=0) If you try putting that into Re(z) you won't get 0

Re: HSC 2014 4U Marathon - Advanced Level I just realised I made a small typo, it should be x^n - x^{n-2} - x + 2 = 0 Edited the original post, apologies for any inconvenience (though as far as I can see the method I had in mind can be used for the original question as well as the typo'd one)

Re: HSC 2014 4U Marathon - Advanced Level \\ $Find all roots with modulus 1 of the polynomial$ \\ x^{n}-x^{n-2} -x + 2 = 0 \\ $where$ \ n \geq 2

May I attend the BOS trials (MX2)? EDIT: =(

Re: MX2 Integration Marathon \\ \int_{\frac{1}{\alpha}}^{\alpha} \frac{\tan^{-1}x}{x} \ dx

Re: HSC 2014 4U Marathon - Advanced Level Does this question need clarification?

Re: HSC 2014 4U Marathon \\ $Let$ \ a,b,c \ $be positive real numbers such that$ \ ab + ac + bc = 1 \\ \\ $ Prove that$ \ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c} \geq \sqrt{3} + \frac{ab}{a+b} + \frac{ac}{a+c} + \frac{bc}{b+c} \\ \\ $and show where equality holds$

Which is why its a side point that you pressed for some strange reason, illustrating your ignorance It is not avoiding the point, its like someone saying, "the fact that apples are red shows that in fact oranges taste nice" The two points have no relevance whatsoever, you must show with proof...

1 - Islamic conception of God exists 2 - In Islam our purpose is to find truth 3 - Therefore our purpose is to find truth Which just shows how what you are trying to press on to is really a side point, because I'm not going to try and prove (1) in this thread I have actually lol. Why does...

I am continuing to respond to you because I thought that you could actually learn something, but instead you have proven yourself to be an ignoramous Yes it is dumbass Its not imaginary, its the reality There is no connection between God's foreknowledge and with free will No it is...

Though I think he has some weird and untenable views, I think William Lane Craig does a good job of explaining these things to a layman, he has a lot of things on youtube and the sort

Though I think he has some weird and untenable views, I think William Lane Craig does a good job of explaining these things to a layman, he has a lot of things on youtube and the sort

The Big Bang is part of the Universe, the Big Bank is an event that marked the beginning of the Universe Hookers aren't immaterial, nor are they timeless, spaceless, nor do they possess perfect Knowledge and Power

If you don't want to seek truth then by definition you are a dumbass and thus I have no desire in talking with you Who said we don't have free-will? That's not what I said

Well this is what philosophers and theologians mean by 'God', it is the 'definition' you can say. I have already proven that God has these attributes Since the causal premise is that everything that begins to exist has a cause. God did not begin to exist and thus does not have a cause...

No I really don't, my life has no meaning without God and without religion, my purpose is to serve and follow God, not my desires, not any other men. Anyone who tries to talk about the meaning of life with reference to the subjective inner feeling has missed the point, subjectivity is all fine...

Oh wow you have a lot more deeper problems that I thought initially, I am going to be a little more sympathetic God exists timelessly, eternally, permanently, so He does not have a beginning Atheists in the past actually said the same thing about the Unvierse, that is, it began to exist...