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Re: HSC 2013 4U Marathon I got: (4*sqrt(2)-5)/3 and am pretty confident about it...

Re: HSC 2013 4U Marathon yeah that sounds about right

Re: HSC 2013 4U Marathon Will check now...I didn't actually do the calculation.

Re: HSC 2013 4U Marathon Nah, the regions won't be circles and spheres.

Re: HSC 2013 4U Marathon A point P is chosen inside the unit square at random. What is the probability that P is closer to the centre of the square than any side thereof? Bonus: Solve this for a cube, replacing the word "side" with "face".

Of course you can! There is a common misconception that innate ability is the most important factor in being good at maths. It may help you grasp some concepts quicker at first, or allow you to perform computations faster and more reliably, but there is NO substitute for hard work...and this...

Re: HSC 2013 3U Marathon Thread I am quite sure the numerator should be 2^n. Edit: In the previous question that is.

Re: HSC 2013 4U Marathon Note: I am sure it can be done without monotone convergence, I just think that it is a stiff ask for nearly any mx2 student.

Re: HSC 2013 4U Marathon Nope, sequences satisfying the properties you provided can either diverge or converge. An example of a convergent one being 1-1/n.

Re: HSC 2013 4U Marathon Was this the only part of the question? If so it will be interesting seeing how people justify that u_n tends to infinity. The easiest way I can see uses monotone convergence.

gyg would have been a better option!

b: - l: schnitty, coleslaw, fries l2: moong dal + rice d: ? am craving mexican though. might fry up some mince for nachos or something.

Re: HSC 2013 4U Marathon $We might as well assume instead that the $x_j$ are non-negative, as this will imply our desired result anyway. \\ For $n=1$, the assertion is trivial. We proceed by induction.\\ \\ Let $P_k(x_1,\ldots,x_k)=\prod_{j=1}^k \frac{1-x_j}{1+x_j}$ for $k$-tuples with each...

Depends on the wording of the question. What your have shown is that: "The most edges a bipartite graph on n vertices can have without containing triangles is blah." where the "without containing triangles" part is redundant because of the "bipartite". Depending on what the question is asking...

Yeah, the method is flawed unfortunately. As I said, a graph without triangles is not necessarily bipartite.

Haven't seen one anywhere.

Haha there are actually functions sn(x), cn(x) and dn(x).

Do you just look through your working or something? I found it best to decide which questions are most likely to induce silly errors and do them a second time from scratch, covering my initial working (doing this for as many as possible given the time constraint, and something even triple...

Hint: The LHS is a geometric series.

Yes, the complete bipartite graph K(m,n) has mn edges, and it is easy to show that for fixed m+n that this is maximised when m and n are as close to equal as possible. However, not every graph without triangles is bipartite. As a counterexample consider the 5-cycle.