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Re: HSC 2014 4U Marathon - Advanced Level No worries.

Re: HSC 2014 4U Marathon - Advanced Level I have already provided counterexamples for problems where you have allowed any positive x and y, you then brought up constrained optimisation in post #3067 (via the constraint xy=4), so I found one for that too.

Re: HSC 2014 4U Marathon - Advanced Level Sorry, my constraint was supposed to be x^2 + y^2 >= 1, not 0.

Re: HSC 2014 4U Marathon - Advanced Level The tweaking I was objecting to was more the tweaking outlined in my second paragraph. He posted a "solution" to a question citing a result that is untrue. He has then successively weakened it many times (but not enough to make it true), and keeps...

Re: HSC 2014 4U Marathon - Advanced Level Apologies for the delay. My family loves to drag on these holiday events. Of course the idea is not baseless...but each of your successive guesses is. You successively strengthen your assumptions and weaken your conclusion hoping that the new claim is...

Re: HSC 2014 4U Marathon - Advanced Level Emphasis on the "good reason" though. Simply tweaking assumptions and hypotheses by guessing is very counterproductive. If you read the last few posts, you will see this is what has been happening. Also it is much more valuable to first at least try...

Re: HSC 2014 4U Marathon - Advanced Level A hunch is nothing without evidence. In trying to prove something you really get to understand a statement. (And counterexamples usually stem from thinking about the step where a logical proof attempt fails). I will provide a counterexample if I can...

Re: HSC 2014 4U Marathon - Advanced Level I really don't think so. You should try to prove it if you think it's true though, thats the only way to actually know. Will look for a counterexample when at home. Sent from my GT-I9000 using Tapatalk 2

Re: HSC 2014 4U Marathon - Advanced Level Well most homogeneous polynomials won't even have global extrema, but even if we just look at those that do, your claim is still incorrect. (x-2y)^2(2x-y)^2 is a homogeneous cyclic polynomial. It doesn't have a maximum, but it is minimised (and is...

Re: HSC 2014 4U Marathon - Advanced Level I assume you mean homogeneous rational function rather than homogenous polynomial. (As that is what Sy's expression is, so we cannot consider x=y=z=0 for example.) So the claim is: "If a homogeneous rational function with cyclic symmetry has a minima...

Re: HSC 2014 4U Marathon - Advanced Level ((x-1)^2+(y-1)^2+(z+1)^2)\cdot ((x-1)^2+(y+1)^2+(z-1)^2) \cdot ((x+1)^2+(y-1)^2+(z-1)^2). The same idea obviously extends to any finite number of variables.

Re: HSC 2014 4U Marathon - Advanced Level So what is the point of the substitution then? You have just changed it from one symmetric counterexample to another. There's no easy answer to that question. The "rule" you stated in your initial response to Sy's question is not valid unless we assume...

Re: HSC 2014 4U Marathon - Advanced Level The constants? Which constants and what about them? What is the purpose of this coordinate change? P is not symmetric in X and Y (and we should not expect it to be, because the relationship between (x,y) and (X,Y) breaks symmetry. Your claim was...

Re: HSC 2014 4U Marathon - Advanced Level So what if some symmetric expressions have extrema when all variables are equal? This does not mean that all do, so we can't use it in proofs. Here is a symmetric expression in two variables, (switching x and y just switches the roles of the two...

Sure, but the speed isn't the main issue. It's the whole physics system. To name a few thing: hitstun in brawl is too low, you can cancel it with airdodges, airdodges don't leave you helpless, ledgesnap distance is way too large, recoveries are way too good in general, pretty much nothing is...

Re: HSC 2014 4U Marathon - Advanced Level Picture it this way in two variables x and y: A real function f can be viewed as a graph z=f(x,y). Saying such a function is symmetric, is just saying that it's graph has reflective symmetry about the plane x=y. This just means that any extrema...

Re: HSC 2014 4U Marathon - Advanced Level Yes, it does equal 3/4 when a=b=c. But this does not prove that this is a minimum, and you cannot assume 3/4 is a minimum because that is what the question is asking you to prove. Symmetry means that a=b=c is often a good candidate for a minimum, but...

Re: HSC 2014 4U Marathon - Advanced Level In this question it happens to occur when x,y and z are equal. But this isn't because of symmetry, this is because Davo has shown it above (the only steps where inequality was introduced was the applications of AM-GM, whose equality conditions we...

Re: HSC 2014 4U Marathon You can use complex numbers to obtain a formula for rotations at an MX2 level.

Re: HSC 2014 4U Marathon - Advanced Level Symmetry in variables does not mean that minima occur when all variables are equal...