SOS article so far

by v_Enhance, Oct 19, 2012, 2:17 AM

What I have so far:
https://www.dropbox.com/s/zxpj8g9wxnda19o/SOS_Dumbass.pdf

Aside from the flipped inequalities and typos that are probably evading my sight, I don't quite like the flow of what I've written. There are some ideas I think I got across well (I'll leave it you to guess which one those are so as to avoid confirmation bias), but a lot of it feels like just splintered thoughts thrown together with no unifying idea.

It's also sort of irritating that I feel like I'm writing about a lost art: most three-variable inequalities this day have square roots and stuff. Finding good examples for this article was a challenge that I certainly didn't have when I wrote, say, my bary article. (But that was a much bigger project than this).

Any thoughts/comments/typos would be highly appreciated. In particular, I'm debating whether this is worth posting in the Olympiad Articles forum; at the moment I'm leaning towards no.

Comment

7 Comments

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I've never taken inequalities very seriously, but IMO you cover the intuition behind CDN and SOS much better than hamster and hungkhtn, whose articles seem to have a lot of decent examples but nothing close to an explicit mention of the key ideas. The other sections effectively develop these ideas at a more practical level (e.g. the $(a-b)^2$ divisibility "Fact"), particularly with the relation between CDN and SOS (which I haven't really seen in other exposition).

Maybe I'm just really inexperienced with inequalities, but I think the only addition worth your time (given that you probably don't want too much overlap with old articles) is adding a few more instructive problems at the end, which shouldn't be hard. Many things are better left to the reader.

Edit: Actually, maybe you shouldn't take my comments very seriously. I haven't nontrivially SOSchurheaded an inequality on a test before, and when I do SOS I usually give up if neither $S_a,S_b,S_c\ge0$ nor ordering+$(a-b)(a-c)=\frac{(a-b)^2+(a-c)^2-(b-c)^2}{2}$ (according to David, the "fundamental equation of inequalities" :P ) works.
This post has been edited 6 times. Last edited by math154, Oct 19, 2012, 3:43 AM

by math154, Oct 19, 2012, 3:24 AM

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Just so you know
Quote:
since we've already done a good chuck
of the work.
on page 6

by pi37, Oct 19, 2012, 9:05 PM

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@pi37 Thanks for the caught typo.

math154, thanks for the good remarks. The things that you mentioned were almost exactly the goals that I had when I started writing, so it looks like I at least managed to get that part right.

And darn your "fundamental equation of inequalities" is pretty powerful. I sort of noticed it when I was writing the Schur section but it didn't sink in until now that it was significantly more general.
Quote:
Actually, maybe you shouldn't take my comments very seriously. I haven't nontrivially SOSchurheaded an inequality on a test before,
Neither have I :P Closest I got was B2.3 this year; once you homogenize with $a+b+c \ge 3$ and expand for about twenty minutes you get $[9,1,0] \ge [7,3,0]$; I don't count this as nontrivial.
This post has been edited 1 time. Last edited by v_Enhance, Oct 20, 2012, 7:11 PM

by v_Enhance, Oct 19, 2012, 10:42 PM

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did you just mistake pi37 for ksun48

by dinoboy, Oct 20, 2012, 6:43 AM

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^i actually do that pretty often

by math154, Oct 20, 2012, 6:55 PM

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oops I read that comment and then read my shoutbox and got confused.
This is how good my memory is :P

by v_Enhance, Oct 20, 2012, 7:12 PM

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wait I'm a pretty different person than ksun

by pi37, Oct 28, 2012, 2:03 AM

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