Timeline for How i show this beautiful inequality :$\frac{x^n}{x^m+y^m}+\frac{y^n}{y^m+z^m}+\frac{z^n}{z^m+x^m}\geq \frac{3} {2}(\frac{1}{\sqrt{3}})^{n-m}$?
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Dec 14, 2015 at 10:40 | comment | added | Peter Mueller | @zeraouliarafik: Are you really unable to check if my examples are counter examples or not? At math.stackexchange.com/questions/1306593/… I've added a second dis-proof which does not require to compute with large integers. | |
| Dec 12, 2015 at 22:17 | comment | added | zeraoulia rafik | @PeterMueller , no, i understood well the reformulation of silvermen , but i'm chocked by the proofs of parfeti and proposer of problem in journal which worked under xy+yz+zx=1.pleas let me know who is true u or journal ? | |
| Dec 11, 2015 at 17:27 | comment | added | Peter Mueller | I don't care about the two ``proofs'' if there is a simple counterexample. As you don't seem to be able to understand the simple reformulation (suggested by Silverman), here is the example in the original setting: Set $x=\frac{3}{4\sqrt{3}}$, $y=\frac{1}{\sqrt{3}}$, $z=\frac{9}{7\sqrt{3}}$. Then $xy+yz+zx=1$, and with $m=7$, $n=8$, $\frac{x^n}{x^m+y^m}+\frac{y^n}{y^m+z^m}+\frac{z^n}{z^m+x^m}-\frac{3}{2}(\frac{1}{\sqrt{3}})^{n-m}=-\frac{30373554930137225}{11449504266834352656}\sqrt{3}<0$. | |
| Dec 11, 2015 at 16:11 | comment | added | zeraoulia rafik | @PeterMueller, i'm regret your solution to this problem is wrong , pleas see the solution under xy+yz+zx=1 ,page 374 by the proposer and paolo parfeti :dropbox.com/s/2wjbj058iq0ffly/Mathproblem.pdf?dl=0 | |
| Oct 5, 2015 at 21:06 | comment | added | Todd Trimble | Yes, the inequality in your post seems to be incorrect as stated. I just went through Peter's example and the arithmetic checks out. | |
| Oct 5, 2015 at 20:52 | comment | added | zeraoulia rafik | @ToddTrimble, so you meant this problem is wrong under :xy+yz+zx=1 condition ? | |
| Oct 5, 2015 at 20:44 | comment | added | Todd Trimble | @zeraouliarafik Peter Mueller's answer uses Joe Silverman's equivalent reformulation of the problem. | |
| Oct 5, 2015 at 20:10 | comment | added | zeraoulia rafik | @ToddTrimble,hop you to reopen the question because it is not a counter-example as i showed to Petermueller | |
| Oct 5, 2015 at 19:59 | comment | added | zeraoulia rafik | @PeterMueller Your choice of X, Y, Z, does not fulfill the auxiliary condition xy+yz+zx=1 as it is required in the problem, so it cannot serve as a counter example. In any case, even with your choice of x,y,z, you still need to show it is less than \sqrt{3}/2, since this is the claim in the problem | |
| Sep 16, 2015 at 23:40 | comment | added | Todd Trimble | @zeraouliarafik I didn't downvote. I don't know who did or why. | |
| Sep 16, 2015 at 22:25 | comment | added | zeraoulia rafik | @ToddTrimble,why this downvote such that i edited the uestion putting it source ? | |
| Sep 16, 2015 at 21:37 | comment | added | Todd Trimble | If you're wondering why I called it a "competition problem": it's a "competition" to see if one can solve the problem by one's self and get one's name in the journal; the "winners" are those that do, and first prize is to have one's solution be the one published as the official solution. We don't discuss that kind of thing here except in rare circumstances (usually because it ties in with a bona fide research question). | |
| Sep 16, 2015 at 21:01 | comment | added | zeraoulia rafik | And i'm confused how competition did a wrong like this however it were published by journal . | |
| Sep 16, 2015 at 20:18 | comment | added | zeraoulia rafik | @ToddTrimble,sorry i was offline for giving a response to GHfrom MO about source of this question and i posted it in MO after the date of submitting it to journal k.s. | |
| Sep 16, 2015 at 17:30 | comment | added | Todd Trimble | Peter is right. We generally don't do competition problems, and there are good reasons we don't. Thus if you submit a competition problem, you should always disclose the source of the problem and explain why an exception should be made, as a courtesy. You were even asked earlier by GH from MO but decided not to respond; why? | |
| Sep 16, 2015 at 13:15 | comment | added | zeraoulia rafik | sorry for this , but since you have got a counterexample and it is work i think no problem to say that is true | |
| Sep 16, 2015 at 13:10 | comment | added | Peter Mueller | Aha, so the math competition problem there isn't correct. By the way, you should have told the source of this question, and given a reason to post it at MO. | |
| Sep 16, 2015 at 12:55 | history | edited | Peter Mueller | CC BY-SA 3.0 |
simpler counterexample
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| Sep 16, 2015 at 12:52 | comment | added | zeraoulia rafik | you can check this :mathproblems-ks.com/?wpfb_dl=59 problem 110 | |
| Sep 16, 2015 at 12:10 | history | answered | Peter Mueller | CC BY-SA 3.0 |