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[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 483: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(9f4c9a621e4456050ebe6dc0c376c8a0.png): failed to open stream: No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(9f4c9a621e4456050ebe6dc0c376c8a0.png): failed to open stream: No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(9f4c9a621e4456050ebe6dc0c376c8a0.png): failed to open stream: No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(9f4c9a621e4456050ebe6dc0c376c8a0.png): failed to open stream: No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 297: unlink([ROOT]/latexrender//tmp/f49ffdd8eebc89f0632f948157ebfb35.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/includes/bbcode.php on line 112: preg_replace(): The /e modifier is no longer supported, use preg_replace_callback instead
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 232: chdir(): No such file or directory (errno 2)
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(71ef4e8bf9409d3d2205e5de4cb8a468.png): failed to open stream: No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(71ef4e8bf9409d3d2205e5de4cb8a468.png): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 281: chmod(): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 297: unlink([ROOT]/latexrender//tmp/71ef4e8bf9409d3d2205e5de4cb8a468.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 302: unlink([ROOT]/latexrender//tmp/71ef4e8bf9409d3d2205e5de4cb8a468.png): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(71ef4e8bf9409d3d2205e5de4cb8a468.png): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 281: chmod(): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 295: chdir(): No such file or directory (errno 2)
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 297: unlink([ROOT]/latexrender//tmp/71ef4e8bf9409d3d2205e5de4cb8a468.tex): No such file or directory
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forum.virtualchallengemeets.com • View topic - POW: Inequalities

POW: Inequalities

What could be so hard. It's a multiple-choice test for Pete's sake.

POW: Inequalities

Postby AuSmith » Sun Apr 01, 2007 1:12 am

Use the Rearrangement Inequality to show that...

(1) For nonnegative a,b,c, [unparseable or potentially dangerous latex formula]

(2) For positive a,b,c, [unparseable or potentially dangerous latex formula]

(3) [unparseable or potentially dangerous latex formula]

(4) For all a,b,c, [unparseable or potentially dangerous latex formula]


From Engel Ch. 7.
Last edited by AuSmith on Sun Apr 01, 2007 4:26 pm, edited 3 times in total.
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Postby polymath25 » Sun Apr 01, 2007 9:07 am

I can do number 1, but I don't know how to use LaTeX and it'd be too complicated without it.
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Postby AuSmith » Sun Apr 01, 2007 10:11 am

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Postby polymath25 » Sun Apr 01, 2007 12:29 pm

Sorry, misread #1, but i can do #2:

Assume that [unparseable or potentially dangerous latex formula] so that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are similarly arranged.

Thus [unparseable or potentially dangerous latex formula]

and

[unparseable or potentially dangerous latex formula]

Adding these together gives us:

[unparseable or potentially dangerous latex formula]

The second part of this inequality gives us 3 (proof to long to type out), and then upon dividing both sides by 2 you get

[unparseable or potentially dangerous latex formula]
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Postby AuSmith » Sun Apr 01, 2007 2:52 pm

it appears #1 was supposed to say nonnegative numbers. Say, [unparseable or potentially dangerous latex formula], then [unparseable or potentially dangerous latex formula], which is clearly untrue. Sorry, 'bout that. You should be able to do it now.
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Postby stupidityismygam » Sun Apr 01, 2007 3:41 pm

1) very trivial but what the heck

WLOG assume [unparseable or potentially dangerous latex formula] then [unparseable or potentially dangerous latex formula]

So then by rearrangement [unparseable or potentially dangerous latex formula] as desired

for 2) i did a combination of Cauchy and Rearrangment, if you want to see my solution tell me
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Postby AuSmith » Sun Apr 01, 2007 4:30 pm

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Postby polymath25 » Sun Apr 01, 2007 4:51 pm

I'm assuming that if I got #3 without using Rearrangement Inequality, then it's cheating
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Re: POW: Inequalities

Postby stupidityismygam » Sun Apr 01, 2007 4:59 pm

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Re: POW: Inequalities

Postby AuSmith » Sun Apr 01, 2007 5:50 pm

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Postby makashi » Sun Apr 01, 2007 9:02 pm

#2 is Nesbitt's Inequality

I believe the standard approach is to add three to both sides:
[unparseable or potentially dangerous latex formula]
If the first factor, [unparseable or potentially dangerous latex formula] is rewritten as [unparseable or potentially dangerous latex formula], then either Cauchy-Schwartz or Rearrangement (Chebyshev, as [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are oppositely sorted) will do the trick:
[unparseable or potentially dangerous latex formula]

#4. Although there is mention of some scalar Rearrangement form, if I remember correctly, Engel never proves it, so we'll resort to other methods first, such as the Trivial Inequality:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
We can use the Rearrangement Inequality (or many others) to show that
[unparseable or potentially dangerous latex formula]
By transitivity, [unparseable or potentially dangerous latex formula] and the desired result follows.
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