[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]/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 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
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(6d7db0f3a803f1d33c413d99d027bd91.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(6d7db0f3a803f1d33c413d99d027bd91.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 297: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): 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/6d7db0f3a803f1d33c413d99d027bd91.tex): 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/6d7db0f3a803f1d33c413d99d027bd91.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(6d7db0f3a803f1d33c413d99d027bd91.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 297: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): 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 297: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): 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 297: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): 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(6d7db0f3a803f1d33c413d99d027bd91.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 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/6d7db0f3a803f1d33c413d99d027bd91.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 301: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.ps): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): failed to open stream: No such file or directory
[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(6d7db0f3a803f1d33c413d99d027bd91.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/6d7db0f3a803f1d33c413d99d027bd91.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 300: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.dvi): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 301: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.ps): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 302: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.png): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): failed to open stream: No such file or directory
[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(6d7db0f3a803f1d33c413d99d027bd91.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/6d7db0f3a803f1d33c413d99d027bd91.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 299: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.log): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 300: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.dvi): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 301: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.ps): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 302: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.png): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 232: chdir(): No such file or directory (errno 2)
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): failed to open stream: No such file or directory
[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(6d7db0f3a803f1d33c413d99d027bd91.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/6d7db0f3a803f1d33c413d99d027bd91.tex): 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/6d7db0f3a803f1d33c413d99d027bd91.tex): 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 297: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): 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 280: copy(6d7db0f3a803f1d33c413d99d027bd91.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 297: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): 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 280: copy(6d7db0f3a803f1d33c413d99d027bd91.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 297: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): 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(6d7db0f3a803f1d33c413d99d027bd91.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/6d7db0f3a803f1d33c413d99d027bd91.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.tex): failed to open stream: No such file or directory
[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(6d7db0f3a803f1d33c413d99d027bd91.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/6d7db0f3a803f1d33c413d99d027bd91.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 301: unlink([ROOT]/latexrender//tmp/6d7db0f3a803f1d33c413d99d027bd91.ps): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/039eaac1c4e584529ca3c288d94bda0d.tex): failed to open stream: No such file or directory
[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(039eaac1c4e584529ca3c288d94bda0d.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/039eaac1c4e584529ca3c288d94bda0d.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 301: unlink([ROOT]/latexrender//tmp/039eaac1c4e584529ca3c288d94bda0d.ps): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/039eaac1c4e584529ca3c288d94bda0d.tex): failed to open stream: No such file or directory
[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(039eaac1c4e584529ca3c288d94bda0d.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/039eaac1c4e584529ca3c288d94bda0d.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 301: unlink([ROOT]/latexrender//tmp/039eaac1c4e584529ca3c288d94bda0d.ps): No such file or directory
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forum.virtualchallengemeets.com • View topic - AIME #7 1995

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AIME #7 1995

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AIME #7 1995

PostPosted: Thu Jul 12, 2007 9:51 pm
by stupidityismygam
Given that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are positive integers with [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] relatively prime, find [unparseable or potentially dangerous latex formula]

PostPosted: Thu Jul 12, 2007 10:52 pm
by Roel
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

After moving things to one side and multiplying everything by [unparseable or potentially dangerous latex formula]:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Quadratic formula:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Since our answer needs to be [unparseable or potentially dangerous latex formula]:
[unparseable or potentially dangerous latex formula]

Back to [unparseable or potentially dangerous latex formula]:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]
From earlier equations we have:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Substituting:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

PostPosted: Thu Jul 12, 2007 11:02 pm
by stupidityismygam
Let [unparseable or potentially dangerous latex formula]

Multiply the first by the second to give [unparseable or potentially dangerous latex formula] Multiply this by 4, [unparseable or potentially dangerous latex formula]

Add the beginning equations together to give [unparseable or potentially dangerous latex formula]. Square this [unparseable or potentially dangerous latex formula]

So this means that [unparseable or potentially dangerous latex formula]

Find the zeros and you win.
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