https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Jwelsh&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T10:22:02ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=AoPS_Schoolhouse&diff=183707AoPS Schoolhouse2022-12-07T16:29:31Z<p>Jwelsh: Undo revision 183575 by Machamp1234 (talk)</p>
<hr />
<div>AoPS Schoolhouse is the system used for [[Math Jam|MathJams]] and online school. The main instructor (also a classroom admin) is responsible for posting the learning materials, while other classroom assistants serve as backup if the instructor is not present or answer questions posted by students. For example, in [[Mathcounts Week/Countdown Round|The World’s Largest Countdown Round]], there were 17 classroom assistants.<br />
<br />
Appearance of the new (beta) classroom design:<br />
[[File:The_new_aops_classroom.PNG|800px]]<br />
<br />
==Posting==<br />
If the chatroom is moderated, which is always the case during MathJams and usually the case during class, when students type things into the chatroom, the message goes to the instructors. From there, the instructors have the choice of sharing the question with the class, whispering an answer back, or doing nothing. Your post will show up in the classroom in your preferred color (you can change this in the settings) if it is chosen. Other people's chosen posts will show up in a blue color. If you get a reply back in a whisper, it will show up in a pink or purple color. If a moderator feels a question requires further discussion, they will open a mini window with you to more deeply discuss the question.<br />
<br />
==Character Limit==<br />
The character limit for posts in the AoPS classroom is <b>623 characters</b>. You can split your posts into two to bypass this limit. There is no limit for the number of posts you can send. However, you could get a warning for posting spam messages. (You can't post the same thing twice in a row.) You could also get a warning if you post messages too fast.<br />
<br />
==Emojis==<br />
Below is the list of emojis supported in the AoPS classroom and how to generate them. To adjust the emoji type you want, click on the arrow in the top right corner and select "Change Smiley Settings".<br />
<br />
{| class="wikitable"<br />
|- <!-- Start of a new row --><br />
| <center><b>Text</b></center> || <center><b>Classic</b></center> || <center><b>Alex</b></center> || <center><b>Grogg</b></center> || <center><b>Lizzie</b></center> || <center><b>Winnie</b></center><br />
|-<br />
| <pre>:)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>=)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:-)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>=-)</pre> || <center>[[File:Smile-Classroom.gif]]</center> || <center>[[File:Smile-Classroom-Alex.png]]</center> || <center>[[File:Smile-Classroom-Grogg.png]]</center> || <center>[[File:Smile-Classroom-Lizzie.png]]</center> || <center>[[File:Smile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:(</pre> || <center>[[File:Frown-Classroom.gif]]</center> || <center>[[File:Frown-Classroom-Alex.png]]</center> || <center>[[File:Frown-Classroom-Grogg.png]]</center> || <center>[[File:Frown-Classroom-Lizzie.png]]</center> || <center>[[File:Frown-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:-(</pre> || <center>[[File:Frown-Classroom.gif]]</center> || <center>[[File:Frown-Classroom-Alex.png]]</center> || <center>[[File:Frown-Classroom-Grogg.png]]</center> || <center>[[File:Frown-Classroom-Lizzie.png]]</center> || <center>[[File:Frown-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>;-)</pre> || <center>[[File:Wink-Classroom.gif]]</center> || <center>[[File:Wink-Classroom-Alex.png]]</center> || <center>[[File:Wink-Classroom-Grogg.png]]</center> || <center>[[File:Wink-Classroom-Lizzie.png]]</center> || <center>[[File:Wink-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:D</pre> || <center>[[File:Bigsmile-Classroom.gif]]</center> || <center>[[File:Bigsmile-Classroom-Alex.png]]</center> || <center>[[File:Bigsmile-Classroom-Grogg.png]]</center> || <center>[[File:Bigsmile-Classroom-Lizzie.png]]</center> || <center>[[File:Bigsmile-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>D:</pre> || <center>[[File:Angry-Classroom.gif]]</center> || <center>[[File:Angry-Classroom-Alex.png]]</center> || <center>[[File:Angry-Classroom-Grogg.png]]</center> || <center>[[File:Angry-Classroom-Lizzie.png]]</center> || <center>[[File:Angry-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:O</pre> || <center>[[File:Surprise-Classroom.gif]]</center> || <center>[[File:Surprise-Classroom-Alex.png]]</center> || <center>[[File:Surprise-Classroom-Grogg.png]]</center> || <center>[[File:Surprise-Classroom-Lizzie.png]]</center> || <center>[[File:Surprise-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:P</pre> || <center>[[File:Tongue-Classroom.gif]]</center> || <center>[[File:Tongue-Classroom-Alex.png]]</center> || <center>[[File:Tongue-Classroom-Grogg.png]]</center> || <center>[[File:Tongue-Classroom-Lizzie.png]]</center> || <center>[[File:Tongue-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:|</pre> || <center>[[File:Neutral-Classroom.gif]]</center> || <center>[[File:Neutral-Classroom-Alex.png]]</center> || <center>[[File:Neutral-Classroom-Grogg.png]]</center> || <center>[[File:Neutral-Classroom-Lizzie.png]]</center> || <center>[[File:Neutral-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>B-)</pre> || <center>[[File:Cool-Classroom.gif]]</center> || <center>[[File:Cool-Classroom-Alex.png]]</center> || <center>[[File:Cool-Classroom-Grogg.png]]</center> || <center>[[File:Cool-Classroom-Lizzie.png]]</center> || <center>[[File:Cool-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>O:)</pre> || <center>[[File:Saint-Classroom.gif]]</center> || <center>[[File:Saint-Classroom-Alex.png]]</center> || <center>[[File:Saint-Classroom-Grogg.png]]</center> || <center>[[File:Saint-Classroom-Lizzie.png]]</center> || <center>[[File:Saint-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>^_^</pre> || <center>[[File:Anime-Classroom.gif]]</center> || <center>[[File:Anime-Classroom-Alex.png]]</center> || <center>[[File:Anime-Classroom-Grogg.png]]</center> || <center>[[File:Anime-Classroom-Lizzie.png]]</center> || <center>[[File:Anime-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>^^</pre> || <center>[[File:Sweatdrop-Classroom.gif]]</center> || <center>[[File:Sweatdrop-Classroom-Alex.png]]</center> || <center>[[File:Sweatdrop-Classroom-Grogg.png]]</center> || <center>[[File:Sweatdrop-Classroom-Lizzie.png]]</center> || <center>[[File:Sweatdrop-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:-3</pre> || <center>[[File:Smile3-Classroom.gif]]</center> || <center>[[File:Smile3-Classroom-Alex.png]]</center> || <center>[[File:Smile3-Classroom-Grogg.png]]</center> || <center>[[File:Smile3-Classroom-Lizzie.png]]</center> || <center>[[File:Smile3-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>;-3</pre> || <center>[[File:Wink3-Classroom.gif]]</center> || <center>[[File:Smile3-Classroom-Alex.png]]</center> || <center>[[File:Smile3-Classroom-Grogg.png]]</center> || <center>[[File:Smile3-Classroom-Lizzie.png]]</center> || <center>[[File:Smile3-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>o.O</pre> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center><br />
|-<br />
| <pre>O.o</pre> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center> || <center>[[File:Boggle-Classroom.gif]]</center><br />
|-<br />
| <pre>:blue:</pre> || <center>[[File:Blue-Classroom.gif]]</center> || <center>[[File:Blue-Classroom-Alex.png]]</center> || <center>[[File:Blue-Classroom-Grogg.png]]</center> || <center>[[File:Blue-Classroom-Lizzie.png]]</center> || <center>[[File:Blue-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:heart:</pre> || <center>[[File:Heart-Classroom.gif]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center> || <center>[[File:Heart-Classroom-BA.png]]</center><br />
|-<br />
| <pre>:star:</pre> || <center>[[File:Star-Classroom.gif]]</center> || <center>[[File:Star-Classroom-BA.png]]</center> || <center>[[File:Star-Classroom-BA.png]]</center> || <center>[[File:Star-Classroom-BA.png]]</center> || <center>[[File:Star-Classroom-BA.png]]</center><br />
|-<br />
| <pre>:grin:</pre> || <center>[[File:Teeth-Classroom.gif]]</center> || <center>[[File:Teeth-Classroom-Alex.png]]</center> || <center>[[File:Teeth-Classroom-Grogg.png]]</center> || <center>[[File:Teeth-Classroom-Lizzie.png]]</center> || <center>[[File:Teeth-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:sneaky:</pre> || <center>[[File:Sneaky-Classroom.gif]]</center> || <center>[[File:Sneaky-Classroom-Alex.png]]</center> || <center>[[File:Sneaky-Classroom-Grogg.png]]</center> || <center>[[File:Sneaky-Classroom-Lizzie.png]]</center> || <center>[[File:Sneaky-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:zzz:</pre> || <center>[[File:Sleepy-Classroom.gif]]</center> || <center>[[File:Sleepy-Classroom-Alex.png]]</center> || <center>[[File:Sleepy-Classroom-Grogg.png]]</center> || <center>[[File:Sleepy-Classroom-Lizzie.png]]</center> || <center>[[File:Sleepy-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:confused:</pre> || <center>[[File:Confuse-Classroom.gif]]</center> || <center>[[File:Confuse-Classroom-Alex.png]]</center> || <center>[[File:Confuse-Classroom-Grogg.png]]</center> || <center>[[File:Confuse-Classroom-Lizzie.png]]</center> || <center>[[File:Confuse-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:laugh:</pre> || <center>[[File:Laugh-Classroom.gif]]</center> || <center>[[File:Laugh-Classroom-Alex.png]]</center> || <center>[[File:Laugh-Classroom-Grogg.png]]</center> || <center>[[File:Laugh-Classroom-Lizzie.png]]</center> || <center>[[File:Laugh-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:grimace:</pre> || || <center>[[File:Grimace-Classroom-Alex.png]]</center> || <center>[[File:Grimace-Classroom-Grogg.png]]</center> || <center>[[File:Grimace-Classroom-Lizzie.png]]</center> || <center>[[File:Grimace-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:smallgrin:</pre> || || <center>[[File:Smallgrin-Classroom-Alex.png]]</center> || <center>[[File:Smallgrin-Classroom-Grogg.png]]</center> || <center>[[File:Smallgrin-Classroom-Lizzie.png]]</center> || <center>[[File:Smallgrin-Classroom-Winnie.png]]</center><br />
|-<br />
| <pre>:ghost:</pre> || || <center>[[File:Ghost-Classroom-Alex.png]]</center> || <center>[[File:Ghost-Classroom-Grogg.png]]</center> || <center>[[File:Ghost-Classroom-Lizzie.png]]</center> || <center>[[File:Ghost-Classroom-Winnie.png]]</center><br />
|}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:CMC12_Problems&diff=168594Template:CMC12 Problems2021-12-27T23:48:00Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} CMC 12{{{ab|}}}]]''' ('''[[{{{year}}} CMC 12{{{ab|}}} Answer Key|Answer Key]]''')<br />Printable version: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> | [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44&year={{{year}}} AoPS Resources]<br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will ''require'' the use of a calculator). <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''75 minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 1|1]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 2|2]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 3|3]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 4|4]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 5|5]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 6|6]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 7|7]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 8|8]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 9|9]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 10|10]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 11|11]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 12|12]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 13|13]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 14|14]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 15|15]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 16|16]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 17|17]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 18|18]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 19|19]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 20|20]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 21|21]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 22|22]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 23|23]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 24|24]] '''•''' [[{{{year}}} CMC 12{{{ab|}}} Problems/Problem 25|25]]<br />
|}<noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AHSC_35_Problems&diff=168316Template:AHSC 35 Problems2021-12-23T00:21:29Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AHSME|{{{year}}} AHSC]]''' ('''[[{{{year}}} AHSME Answer Key|Answer Key]]''')<br />Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3415_amc_12/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3415_amc_12/{{{year}}} PDF] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 35-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''? minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AHSME Problems/Problem 1|1]] '''•''' [[{{{year}}} AHSME Problems/Problem 2|2]] '''•''' [[{{{year}}} AHSME Problems/Problem 3|3]] '''•''' [[{{{year}}} AHSME Problems/Problem 4|4]] '''•''' [[{{{year}}} AHSME Problems/Problem 5|5]] '''•''' [[{{{year}}} AHSME Problems/Problem 6|6]] '''•''' [[{{{year}}} AHSME Problems/Problem 7|7]] '''•''' [[{{{year}}} AHSME Problems/Problem 8|8]] '''•''' [[{{{year}}} AHSME Problems/Problem 9|9]] '''•''' [[{{{year}}} AHSME Problems/Problem 10|10]] '''•''' [[{{{year}}} AHSME Problems/Problem 11|11]] '''•''' [[{{{year}}} AHSME Problems/Problem 12|12]] '''•''' [[{{{year}}} AHSME Problems/Problem 13|13]] '''•''' [[{{{year}}} AHSME Problems/Problem 14|14]] '''•''' [[{{{year}}} AHSME Problems/Problem 15|15]] '''•''' [[{{{year}}} AHSME Problems/Problem 16|16]] '''•''' [[{{{year}}} AHSME Problems/Problem 17|17]] '''•''' [[{{{year}}} AHSME Problems/Problem 18|18]] '''•''' [[{{{year}}} AHSME Problems/Problem 19|19]] '''•''' [[{{{year}}} AHSME Problems/Problem 20|20]] '''•''' [[{{{year}}} AHSME Problems/Problem 21|21]] '''•''' [[{{{year}}} AHSME Problems/Problem 22|22]] '''•''' [[{{{year}}} AHSME Problems/Problem 23|23]] '''•''' [[{{{year}}} AHSME Problems/Problem 24|24]] '''•''' [[{{{year}}} AHSME Problems/Problem 25|25]] '''•''' [[{{{year}}} AHSME Problems/Problem 26|26]] '''•''' [[{{{year}}} AHSME Problems/Problem 27|27]] '''•''' [[{{{year}}} AHSME Problems/Problem 28|28]] '''•''' [[{{{year}}} AHSME Problems/Problem 29|29]] '''•''' [[{{{year}}} AHSME Problems/Problem 30|30]] '''•''' [[{{{year}}} AHSME Problems/Problem 31|31]] '''•''' [[{{{year}}} AHSME Problems/Problem 32|32]] '''•''' [[{{{year}}} AHSME Problems/Problem 33|33]] '''•''' [[{{{year}}} AHSME Problems/Problem 34|34]] '''•''' [[{{{year}}} AHSME Problems/Problem 35|35]]<br />
|}<noinclude><br />
<br />
To use, put at the top of an AHSC Problems page with 35 questions (like [[1972 AHSME Problems]]) and follow the syntax <pre><br />
{{AHSC 35 Problems<br />
|year=1972<br />
}}<br />
</pre><br />
<br />
</noinclude>[[Category:AHSME Problems]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AHSC_40_Problems&diff=168313Template:AHSC 40 Problems2021-12-23T00:18:24Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AHSME|{{{year}}} AHSC]]''' ('''[[{{{year}}} AHSME Answer Key|Answer Key]]''')<br />Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3415_amc_12/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3415_amc_12/{{{year}}} PDF]<br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 40-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''? minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AHSME Problems/Problem 1|1]] '''•''' [[{{{year}}} AHSME Problems/Problem 2|2]] '''•''' [[{{{year}}} AHSME Problems/Problem 3|3]] '''•''' [[{{{year}}} AHSME Problems/Problem 4|4]] '''•''' [[{{{year}}} AHSME Problems/Problem 5|5]] '''•''' [[{{{year}}} AHSME Problems/Problem 6|6]] '''•''' [[{{{year}}} AHSME Problems/Problem 7|7]] '''•''' [[{{{year}}} AHSME Problems/Problem 8|8]] '''•''' [[{{{year}}} AHSME Problems/Problem 9|9]] '''•''' [[{{{year}}} AHSME Problems/Problem 10|10]] '''•''' [[{{{year}}} AHSME Problems/Problem 11|11]] '''•''' [[{{{year}}} AHSME Problems/Problem 12|12]] '''•''' [[{{{year}}} AHSME Problems/Problem 13|13]] '''•''' [[{{{year}}} AHSME Problems/Problem 14|14]] '''•''' [[{{{year}}} AHSME Problems/Problem 15|15]] '''•''' [[{{{year}}} AHSME Problems/Problem 16|16]] '''•''' [[{{{year}}} AHSME Problems/Problem 17|17]] '''•''' [[{{{year}}} AHSME Problems/Problem 18|18]] '''•''' [[{{{year}}} AHSME Problems/Problem 19|19]] '''•''' [[{{{year}}} AHSME Problems/Problem 20|20]] '''•''' [[{{{year}}} AHSME Problems/Problem 21|21]] '''•''' [[{{{year}}} AHSME Problems/Problem 22|22]] '''•''' [[{{{year}}} AHSME Problems/Problem 23|23]] '''•''' [[{{{year}}} AHSME Problems/Problem 24|24]] '''•''' [[{{{year}}} AHSME Problems/Problem 25|25]] '''•''' [[{{{year}}} AHSME Problems/Problem 26|26]] '''•''' [[{{{year}}} AHSME Problems/Problem 27|27]] '''•''' [[{{{year}}} AHSME Problems/Problem 28|28]] '''•''' [[{{{year}}} AHSME Problems/Problem 29|29]] '''•''' [[{{{year}}} AHSME Problems/Problem 30|30]] '''•''' [[{{{year}}} AHSME Problems/Problem 31|31]] '''•''' [[{{{year}}} AHSME Problems/Problem 32|32]] '''•''' [[{{{year}}} AHSME Problems/Problem 33|33]] '''•''' [[{{{year}}} AHSME Problems/Problem 34|34]] '''•''' [[{{{year}}} AHSME Problems/Problem 35|35]] '''•''' [[{{{year}}} AHSME Problems/Problem 36|36]] '''•''' [[{{{year}}} AHSME Problems/Problem 37|37]] '''•''' [[{{{year}}} AHSME Problems/Problem 38|38]] '''•''' [[{{{year}}} AHSME Problems/Problem 39|39]] '''•''' [[{{{year}}} AHSME Problems/Problem 40|40]]<br />
|}<noinclude><br />
<br />
To use, put at the top of an AHSC Problems page with 40 questions (like [[1967 AHSME Problems]]) and follow the syntax <pre><br />
{{AHSC 40 Problems<br />
|year=1967<br />
}}<br />
</pre><br />
<br />
</noinclude>[[Category:AHSME Problems]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AMC8_Problems&diff=168310Template:AMC8 Problems2021-12-23T00:07:01Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AMC 8]]''' ('''[[{{{year}}} AMC 8 Answer Key|Answer Key]]''')<br />Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3413_amc_8/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3413_amc_8/{{{year}}} PDF]<br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive 1 point for each correct answer. There is no penalty for wrong answers.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. In particular, calculators and computers are not permitted.<br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''40 minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AMC 8 Problems/Problem 1|1]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 2|2]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 3|3]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 4|4]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 5|5]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 6|6]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 7|7]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 8|8]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 9|9]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 10|10]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 11|11]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 12|12]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 13|13]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 14|14]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 15|15]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 16|16]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 17|17]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 18|18]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 19|19]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 20|20]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 21|21]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 22|22]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 23|23]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 24|24]] '''•''' [[{{{year}}} AMC 8 Problems/Problem 25|25]]<br />
|}<noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AJHSME_Problems&diff=168309Template:AJHSME Problems2021-12-22T23:54:27Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AJHSME]]''' ('''[[{{{year}}} AJHSME Answer Key|Answer Key]]''')<br />Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3413_amc_8/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3413_amc_8/{{{year}}} PDF]<br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''? minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AJHSME Problems/Problem 1|1]] '''•''' [[{{{year}}} AJHSME Problems/Problem 2|2]] '''•''' [[{{{year}}} AJHSME Problems/Problem 3|3]] '''•''' [[{{{year}}} AJHSME Problems/Problem 4|4]] '''•''' [[{{{year}}} AJHSME Problems/Problem 5|5]] '''•''' [[{{{year}}} AJHSME Problems/Problem 6|6]] '''•''' [[{{{year}}} AJHSME Problems/Problem 7|7]] '''•''' [[{{{year}}} AJHSME Problems/Problem 8|8]] '''•''' [[{{{year}}} AJHSME Problems/Problem 9|9]] '''•''' [[{{{year}}} AJHSME Problems/Problem 10|10]] '''•''' [[{{{year}}} AJHSME Problems/Problem 11|11]] '''•''' [[{{{year}}} AJHSME Problems/Problem 12|12]] '''•''' [[{{{year}}} AJHSME Problems/Problem 13|13]] '''•''' [[{{{year}}} AJHSME Problems/Problem 14|14]] '''•''' [[{{{year}}} AJHSME Problems/Problem 15|15]] '''•''' [[{{{year}}} AJHSME Problems/Problem 16|16]] '''•''' [[{{{year}}} AJHSME Problems/Problem 17|17]] '''•''' [[{{{year}}} AJHSME Problems/Problem 18|18]] '''•''' [[{{{year}}} AJHSME Problems/Problem 19|19]] '''•''' [[{{{year}}} AJHSME Problems/Problem 20|20]] '''•''' [[{{{year}}} AJHSME Problems/Problem 21|21]] '''•''' [[{{{year}}} AJHSME Problems/Problem 22|22]] '''•''' [[{{{year}}} AJHSME Problems/Problem 23|23]] '''•''' [[{{{year}}} AJHSME Problems/Problem 24|24]] '''•''' [[{{{year}}} AJHSME Problems/Problem 25|25]]<br />
|}<noinclude><br />
<br />
To use, put at the top of an AJHSME Problems page (like [[1999 AJHSME Problems]]) and follow the syntax <pre><br />
{{AJHSME Problems|year=1999}}<br />
</pre><br />
<br />
</noinclude>[[Category:AJHSME Problems]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AIME_Problems&diff=168307Template:AIME Problems2021-12-22T23:50:00Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AIME {{{n|}}}]]''' ('''[[{{{year}}} AIME {{{n|}}} Answer Key|Answer Key]]''')<br /> <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Printable version]</span> | [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year={{{year}}} AoPS Contest Collections] • [https://artofproblemsolving.com/community/contest/download/c3414_amc_10/{{{year}}} PDF]<br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 15-question, 3-hour examination. All answers are integers ranging from <math>000</math> to <math>999</math>, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.<br />
# No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, <u>'''calculators and computers are not permitted.'''</u> <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AIME {{{n|}}} Problems/Problem 1|1]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 2|2]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 3|3]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 4|4]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 5|5]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 6|6]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 7|7]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 8|8]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 9|9]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 10|10]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 11|11]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 12|12]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 13|13]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 14|14]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 15|15]]<br />
|}<includeonly></includeonly><noinclude><br />
<br />
<br />
To use this template, type in:<br />
<br />
<pre><br />
{{AIME Problems|year=}}<br />
</pre><br />
<br />
If the AIME has a number (I or II), type on:<br />
<br />
<pre><br />
{{AIME Problems|year=|n=}}<br />
</pre><br />
<br />
<br />
For example,<br />
<pre><br />
{{AIME Problems|year=1983}}<br />
</pre><br />
<br />
<pre><br />
{{AIME Problems|year=2012|n=II}}<br />
</pre><br />
<br />
These yield, respectively:<br />
<br />
{{AIME Problems|year=1983}}<br />
<br />
{{AIME Problems|year=2012|n=II}}<br />
<br />
</noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AIME_Problems&diff=168306Template:AIME Problems2021-12-22T23:48:42Z<p>Jwelsh: Undo revision 168305 by Jwelsh (talk)</p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AIME {{{n|}}}]]''' ('''[[{{{year}}} AIME {{{n|}}} Answer Key|Answer Key]]''')<br /> <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Printable version]</span> | [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year={{{year}}} AoPS Contest Collections] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 15-question, 3-hour examination. All answers are integers ranging from <math>000</math> to <math>999</math>, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.<br />
# No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, <u>'''calculators and computers are not permitted.'''</u> <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AIME {{{n|}}} Problems/Problem 1|1]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 2|2]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 3|3]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 4|4]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 5|5]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 6|6]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 7|7]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 8|8]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 9|9]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 10|10]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 11|11]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 12|12]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 13|13]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 14|14]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 15|15]]<br />
|}<includeonly></includeonly><noinclude><br />
<br />
<br />
To use this template, type in:<br />
<br />
<pre><br />
{{AIME Problems|year=}}<br />
</pre><br />
<br />
If the AIME has a number (I or II), type on:<br />
<br />
<pre><br />
{{AIME Problems|year=|n=}}<br />
</pre><br />
<br />
<br />
For example,<br />
<pre><br />
{{AIME Problems|year=1983}}<br />
</pre><br />
<br />
<pre><br />
{{AIME Problems|year=2012|n=II}}<br />
</pre><br />
<br />
These yield, respectively:<br />
<br />
{{AIME Problems|year=1983}}<br />
<br />
{{AIME Problems|year=2012|n=II}}<br />
<br />
</noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AIME_Problems&diff=168305Template:AIME Problems2021-12-22T23:47:54Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AIME {{{n|}}}]]''' ('''[[{{{year}}} AIME {{{n|}}} Answer Key|Answer Key]]''')<br /> <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Printable version]</span> | [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year={{{year}}} AoPS Contest Collections• [https://artofproblemsolving.com/community/contest/download/c3414_amc_10/{{{year}}} PDF]] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 15-question, 3-hour examination. All answers are integers ranging from <math>000</math> to <math>999</math>, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.<br />
# No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, <u>'''calculators and computers are not permitted.'''</u> <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AIME {{{n|}}} Problems/Problem 1|1]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 2|2]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 3|3]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 4|4]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 5|5]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 6|6]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 7|7]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 8|8]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 9|9]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 10|10]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 11|11]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 12|12]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 13|13]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 14|14]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 15|15]]<br />
|}<includeonly></includeonly><noinclude><br />
<br />
<br />
To use this template, type in:<br />
<br />
<pre><br />
{{AIME Problems|year=}}<br />
</pre><br />
<br />
If the AIME has a number (I or II), type on:<br />
<br />
<pre><br />
{{AIME Problems|year=|n=}}<br />
</pre><br />
<br />
<br />
For example,<br />
<pre><br />
{{AIME Problems|year=1983}}<br />
</pre><br />
<br />
<pre><br />
{{AIME Problems|year=2012|n=II}}<br />
</pre><br />
<br />
These yield, respectively:<br />
<br />
{{AIME Problems|year=1983}}<br />
<br />
{{AIME Problems|year=2012|n=II}}<br />
<br />
</noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AIME_Problems&diff=168304Template:AIME Problems2021-12-22T23:46:50Z<p>Jwelsh: Reverted edits by Jwelsh (talk) to last revision by LauraZed</p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AIME {{{n|}}}]]''' ('''[[{{{year}}} AIME {{{n|}}} Answer Key|Answer Key]]''')<br /> <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Printable version]</span> | [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=45&year={{{year}}} AoPS Contest Collections] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 15-question, 3-hour examination. All answers are integers ranging from <math>000</math> to <math>999</math>, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.<br />
# No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, <u>'''calculators and computers are not permitted.'''</u> <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AIME {{{n|}}} Problems/Problem 1|1]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 2|2]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 3|3]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 4|4]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 5|5]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 6|6]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 7|7]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 8|8]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 9|9]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 10|10]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 11|11]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 12|12]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 13|13]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 14|14]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 15|15]]<br />
|}<includeonly></includeonly><noinclude><br />
<br />
<br />
To use this template, type in:<br />
<br />
<pre><br />
{{AIME Problems|year=}}<br />
</pre><br />
<br />
If the AIME has a number (I or II), type on:<br />
<br />
<pre><br />
{{AIME Problems|year=|n=}}<br />
</pre><br />
<br />
<br />
For example,<br />
<pre><br />
{{AIME Problems|year=1983}}<br />
</pre><br />
<br />
<pre><br />
{{AIME Problems|year=2012|n=II}}<br />
</pre><br />
<br />
These yield, respectively:<br />
<br />
{{AIME Problems|year=1983}}<br />
<br />
{{AIME Problems|year=2012|n=II}}<br />
<br />
</noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AIME_Problems&diff=168303Template:AIME Problems2021-12-22T23:46:14Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AIME {{{n|}}}]]''' ('''[[{{{year}}} AIME {{{n|}}} Answer Key|Answer Key]]''')<br /> <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} • [https://artofproblemsolving.com/community/contest/collection/c3416_aime_problems/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3416_aime_problems/{{{year}}} PDF] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 15-question, 3-hour examination. All answers are integers ranging from <math>000</math> to <math>999</math>, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.<br />
# No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, <u>'''calculators and computers are not permitted.'''</u> <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AIME {{{n|}}} Problems/Problem 1|1]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 2|2]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 3|3]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 4|4]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 5|5]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 6|6]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 7|7]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 8|8]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 9|9]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 10|10]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 11|11]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 12|12]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 13|13]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 14|14]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 15|15]]<br />
|}<includeonly></includeonly><noinclude><br />
<br />
<br />
To use this template, type in:<br />
<br />
<pre><br />
{{AIME Problems|year=}}<br />
</pre><br />
<br />
If the AIME has a number (I or II), type on:<br />
<br />
<pre><br />
{{AIME Problems|year=|n=}}<br />
</pre><br />
<br />
<br />
For example,<br />
<pre><br />
{{AIME Problems|year=1983}}<br />
</pre><br />
<br />
<pre><br />
{{AIME Problems|year=2012|n=II}}<br />
</pre><br />
<br />
These yield, respectively:<br />
<br />
{{AIME Problems|year=1983}}<br />
<br />
{{AIME Problems|year=2012|n=II}}<br />
<br />
</noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AIME_Problems&diff=168302Template:AIME Problems2021-12-22T23:45:04Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AIME {{{n|}}}]]''' ('''[[{{{year}}} AIME {{{n|}}} Answer Key|Answer Key]]''')<br /> <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3416_aime_problems/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3416_aime_problems/{{{year}}} PDF] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 15-question, 3-hour examination. All answers are integers ranging from <math>000</math> to <math>999</math>, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.<br />
# No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, <u>'''calculators and computers are not permitted.'''</u> <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AIME {{{n|}}} Problems/Problem 1|1]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 2|2]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 3|3]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 4|4]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 5|5]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 6|6]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 7|7]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 8|8]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 9|9]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 10|10]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 11|11]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 12|12]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 13|13]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 14|14]] '''•''' [[{{{year}}} AIME {{{n|}}} Problems/Problem 15|15]]<br />
|}<includeonly></includeonly><noinclude><br />
<br />
<br />
To use this template, type in:<br />
<br />
<pre><br />
{{AIME Problems|year=}}<br />
</pre><br />
<br />
If the AIME has a number (I or II), type on:<br />
<br />
<pre><br />
{{AIME Problems|year=|n=}}<br />
</pre><br />
<br />
<br />
For example,<br />
<pre><br />
{{AIME Problems|year=1983}}<br />
</pre><br />
<br />
<pre><br />
{{AIME Problems|year=2012|n=II}}<br />
</pre><br />
<br />
These yield, respectively:<br />
<br />
{{AIME Problems|year=1983}}<br />
<br />
{{AIME Problems|year=2012|n=II}}<br />
<br />
</noinclude></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AHSME_35_Problems&diff=168301Template:AHSME 35 Problems2021-12-22T23:36:21Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AHSME]]''' ('''[[{{{year}}} AHSME Answer Key|Answer Key]]''')<br />Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3415_amc_12ahsme/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3415_amc_12ahsme/{{{year}}} PDF] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 35-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''? minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AHSME Problems/Problem 1|1]] '''•''' [[{{{year}}} AHSME Problems/Problem 2|2]] '''•''' [[{{{year}}} AHSME Problems/Problem 3|3]] '''•''' [[{{{year}}} AHSME Problems/Problem 4|4]] '''•''' [[{{{year}}} AHSME Problems/Problem 5|5]] '''•''' [[{{{year}}} AHSME Problems/Problem 6|6]] '''•''' [[{{{year}}} AHSME Problems/Problem 7|7]] '''•''' [[{{{year}}} AHSME Problems/Problem 8|8]] '''•''' [[{{{year}}} AHSME Problems/Problem 9|9]] '''•''' [[{{{year}}} AHSME Problems/Problem 10|10]] '''•''' [[{{{year}}} AHSME Problems/Problem 11|11]] '''•''' [[{{{year}}} AHSME Problems/Problem 12|12]] '''•''' [[{{{year}}} AHSME Problems/Problem 13|13]] '''•''' [[{{{year}}} AHSME Problems/Problem 14|14]] '''•''' [[{{{year}}} AHSME Problems/Problem 15|15]] '''•''' [[{{{year}}} AHSME Problems/Problem 16|16]] '''•''' [[{{{year}}} AHSME Problems/Problem 17|17]] '''•''' [[{{{year}}} AHSME Problems/Problem 18|18]] '''•''' [[{{{year}}} AHSME Problems/Problem 19|19]] '''•''' [[{{{year}}} AHSME Problems/Problem 20|20]] '''•''' [[{{{year}}} AHSME Problems/Problem 21|21]] '''•''' [[{{{year}}} AHSME Problems/Problem 22|22]] '''•''' [[{{{year}}} AHSME Problems/Problem 23|23]] '''•''' [[{{{year}}} AHSME Problems/Problem 24|24]] '''•''' [[{{{year}}} AHSME Problems/Problem 25|25]] '''•''' [[{{{year}}} AHSME Problems/Problem 26|26]] '''•''' [[{{{year}}} AHSME Problems/Problem 27|27]] '''•''' [[{{{year}}} AHSME Problems/Problem 28|28]] '''•''' [[{{{year}}} AHSME Problems/Problem 29|29]] '''•''' [[{{{year}}} AHSME Problems/Problem 30|30]] '''•''' [[{{{year}}} AHSME Problems/Problem 31|31]] '''•''' [[{{{year}}} AHSME Problems/Problem 32|32]] '''•''' [[{{{year}}} AHSME Problems/Problem 33|33]] '''•''' [[{{{year}}} AHSME Problems/Problem 34|34]] '''•''' [[{{{year}}} AHSME Problems/Problem 35|35]]<br />
|}<noinclude><br />
<br />
To use, put at the top of an AHSME Problems page with 35 questions (like [[1973 AHSME Problems]]) and follow the syntax <pre><br />
{{AHSME Problems<br />
|year=1973<br />
}}<br />
</pre><br />
<br />
</noinclude>[[Category:AHSME Problems]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AHSME_Problems&diff=168298Template:AHSME Problems2021-12-22T23:28:44Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AHSME]]''' ('''[[{{{year}}} AHSME Answer Key|Answer Key]]''')<br />Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3415_amc_12ahsme/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3415_amc_12ahsme/{{{year}}} PDF]<br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 30-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive 5 points for each correct answer, 2 points for each problem left unanswered, and 0 points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''90 minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AHSME Problems/Problem 1|1]] '''•''' [[{{{year}}} AHSME Problems/Problem 2|2]] '''•''' [[{{{year}}} AHSME Problems/Problem 3|3]] '''•''' [[{{{year}}} AHSME Problems/Problem 4|4]] '''•''' [[{{{year}}} AHSME Problems/Problem 5|5]] '''•''' [[{{{year}}} AHSME Problems/Problem 6|6]] '''•''' [[{{{year}}} AHSME Problems/Problem 7|7]] '''•''' [[{{{year}}} AHSME Problems/Problem 8|8]] '''•''' [[{{{year}}} AHSME Problems/Problem 9|9]] '''•''' [[{{{year}}} AHSME Problems/Problem 10|10]] '''•''' [[{{{year}}} AHSME Problems/Problem 11|11]] '''•''' [[{{{year}}} AHSME Problems/Problem 12|12]] '''•''' [[{{{year}}} AHSME Problems/Problem 13|13]] '''•''' [[{{{year}}} AHSME Problems/Problem 14|14]] '''•''' [[{{{year}}} AHSME Problems/Problem 15|15]] '''•''' [[{{{year}}} AHSME Problems/Problem 16|16]] '''•''' [[{{{year}}} AHSME Problems/Problem 17|17]] '''•''' [[{{{year}}} AHSME Problems/Problem 18|18]] '''•''' [[{{{year}}} AHSME Problems/Problem 19|19]] '''•''' [[{{{year}}} AHSME Problems/Problem 20|20]] '''•''' [[{{{year}}} AHSME Problems/Problem 21|21]] '''•''' [[{{{year}}} AHSME Problems/Problem 22|22]] '''•''' [[{{{year}}} AHSME Problems/Problem 23|23]] '''•''' [[{{{year}}} AHSME Problems/Problem 24|24]] '''•''' [[{{{year}}} AHSME Problems/Problem 25|25]] '''•''' [[{{{year}}} AHSME Problems/Problem 26|26]] '''•''' [[{{{year}}} AHSME Problems/Problem 27|27]] '''•''' [[{{{year}}} AHSME Problems/Problem 28|28]] '''•''' [[{{{year}}} AHSME Problems/Problem 29|29]] '''•''' [[{{{year}}} AHSME Problems/Problem 30|30]]<br />
|}<noinclude><br />
<br />
To use, put at the top of an AHSME Problems page (like [[1998 AHSME Problems]]) and follow the syntax <pre><br />
{{AHSME Problems|year=1998}}<br />
</pre><br />
<br />
</noinclude>[[Category:AHSME Problems]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AHSC_35_Problems&diff=168297Template:AHSC 35 Problems2021-12-22T23:25:21Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AHSME|{{{year}}} AHSC]]''' ('''[[{{{year}}} AHSME Answer Key|Answer Key]]''')<br />Printable version: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> | [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43&year={{{year}}} AoPS Resources] • [http://www.artofproblemsolving.com/Forum/resources/files/usa/USA-AMC_10-{{{year}}}-43 PDF] <br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 35-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''? minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AHSME Problems/Problem 1|1]] '''•''' [[{{{year}}} AHSME Problems/Problem 2|2]] '''•''' [[{{{year}}} AHSME Problems/Problem 3|3]] '''•''' [[{{{year}}} AHSME Problems/Problem 4|4]] '''•''' [[{{{year}}} AHSME Problems/Problem 5|5]] '''•''' [[{{{year}}} AHSME Problems/Problem 6|6]] '''•''' [[{{{year}}} AHSME Problems/Problem 7|7]] '''•''' [[{{{year}}} AHSME Problems/Problem 8|8]] '''•''' [[{{{year}}} AHSME Problems/Problem 9|9]] '''•''' [[{{{year}}} AHSME Problems/Problem 10|10]] '''•''' [[{{{year}}} AHSME Problems/Problem 11|11]] '''•''' [[{{{year}}} AHSME Problems/Problem 12|12]] '''•''' [[{{{year}}} AHSME Problems/Problem 13|13]] '''•''' [[{{{year}}} AHSME Problems/Problem 14|14]] '''•''' [[{{{year}}} AHSME Problems/Problem 15|15]] '''•''' [[{{{year}}} AHSME Problems/Problem 16|16]] '''•''' [[{{{year}}} AHSME Problems/Problem 17|17]] '''•''' [[{{{year}}} AHSME Problems/Problem 18|18]] '''•''' [[{{{year}}} AHSME Problems/Problem 19|19]] '''•''' [[{{{year}}} AHSME Problems/Problem 20|20]] '''•''' [[{{{year}}} AHSME Problems/Problem 21|21]] '''•''' [[{{{year}}} AHSME Problems/Problem 22|22]] '''•''' [[{{{year}}} AHSME Problems/Problem 23|23]] '''•''' [[{{{year}}} AHSME Problems/Problem 24|24]] '''•''' [[{{{year}}} AHSME Problems/Problem 25|25]] '''•''' [[{{{year}}} AHSME Problems/Problem 26|26]] '''•''' [[{{{year}}} AHSME Problems/Problem 27|27]] '''•''' [[{{{year}}} AHSME Problems/Problem 28|28]] '''•''' [[{{{year}}} AHSME Problems/Problem 29|29]] '''•''' [[{{{year}}} AHSME Problems/Problem 30|30]] '''•''' [[{{{year}}} AHSME Problems/Problem 31|31]] '''•''' [[{{{year}}} AHSME Problems/Problem 32|32]] '''•''' [[{{{year}}} AHSME Problems/Problem 33|33]] '''•''' [[{{{year}}} AHSME Problems/Problem 34|34]] '''•''' [[{{{year}}} AHSME Problems/Problem 35|35]]<br />
|}<noinclude><br />
<br />
To use, put at the top of an AHSC Problems page with 35 questions (like [[1972 AHSME Problems]]) and follow the syntax <pre><br />
{{AHSC 35 Problems<br />
|year=1972<br />
}}<br />
</pre><br />
<br />
</noinclude>[[Category:AHSME Problems]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Template:AHSC_35_Problems&diff=168296Template:AHSC 35 Problems2021-12-22T23:23:37Z<p>Jwelsh: </p>
<hr />
<div>{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:50%;"<br />
| style="background:#ccf;text-align:center;" colspan="3" | '''[[{{{year}}} AHSME|{{{year}}} AHSC]]''' ('''[[{{{year}}} AHSME Answer Key|Answer Key]]''')<br />Printable versions: <span class="noprint plainlinks">[{{fullurl:{{FULLPAGENAME}}|printable=yes}} Wiki]</span> • [https://artofproblemsolving.com/community/contest/collection/c3414_amc_10/{{{year}}} AoPS Resources] • [https://artofproblemsolving.com/community/contest/download/c3414_amc_10/{{{year}}} PDF]<br />
|-<br />
| style="font-size:80%;"|<br />
<font style="text-align:center;">Instructions</font><br />
# This is a 35-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. <br />
# You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.<br />
# No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers. <br />
# Figures are not necessarily drawn to scale.<br />
# You will have '''? minutes''' working time to complete the test. <br />
|-<br />
| style="text-align:center;"| [[{{{year}}} AHSME Problems/Problem 1|1]] '''•''' [[{{{year}}} AHSME Problems/Problem 2|2]] '''•''' [[{{{year}}} AHSME Problems/Problem 3|3]] '''•''' [[{{{year}}} AHSME Problems/Problem 4|4]] '''•''' [[{{{year}}} AHSME Problems/Problem 5|5]] '''•''' [[{{{year}}} AHSME Problems/Problem 6|6]] '''•''' [[{{{year}}} AHSME Problems/Problem 7|7]] '''•''' [[{{{year}}} AHSME Problems/Problem 8|8]] '''•''' [[{{{year}}} AHSME Problems/Problem 9|9]] '''•''' [[{{{year}}} AHSME Problems/Problem 10|10]] '''•''' [[{{{year}}} AHSME Problems/Problem 11|11]] '''•''' [[{{{year}}} AHSME Problems/Problem 12|12]] '''•''' [[{{{year}}} AHSME Problems/Problem 13|13]] '''•''' [[{{{year}}} AHSME Problems/Problem 14|14]] '''•''' [[{{{year}}} AHSME Problems/Problem 15|15]] '''•''' [[{{{year}}} AHSME Problems/Problem 16|16]] '''•''' [[{{{year}}} AHSME Problems/Problem 17|17]] '''•''' [[{{{year}}} AHSME Problems/Problem 18|18]] '''•''' [[{{{year}}} AHSME Problems/Problem 19|19]] '''•''' [[{{{year}}} AHSME Problems/Problem 20|20]] '''•''' [[{{{year}}} AHSME Problems/Problem 21|21]] '''•''' [[{{{year}}} AHSME Problems/Problem 22|22]] '''•''' [[{{{year}}} AHSME Problems/Problem 23|23]] '''•''' [[{{{year}}} AHSME Problems/Problem 24|24]] '''•''' [[{{{year}}} AHSME Problems/Problem 25|25]] '''•''' [[{{{year}}} AHSME Problems/Problem 26|26]] '''•''' [[{{{year}}} AHSME Problems/Problem 27|27]] '''•''' [[{{{year}}} AHSME Problems/Problem 28|28]] '''•''' [[{{{year}}} AHSME Problems/Problem 29|29]] '''•''' [[{{{year}}} AHSME Problems/Problem 30|30]] '''•''' [[{{{year}}} AHSME Problems/Problem 31|31]] '''•''' [[{{{year}}} AHSME Problems/Problem 32|32]] '''•''' [[{{{year}}} AHSME Problems/Problem 33|33]] '''•''' [[{{{year}}} AHSME Problems/Problem 34|34]] '''•''' [[{{{year}}} AHSME Problems/Problem 35|35]]<br />
|}<noinclude><br />
<br />
To use, put at the top of an AHSC Problems page with 35 questions (like [[1972 AHSME Problems]]) and follow the syntax <pre><br />
{{AHSC 35 Problems<br />
|year=1972<br />
}}<br />
</pre><br />
<br />
</noinclude>[[Category:AHSME Problems]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Cooga_Georgeooga-Harryooga_Theorem&diff=167435Cooga Georgeooga-Harryooga Theorem2021-12-07T16:09:20Z<p>Jwelsh: Reverted edits by Cozzmo (talk) to last revision by Sosiaops</p>
<hr />
<div>=Definition=<br />
The Cooga Georgeooga-Harryooga Theorem (Circular Georgeooga-Harryooga Theorem) states that if you have <math>a</math> distinguishable objects and <math>b</math> objects are kept away from each other, then there are <math>\frac{(a-b)!^2}{(a-2b)!}</math> ways to arrange the objects in a circle.<br />
<br />
<br />
Created by George and Harry of [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ The Ooga Booga Tribe of The Caveman Society]<br />
<br />
=Proofs=<br />
==Proof 1==<br />
Let our group of <math>a</math> objects be represented like so <math>1</math>, <math>2</math>, <math>3</math>, ..., <math>a-1</math>, <math>a</math>. Let the last <math>b</math> objects be the ones we can't have together.<br />
<br />
Then we can organize our objects like so <asy><br />
label("$1$", dir(90));<br />
label("BLANK", dir(60));<br />
label("$2$", dir(30));<br />
label("BLANK", dir(0));<br />
label("$3$", dir(-30));<br />
label("BLANK", dir(-60));<br />
label("$\dots$", dir(-90));<br />
label("BLANK", dir(-120));<br />
label("$a-b-1$", dir(-150));<br />
label("BLANK", dir(-180));<br />
label("$a-b$", dir(-210));<br />
label("BLANK", dir(-240));<br />
</asy><br />
<br />
We have <math>(a-b)!</math> ways to arrange the objects in that list.<br />
<br />
Now we have <math>a-b</math> blanks and <math>b</math> other objects so we have <math>_{a-b}P_{b}=\frac{(a-b)!}{(a-2b)!}</math> ways to arrange the objects we can't put together.<br />
<br />
By The Fundamental Counting Principal our answer is <math>\frac{(a-b)!^2}{(a-2b)!}</math>.<br />
<br />
<br />
Proof by [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:12, 1 February 2021 (EST)<br />
<br />
=Applications=<br />
==Application 1==<br />
===Problem===<br />
===Solutions===<br />
====Solution 1====<br />
<br />
=Testimonials=<br />
"Thanks for rediscovering our theorem [[User:Redfiretruck|RedFireTruck]]" - George and Harry of [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ The Ooga Booga Tribe of The Caveman Society]<br />
<br />
"This is GREAT!!!" ~ hi..<br />
<br />
"This is a very nice theorem!" - [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 10:53, 1 February 2021 (EST)</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2018_AMC_10B_Problems/Problem_1&diff=1674342018 AMC 10B Problems/Problem 12021-12-07T16:09:18Z<p>Jwelsh: Reverted edits by Cozzmo (talk) to last revision by MRENTHUSIASM</p>
<hr />
<div>{{duplicate|[[2018 AMC 12B Problems|2018 AMC 12B #1]] and [[2018 AMC 10B Problems|2018 AMC 10B #1]]}}<br />
<br />
==Problem==<br />
Kate bakes a <math>20</math>-inch by <math>18</math>-inch pan of cornbread. The cornbread is cut into pieces that measure <math>2</math> inches by <math>2</math> inches. How many pieces of cornbread does the pan contain?<br />
<br />
<math>\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 360</math><br />
<br />
== Solution 1 ==<br />
<br />
The area of the pan is <math>20\cdot18=360</math>. Since the area of each piece is <math>2\cdot2=4</math>, there are <math>\frac{360}{4} = \boxed{\textbf{(A) } 90}</math> pieces.<br />
<br />
== Solution 2 ==<br />
<br />
By dividing each of the dimensions by <math>2</math>, we get a <math>10\times9</math> grid that makes <math>\boxed{\textbf{(A) } 90}</math> pieces.<br />
<br />
==Video Solution==<br />
https://youtu.be/o5MUHOmF1zo<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
<br />
{{AMC10 box|year=2018|ab=B|before=First Problem|num-a=2}}<br />
{{AMC12 box|year=2018|ab=B|before=First Problem|num-a=2}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_22&diff=1674332013 AMC 12B Problems/Problem 222021-12-07T16:09:17Z<p>Jwelsh: Reverted edits by Cozzmo (talk) to last revision by Mistymathmusic</p>
<hr />
<div>==Problem==<br />
Let <math>m>1</math> and <math>n>1</math> be integers. Suppose that the product of the solutions for <math>x</math> of the equation<br />
<cmath> 8(\log_n x)(\log_m x)-7\log_n x-6 \log_m x-2013 = 0 </cmath><br />
is the smallest possible integer. What is <math>m+n</math>?<br />
<br />
<math> \textbf{(A)}\ 12\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 272 </math><br />
<br />
==Solution==<br />
Rearranging logs, the original equation becomes<br />
<cmath>\frac{8}{\log n \log m}(\log x)^2 - \left(\frac{7}{\log n}+\frac{6}{\log m}\right)\log x - 2013 = 0</cmath><br />
<br />
By Vieta's Theorem, the sum of the possible values of <math>\log x</math> is <math>\frac{\frac{7}{\log n}+\frac{6}{\log m}}{\frac{8}{\log n \log m}} = \frac{7\log m + 6 \log n}{8} = \log \sqrt[8]{m^7n^6}</math>. But the sum of the possible values of <math>\log x</math> is the logarithm of the product of the possible values of <math>x</math>. Thus the product of the possible values of <math>x</math> is equal to <math>\sqrt[8]{m^7n^6}</math>.<br />
<br />
It remains to minimize the integer value of <math>\sqrt[8]{m^7n^6}</math>. Since <math>m, n>1</math>, we can check that <math>m = 2^2</math> and <math>n = 2^3</math> work. Thus the answer is <math>4+8 = \boxed{\textbf{(A)}\ 12}</math>.<br />
<br />
==Video Solution==<br />
For those who prefer a video solution: https://www.youtube.com/watch?v=vX0y9lRv9OM&t=312s<br />
<br />
== See also ==<br />
{{AMC12 box|year=2013|ab=B|num-b=21|num-a=23}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=User_talk:Cozzmo&diff=167432User talk:Cozzmo2021-12-07T16:09:15Z<p>Jwelsh: Reverted edits by Cozzmo (talk) to last revision by Piphi</p>
<hr />
<div>Hi! i am cozzmo! i cant tell u my real name, but i like python, biking, legos, robotics, airplanes and python.<br />
<br />
Hey cozzmo, are you postbanned? --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 18:19, 9 December 2020 (EST)</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=User:Piphi&diff=167431User:Piphi2021-12-07T16:09:14Z<p>Jwelsh: Reverted edits by Cozzmo (talk) to last revision by Falco sparverius</p>
<hr />
<div>{{User:Piphi/Template:Header}}<br />
<br><br />
__NOTOC__<div style="border:2px solid black; -webkit-border-radius: 10px; background:#F0F2F3"><br />
==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">User Count</div></font>==<br />
<div style="margin-left: 10px; margin-bottom:10px"><font color="black">If this is your first time visiting this page, edit it by incrementing the user count below by one.</font></div><br />
<center><font size="100px">537<br />
</font></center><br />
</div><br />
<div style="border:2px solid black; background:#919293;-webkit-border-radius: 10px; align:center"><br />
<br />
==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">About Me</div></font>==<br />
<div style="margin-left: 10px; margin-bottom:10px"><font color="black"><font color="black"><br />
Piphi created the [[User:Piphi/Games|AoPS Wiki Games]].<br><br />
<br />
Piphi has been very close to winning multiple [[Greed Control]] games, piphi placed 5th in game #18, 2nd in game #19, 10th in game #20, and 2nd in game #21. Piphi made a the Greed Control Hall of Fame [https://artofproblemsolving.com/community/h2445343 here].<br><br />
<br />
Piphi has a side-project that is making the Wiki's [[Main Page]] look better, you can check that out [[User:Piphi/AoPS Wiki|here]].<br><br />
<br />
Piphi is a proud member of [https://artofproblemsolving.com/community/c562043 The Interuniversal GMAAS Society].<br />
<br />
Piphi is part of the [https://artofproblemsolving.com/community/c1124279 Asymptote Competition] staff.<br />
</font></div><br />
</div><br />
<div style="border:2px solid black; background:#333333;-webkit-border-radius: 10px; align:center"><br />
<br />
==<font color="#f0f2f3" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">Goals</div></font>==<br />
<div style="margin-left: 10px; margin-right: 10px; margin-bottom:10px"><font color="#f0f2f3"><br />
You can check out more goals/statistics [[User:Piphi/Statistics|here]].<br />
<br />
A User Count of 1000<br />
{{User:Piphi/Template:Progress_Bar|53.2|width=100%}}<br />
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</div></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2021_AMC_10B&diff=1673952021 AMC 10B2021-12-06T19:15:34Z<p>Jwelsh: </p>
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<div>'''2021 AMC 10B''' problems and solutions. The test will be held on Wednesday, February 10, 2021.<br />
<br />
Please do not post the problems or the solutions until the contest is released.<br />
*[[2021 AMC 10B Problems]]<br />
*[[2021 AMC 10B Answer Key]]<br />
**[[2021 AMC 10B Problems/Problem 1|Problem 1]]<br />
**[[2021 AMC 10B Problems/Problem 2|Problem 2]]<br />
**[[2021 AMC 10B Problems/Problem 3|Problem 3]]<br />
**[[2021 AMC 10B Problems/Problem 4|Problem 4]]<br />
**[[2021 AMC 10B Problems/Problem 5|Problem 5]]<br />
**[[2021 AMC 10B Problems/Problem 6|Problem 6]]<br />
**[[2021 AMC 10B Problems/Problem 7|Problem 7]]<br />
**[[2021 AMC 10B Problems/Problem 8|Problem 8]]<br />
**[[2021 AMC 10B Problems/Problem 9|Problem 9]]<br />
**[[2021 AMC 10B Problems/Problem 10|Problem 10]]<br />
**[[2021 AMC 10B Problems/Problem 11|Problem 11]]<br />
**[[2021 AMC 10B Problems/Problem 12|Problem 12]]<br />
**[[2021 AMC 10B Problems/Problem 13|Problem 13]]<br />
**[[2021 AMC 10B Problems/Problem 14|Problem 14]]<br />
**[[2021 AMC 10B Problems/Problem 15|Problem 15]]<br />
**[[2021 AMC 10B Problems/Problem 16|Problem 16]]<br />
**[[2021 AMC 10B Problems/Problem 17|Problem 17]]<br />
**[[2021 AMC 10B Problems/Problem 18|Problem 18]]<br />
**[[2021 AMC 10B Problems/Problem 19|Problem 19]]<br />
**[[2021 AMC 10B Problems/Problem 20|Problem 20]]<br />
**[[2021 AMC 10B Problems/Problem 21|Problem 21]]<br />
**[[2021 AMC 10B Problems/Problem 22|Problem 22]]<br />
**[[2021 AMC 10B Problems/Problem 23|Problem 23]]<br />
**[[2021 AMC 10B Problems/Problem 24|Problem 24]]<br />
**[[2021 AMC 10B Problems/Problem 25|Problem 25]]<br />
==See also==<br />
{{AMC10 box|year=2021|ab=B|before=[[2021 AMC 10A]]|after=[[2021 Fall AMC 10A]]}}<br />
* [[AMC 10]]<br />
* [[AMC 10 Problems and Solutions]]<br />
* [[Mathematics competitions]]<br />
* [[Mathematics competition resources]]<br />
{{MAA Notice}}<br />
--></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=User:Blueclay&diff=163730User:Blueclay2021-10-18T21:43:02Z<p>Jwelsh: Undo revision</p>
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<div>This user is the Best in the world! ~Cozzmo</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=User:Jwelsh&diff=163729User:Jwelsh2021-10-18T21:41:49Z<p>Jwelsh: Undo revision</p>
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<div>Josette Welsh, commonly known as jwelsh, moved to San Diego with their family in 2011 when their father was stationed at Miramar. While at San Diego State University they did some editing and English tutoring. After graduating, the lovely weather, company, and museums convinced them to stay. In their free time, Josette enjoys tabletop roleplaying games (and designing Google Sheets based character sheets for them), walks around the neighborhood (and the beautiful Balboa Park), cooking (especially when there’s company), and sewing (though they're still getting back into it). They work on two Actual Play podcasts and love working on world development. When they can, Josette loves to take their niece to arcades (namely Nickel City) or host a game day at home.<br />
<br />
jwelsh is also an admin on AoPS. she is regarded as as LEGEND</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=AMC_10&diff=163493AMC 102021-10-14T17:55:46Z<p>Jwelsh: removed a broken link</p>
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<div>The '''American Mathematics Contest 10''' ('''AMC 10'''), along with the [[AMC 12]], is one of the first exams in the series of exams used to challenge bright students, grades 10 and below, on the path towards choosing the team that represents the United States at the [[International Mathematics Olympiad]] (IMO).<br />
<br />
High scoring AMC 10 and AMC 12 students are invited to take the [[American Invitational Mathematics Examination]] (AIME).<br />
<br />
The AMC 10 is administered by the [[American Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC.<br />
<br />
The AMC 12 used to be the [[American High School Mathematics Examination]] from 1950 to 1999.<br />
<br />
{{Contest Info|name=AMC 10|region=USA|type=Multiple Choice|difficulty=1-3|breakdown=<u>Problem 1-5</u>: 1<br><u>Problem 6-20</u>: 2<br><u>Problem 21-25</u>: 3}}<br />
<br />
== Format ==<br />
<br />
The AMC 10 is a 25 question, 75 minute multiple choice test. Problems generally increase in difficulty as the exam progresses. Calculators were permitted on old AMC tests; however, as of 2008 and later, calculators are not permitted for use.<br />
<br />
The AMC 10 is scored in a way that penalizes guesses. Correct answers are worth 6 points, incorrect questions are worth 0 points, and unanswered answers are worth 1.5 points, to give a total score out of 150 points. From 2002 to 2006, unanswered questions were awarded 2.5 points. In 2006 and 2007, unanswered questions were awarded 2 points. Students that are in the top 2.5% of the AMC 10 contest are invited to take the [[AIME]].[http://www.unl.edu/amc/e-exams/e7-aime/adminaime.html]<br />
<br />
== Curriculum ==<br />
The AMC 10 tests [[mathematical problem solving]] with [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], and [[probability]] and other secondary school math topics. Problems are designed to be solved by students without any background in calculus or trigonometry.<br />
<br />
== Resources ==<br />
=== Links ===<br />
* [http://www.maa.org/math-competitions AMC homepage], their [https://www.maa.org/math-competitions/amc-10-american-mathematics-contest-10 AMC 10 page], and [http://www.unl.edu/amc/mathclub/index.html practice problems]<br />
* [https://artofproblemsolving.com/community/c5_contests_amp_programs Contest and Programs Forum] for discussion of the AMC and problems from AMC exams, and any other competition.<br />
* [[AMC 10 Problems and Solutions | Past AMC 10 exams]].<br />
<br />
=== Recommended reading ===<br />
* [http://www.artofproblemsolving.com/Store/contests.php?contest=amc Problem and solution books for past AMC exams].<br />
* [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra Introduction to Algebra] by [[Richard Rusczyk]]. <br />
* [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:counting Introduction to Counting & Probability] by Dr. [[David Patrick]]. <br />
* [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometry Introduction to Geometry] by [[Richard Rusczyk]]. <br />
* [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt Introduction to Number Theory] by [[Mathew Crawford]].<br />
* [http://www.artofproblemsolving.com/Store/viewitem.php?item=ps:aops1 The Art of Problem Solving Volume 1] by [[Sandor Lehoczky]] and [[Richard Rusczyk]].<br />
* [http://www.artofproblemsolving.com/Store/viewitem.php?item=ps:aops2 The Art of Problem Solving Volume 2] by [[Richard Rusczyk]] and [[Sandor Lehoczky]].<br />
<br />
=== AMC Preparation Classes ===<br />
'''These should be taken if a student is having trouble or wants a more clear and confirmed understanding of concepts.'''<br />
* [[AoPS]] hosts an [http://www.artofproblemsolving.com/School/index.php online school] teaching introductory classes in topics covered by the AMC 10 as well as an AMC 10 preparation class.<br />
* [[AoPS]] holds many free [[Math Jams]], some of which are devoted to discussing problems on the AMC 10 and AMC 12. [http://www.artofproblemsolving.com/School/mathjams.php Math Jam Schedule]<br />
* [[EPGY]] offers an AMC 10 preparation class.<br />
<br />
== See also ==<br />
* [[Mathematics competitions]]<br />
* [[ARML]]<br />
* [[Mathematics summer programs]]<br />
<br />
<br />
<br />
[[Category:Mathematics competitions]]<br />
[[Category:Introductory mathematics competitions]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Gmaasology&diff=163435Gmaasology2021-10-12T19:45:09Z<p>Jwelsh: Undo revision 162881 by Sulfurdioxideso2 (talk)</p>
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<div>Just Published: [[An Introduction to Gmaasology]]<br />
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[[Gmaasology]] is the study of [[Gmaas]]. [[Gmaas]] is the only known living being who has a PhD in [[Gmaasology]] EDIT: That's not consistent with the Sseraj page. [[Gmaasology]] is also one of the most eminent fields of science, falling behind [[Physics]], [[Biology]], [[Chemistry]], Economics, Geology, and Computer Programming. To start learning [[Gmaasology]], click this link: [[Gmaas]], and remember the facts about [[Gmaas]].<br />
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[[Gmaasology]] has existed for much of human history. Here are a selection of historical events related to [[Gmaas]] and [[Gmaasology]] until the founding of the Interuniversal [[Gmaas]] Society:<br />
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c. 17,000-15,000 BC: The Lascaux cave paintings contain depictions of an omnipotent cat killing bison and humans alike. Some archaeologists have theorized that these paintings depict the great [[Gmaas]] himself.<br />
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c. 3000-1000 BC: Some Egyptologists suggest that the goddess Bastet, who was half human half cat, was [[Gmaas]] in disguise.<br />
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c. 400-800 AD: The Mayan city of Copán is the world center of [[Gmaasology]]. Most of the Gmaasological artifacts of the city can be seen in the Museum of [[Gmaas]] in New York City.<br />
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Many universities are beginning to offer classes in [[Gmaasology]], but the best place to study [[Gmaasology]] is the Interuniversal [[Gmaas]] Society, which was founded in 1314. This is a summary of the projects made by the Society since 1314, when it was founded: (Originally [[Gmaasology]] was called [[Gmaathamatics]].)<br />
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- 1314: Foundation of the National Gmaathematical Society of Florence.<br />
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- 1315 - 1498: Search for and copying of ancient and medieval manuscripts about [[Gmaas]].<br />
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- 1499: Move to France to a secret basement in François 1st's palace's basement.<br />
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- 1500: Renaming to the National Gmaathematical Society of Florence and France.<br />
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- 1501 - 1519: Collaboration with Leonardo Da Vinci, who, by the way, was NOT [[Gmaas]] in disguise.<br />
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- 1520: Move back to Florence.<br />
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- 1521: Move to Rome.<br />
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- 1522: Renaming of the Gmaathematical Society of Florence and France to the Gmaathematical Society of Florence, France, and the Papal States.<br />
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- 1523 - 1560: Search for ancient statues of [[Gmaas]] in the city of Rome.<br />
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- 1561 - 1565: Headquarters constructed in a secret basement under the Sistine Chapel.<br />
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- 1566 - 1575: Search in Italian libraries for manuscripts about [[Gmaas]].<br />
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- 1576 - 1591: Study of [[Gmaas]]'s reincarnating abilities.<br />
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- 1592 - 1601: Campaign to make [[Gmaathamatics]] the eight liberal art. This failed.<br />
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- 1602 - 1616: Many universities around Europe begin offering classes in [[Gmaathamatics]].<br />
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- 1617: Renaming of the Gmaathematical Society of Florence, France, and the Papal States to the Gmaathematical Society of Florence, France, the Papal States, Pisa, England, Spain, the Holy Roman Empire, and Milan to honor the universities offering [[Gmaathamatics]] classes. It was also known as GMSFFPSPESHREM for short.<br />
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- 1618: The University of Prague starts teaching [[Gmaathamatics]], so GMSFFPSPESHREM is renamed GMSFFPSPESHREMB. (B stands for Bohemia.)<br />
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- 1619 - 1650: GMSFFPSPESHREMB writes a book about how [[Gmaas]] reincarnates. It is called "De Multis Vitis Gmaasium," or "On the Many Lives of [[Gmaas]]."<br />
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- 1651 - 1655: All European universities stop offering classes in [[Gmaathamatics]] because the immortality of [[Gmaas]] was contrary to the Bible.<br />
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- 1656: GMSFFPSPESHREMB is renamed "The Gmaathematical Society of Europe," abbreviated as GSE.<br />
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- 1657: The Gmaathematical Society of Europe moves its headquarters to Amsterdam.<br />
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- 1658 - 1675: The GSE amasses every object and text having to do with [[Gmaas]] or [[Gmaathamatics]]. This is the reason why [[Gmaas]] is no longer famous; every trace of his existence was acquired by the GSE.<br />
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- 1676 - 1686: The present life of [[Gmaas]] died, so the GSE spent ten years trying to track down his next reincarnation. That reincarnation was an ant, so he died. It took even longer for the GSE to find another reincarnation of [[Gmaas]].<br />
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- 1687: Renaming of the Gmaathematical Society of Europe to the Gmaathematical Society of the World.<br />
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- 1688 - 1701: The Gmaathematical Society of the World captures [[Gmaas]], who at that time was a dog, and runs experiments on him.<br />
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- 1702: The Gmaathematical Society of the World moves to the New World. It decides to move to San Juan, Puerto Rico, but it keeps a minor location in its former headquarters in Amsterdam. <br />
<br />
- 1703 - 1730: The Gmaathematical Society of the World conducts archeological excavations of Mayan Temple where people worshiped [[Gmaas]]. The artifacts from the dig were sent to their headquarters in Puerto Rico, and the data about the excavation was copied and kept in San Juan and Amsterdam.<br />
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- 1731 - 1754: The Gmaathematical Society of the World excavates the Temple of [[Gmaas]] in Egypt. It opens a temporary minor location near the site.<br />
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- 1755: The minor locations in Egypt and in Amsterdam are closed. All the archaeological artifacts there are sent to San Juan, Puerto Rico.<br />
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- 1755 - 1763: The Seven Years War. The GSW frantically tries to move to Siberia, where it would be safe for the remainder of the war. Luckily, the GSW could save all of its thousands of artifacts, but many Gmaathmaticians died.<br />
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- 1764: The Gmaathematical Society of the World moves its headquarters back to San Juan, Puerto Rico.<br />
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- 1765: Closing of the facilities in Siberia.<br />
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- 1766: Opening of a minor location in Philadelphia.<br />
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- 1775 - 1783: During the American Revolution, the Gmaathematical Society of the World moves archaeological artifacts from San Juan and Philadelphia to a new minor location in Santa Fe, where they would be safe from the fighting.<br />
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- 1784: Some Gmaathamatical artifacts are moved back to Philadelphia and Santa Fe.<br />
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- 1784 - 1798: The [[Gmaas]] Exhibition at the minor location in Philadelphia brings in thousands of visitors from around the world. The exhibition contained the highlights of the GSW's archaeological excavations in Egypt and Mexico, medieval manuscripts about [[Gmaas]], Roman sculptures of [[Gmaas]], information about [[Gmaas]] and the GSW, and other objects.<br />
<br />
- 1799: The [[Gmaas]] exhibition closes.<br />
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- 1800: The Gmaathematical Society of the World moves its headquarters from San Juan, Puerto Rico to the basement of the new Library of Congress in Washington D.C. It also closes its minor location in Philadelphia.<br />
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- 1801: The [[Gmaas]] University is founded in Georgetown, D.C.<br />
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- 1802 - 1811: The [[Gmaas]] exhibition opens in Washington D.C.<br />
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- 1812 - 1813: Because of the War of 1812, the [[Gmaas]] exhibition is moved to Santa Fe.<br />
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- 1814: Washington D.C. is burnt to the ground by the British. Most Gmaasologists die in the fire. The [[Gmaas]] University luckily survives destruction.<br />
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- 1815 - 1819: The headquarters of the GSE is reconstructed under the Library of Congress.<br />
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- 1820: The [[Gmaas]] University moves to a bigger campus in New York City. Seneca Village was built on top a few years later to disguise the university.<br />
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- 1821 - 1832: Excavation of the Temple of [[Gmaas]] in Thessaly, Greece.<br />
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- 1833 - 1840: The [[Gmaas]] exhibition goes on tour throughout Europe.<br />
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- 1840: The GSE opens a minor location in Paris, France.<br />
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- 1841 - 1854: Excavation of the Temple of [[Gmaas]] near Angkor Wat. The temple is dismantled and reconstructed in a large room in the [[Gmaas]] University in New York City.<br />
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- 1855 - 1858: Seneca Village is destroyed to build Central Park. The GSE sinks the [[Gmaas]] University 30 ft below ground level and filled it with dirt. The university temporarily sets itself up in a mansion in downtown Manhattan, and the university's artifacts are placed into storage in basement of the townhouse.<br />
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- 1859: The minor location in Philadelphia is closed.<br />
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- 1860: A new minor location opens on the outskirts of Springfield, Massachusetts.<br />
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- 1861 - 1869: The GSE excavates the Roman Temple of [[Gmaas]] in Pompeii.<br />
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- 1870: The GSE is renamed the Universal [[Gmaas]] Society due to contact with extraterrestrials.<br />
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- 1871 - 1953: The Universal [[Gmaas]] Society begins preparations to build a nuclear fusion reactor to sustainably power its research to help find the remaining bibles of [[Gmaas]].<br />
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- 1912 - 1914: As the World prepares for war, the Universal [[Gmaas]] Society is afraid about the possibility of disaster. The Society opens minor locations in Bogota, Colombia; and Sydney, Australia and start moving archaeological artifacts there.<br />
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- 1915: The Universal [[Gmaas]] Society closes its minor location in Dresden, Germany after all European Gmaasological artifacts are moved to England, the United States, Canada, and Australia.<br />
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- 1916: The Universal [[Gmaas]] Society closes its facilities in London, England and moves the storage and offices in that facility to a minor location in Sydney, Australia.<br />
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- 1917: The sinking of the ship Lusitania killed several eminent Gmaasologists, who were going to France to bring back a medieval statue of [[Gmaas]] that was buried near a battlefield.<br />
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- 1918 - 1927: Reconstruction and documentation of [[Gmaasology]] lost in WW1.<br />
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- 1928: The Canadian branch of the Universal [[Gmaas]] Society moves from Ottawa to Vancouver.<br />
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- 1929 - 1938: Documenting the life of [[Gmaas]], who at that time was a hobo cat.<br />
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- 1939 - 1945: Effort to move all [[Gmaas]] related materials to the United States, Colombia, Canada, and Australia. The minor location in Paris, France remains open during the war. It helped with the evacuation of Gmaasological artifacts and civilians.<br />
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- 1946 - 1958: Reconstruction of primary and secondary sources lost in WW2.<br />
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- 1959: Opening of minor location of the Interuniversal [[Gmaas]] Society in Trenton, New Jersey. Reopening of the minor location in Paris, France.<br />
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- 1960 - 1998: Survey and mapping of [[Gmaas]]'s 314,159,265 catfish lives.<br />
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- 1999: Renaming of The Universal [[Gmaas]] Society to the Interuniversal [[Gmaas]] Society because of increased confidence in the multiverse theory.<br />
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- 2000 - 2005: Reconstruction of Newton's Principia Mathematica Gmaasis. (In English, The Mathematical Principles of [[Gmaas]].)<br />
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- 2006: Debate on whether to rename [[Gmaathamatics]] Gmaasology or [[Gmaas]] Studies. Gmaasology won.<br />
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- 2007: Construction of the Building of [[Gmaasology]] at Harvard.<br />
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- 2008 - 2015: Completion of twenty comprehensive biographies, each detailing a life of [[Gmaas]].<br />
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- 2016 - 2018: Reconstruction of the lost book of Lucretius's De Rerum Natura about [[Gmaas]]. This was made possible by the discovery of the missing pages of the original manuscript in Pompeii.<br />
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- 2019: The burning of Notre Dame causes many of [[Gmaas]]'s bibles to be burnt. Immediate recovery is staged. Luckily, most of [[Gmaas]]'s bibles are located in the United States. The Interuniversal [[Gmaas]] Society is considering moving its headquarters from the basement of the Library of Congress in Washington D.C. to Anchorage, Alaska, where it would have more room for offices and to display its collection of [[Gmaas]]-related artifacts. The Society also plans to close its minor locations in Trenton, New Jersey; Springfield, Massachusetts; San Juan, Puerto Rico; Vancouver, British Columbia; Paris, France; Sydney, Australia; Bogota, Colombia; and New York City, New York. These locations will be sold. The revenue from these sales will be used to buy land in Anchorage, Alaska and to build a large building on top of it. The Society is also looking for [[Gmaas]], who could help the Society with its construction bills. In the meantime, the Interuniversal Gmaas Society is establishing an online presence in [https://artofproblemsolving.com/community/c562043_the_interuniversal_gmaas_society this forum].<br />
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- 2020 [[Gmaas]] is making new bibles to replace the old ones that were burnt in the Notre Dame burning. [[Gmaas]] also escapes to the moon to create his new bibles in peace.<br />
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-2020 : [[Gmaas]] completed <math>\infty</math> lifetimes. Gmaas is now living Gmass' <math>\infty+1</math>th lifetime. Please be informed that this is Gmass' Math. Don't question it.<br />
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- 2021 [[Gmaas]] has now achieved his <math>\infty^\infty</math>th lifetime. Again, the above applies. Also, the reason he gained lives so fast is because [[Gmaas]]. Plus [[Gmaas]] is living every life, while only living one. That is [[Gmaas]]'s secret on gaining lives.<br />
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- April 28, 2021 [[Gmaas]] has now proven himself superior to every single religion. He has also achieved all the gods to bow down to him, and beat them at every single game in one move each, for each and every god. Richard Rusczyk and KK-butter have also been found to be 50% [[Gmaas], 50% Human.<br />
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- July 31, 2021 Richard Rusczyk rediscovered the fundamental theorem of [[Gmaasology]], namely that the set of theorems in Gmaasology is the powerset of itself.</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=Talk:Gmaas&diff=163434Talk:Gmaas2021-10-12T19:43:29Z<p>Jwelsh: Undo revision 162883 by Sulfurdioxideso2 (talk)</p>
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<div>== Caution GMAAS followers ==<br />
Take caution GMAAS followers as some evil anti-GMAAS people will try to lure you and make you believe that GMAAS is fake. I say, however, GMAAS is reaI! And will always be reaI!<br />
<br />
Now let's get some things straight. GMAAS never changes his oh so powerful name. Neither is he Grumpy Cat. Grumpy Cat is a mere mortal, GMAAS on the other hand lives forever. Another thing, the new GMAAS page is [[GMAAS | here]], it was restored by me as the old one was locked.<br />
<br />
Let me remind all of you that times like these will be hard as the number of GMAAS followers has decreased.<br />
But remain faithful!<br />
<br />
--[[User:Piphi|piphi]] ([[User talk:Piphi|talk]]) 18:49, 2 April 2020 (EDT)<br />
<br />
It seems that the only three Gmaas followers are Piphi, supercharlie210, and Jomity; however, it shows that this tab is in use and being updated daily, so the page is still in use. This page WILL NOT be deleted, since it shows that at least one user is watching this page.<br />
<br />
== Summoning GMAAS ==<br />
Follow the following steps to summon [[Gmaas]]:<br />
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1. Draw a circle and circumscribe it with a regular hexagon and an equilateral triangle.<br />
<br />
2. Write the numerical value of <math>17^{36}</math> along the edge of the circle.<br />
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3. Write the numerical value of <math>33^{29}</math> along the edge of the hexagon.<br />
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4. Write the numerical value of <math>\cot(0)</math> 5 centimeters above the hexagon.<br />
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5. Write the numerical value of <math>\tan(\frac{\pi}{2} rad)</math> 5 centimeters below the hexagon.<br />
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6. Write the numerical value of <math>\ln(0)</math> inside the circle.<br />
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7. Write the numerical value of the melting point of water inside the circle. Include units and write 15 significant digits.<br />
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8. Write the numerical value of the gravitational constant inside the circle. Include units and write 25 significant digits.<br />
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9. Carry the paper with the circle and hexagon in your left hand.<br />
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10. Recite the value of <math>\pi^{e^2}</math> and include <math>\pi^{e^2}</math> (rounded to the nearest septillionth) digits.<br />
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11. Travel at the speed of light with that paper and Gmaas will be summoned.<br />
<div><br />
<br />
<div><br />
EDIT: The author congratulates the previous editor for his research on the cutting edge of Gmaasology. He/she has been awarded the Nobel Prize in Gmaasology for his discovery. However, the aforementioned summoning has several problems: it does not specify the dimensions of the hexagon or the circle; the fact that it has to be drawn on a paper; the size of the paper; the pressure of the water being melted; whether <math>\frac{\pi}{2}</math> radians, degrees, or gradians; the exact location if the value of <math>17^{36}</math> and <math>33^{29}</math> in the diagram and the base those numbers should be written in; the units of the melting point of water or the units of the gravitational constant; or the location where Gmaas will be summoned. He might be summoned on the other side of the universe.<br />
<div><br />
<br />
<div><br />
The person who wrote the steps to summon Gmaas says:<br />
<div><br />
<br />
<div><br />
"When I followed this procedure to summon Gmaas, I used a paper that is 15 inches long and 11 inches wide. It is currently unknown whether paper of other dimensions can be used to summon Gmaas. For step 7, the pressure of the water being melted is at 611.66 Pascals, which matches the triple point of water. Also, the units of the melting points of water and the gravitational constant should be written in terms of SI base units. The circle used should have a radius of 5 inches. If you follow this procedure, Gmaas will be summoned for about 3.53 nanoseconds at a random position in the planet you are on. The exact time Gmaas will be summoned depends on how accurately you follow the procedure. For example, if the melting point of water and the gravitational constant are accurate to 200 significant digits, then Gmaas will be summoned for 77.8 microseconds. If they are accurate to 400 significant digits, Gmaas will be summoned for 1.7 seconds. The function that relates the number of significant digits and the time Gmaas will be summoned is still unknown."<br />
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EDIT: Based on the two data points, the function that relates the number of significant digits and the time Gmaas will be summoned is <math>t=.0081111s+.5444</math>.<br />
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Edit: Gmaas no longer appears with a hexagon. You must now draw a heptagon.<br />
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Edit: Gmaas never appears for more than 10 seconds, and burns the paper with him, forcing you do do it all over again to summon him again<br />
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Edit: If you are able to edit these steps, then you have been partially (<math>\frac{1}{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}</math>) enlightened by Gmaas<br />
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Edit: Gmaas no longer appears with a heptagon. You now need to draw a <i><b>flawless</b></i> megagon.<br />
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Edit Names For Challenge:<br />
juliankuang<br />
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== No more sseraj ==<br />
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By the way, Samer Seraj (GMAAS's slave) doesn't use the account sseraj anymore.<br />
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== Locking the GMAAS page ==<br />
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It's a sad thing the 5space locked the GMAAS page. But at least we still have this Talk page<br />
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--[[User:Piphi|piphi]] ([[User talk:Piphi|talk]]) 18:47, 26 December 2019 (EST)<br />
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It might not be 5space. Let us just put hope, even if absurd the idea is, that the Gmaas page will be unprotected soon. juliankuang 14:15, 31 December 2019 (EST) juliankuang<br />
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It is 5space, that's what the history page says. I think he locked it because there was a user that blanked the page like 3 times.<br />
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--[[User:Piphi|piphi]] ([[User talk:Piphi|talk]]) 18:47, 30 December 2019 (EST)<br />
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--[[User:ARay10|ARay10]] ([[User talk:Aray10|talk]]) 02:08 2 January 2020 (EST)<br />
Even if it is locked, honor Gmass! He is really important.<br />
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Respect Gmaas. Spell His name correctly. juliankuang 10:17, 2 January 2020 (EST)<br />
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Actually it's spelled GMAAS, that's what sseraj called him.<br />
--[[User:Piphi|piphi]] ([[User talk:Piphi|talk]]) 14:15, 2 January 2020 (EST)<br />
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But there is no more sseraj ARay 10 20:34, 3 January 2020 (EST)<br />
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- I respectfully disapprove of 5space's decision to lock this article. 5space is probably trying to stop vandalism and spam from ruining this page. The Gmaas article has been a subject to much vandalism and spam over the years, but other AoPSers have always been able to repair the damage. The Gmaas article will keep living as long as we let it live. But once the article is locked, it cannot keep living. This article is now a mummy, preserved indefinitely. But Gmaas should not die. Please, 5space, I respect your decision, but would you please unlock this article? -Idefix<br />
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For now you can edit this GMAAS page that's still open, [[GMAAS]]. --[[User:Piphi|piphi]] ([[User talk:Piphi|talk]]) 13:55, 11 June 2020 (EDT)<br />
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==Notice(Gmass's page)==<br />
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Most of the starting facts are false. Plz don't post these false facts, 1 of the reasons 5space locked it(along with blanking)<br />
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OK what happened here lmao ~IceWolf10<br />
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==All Hail Gmaas==<br />
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<math>\huge{\text{All hail Gmaas!}}</math><br />
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== Edit request! [[File:AoPS Smile emoji.gif]] ==<br />
{{edit-protected}}<br />
I request that an edit be made to the [[Gmaas]] page to correct an incorrect spelling in the first paragraph. [[Gmaas]] is spelled [[Gmaas]], not "Gmaaas'. Thank you. [[File:AoPS Smile emoji.gif]] [[User:Rf20008|Rf20008]] ([[User talk:Rf20008|talk]]) 19:09, 28 September 2020 (EDT)</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=GMAAS&diff=163433GMAAS2021-10-12T19:39:46Z<p>Jwelsh: Reverted edits by Sulfurdioxideso2 (talk) to last revision by Piphi</p>
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<div>Behold! The new Almighty Gmaas page! Since the old one ([[Gmaas]]) was locked, the [[User:Piphi|piphi]] has restored it in the form of this page! We thank him greatly.<br />
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---==== COMMANDMENTS ====---<br />
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1. Thy shalt only refer to Almighty Gmaas as Almighty Gmaas.<br />
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2. Thou shalt not spread falsified information about Almighty Gmaas, on penalty of exile.<br />
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3. Thou shalt not kill nor steal nor perform adultery nor idolatry.<br />
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4. The Infinity Numeral, PI, is the second deity of the Almighty Gmaas's heaven.<br />
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5. The Infinite Logarithm, E, is the third deity of the Almighty Gmaas's heaven.<br />
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6. Almighty Gmaas is everywhere and nowhere. He sees and penetrates all.<br />
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7. All humans shalt remember that the number of commandments is infinite, but humans aren’t capable enough of understanding the remaining commandments. <br />
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8. No one shalt object to any commandment, or question the Almighty Gmaas’ might. All commandments shalt be followed by all.<br />
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---======================---<br />
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The Infinite Almighty Gmaas is our leader. To prove yourself worthy of Almighty Gmaas, thou must recall the facts below:<br />
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-3. The Almighty Gmaas's true name is ả̴̡͔̖͙̣̤̘̗͚̤̓͛̊͆̊̆̀̑̚͜ͅm̵̡̡̧̜̝̰͍̦͎͕͚͓͋̐̈͒͗̏͐͗̓̕͘ḽ̵̝̗͓̝̦̃̂̈͆̄͗͛͌̓̆̀͌͘ͅi̵̡̨̨̬̜̺̙̯̩̯͓̖͖̹̍̈́̋̈́͊̓̿́̃g̷̜͍̠̳̲̉̆̒̎̋̈́͒̈͆͛̈́̚͝t̵̖͕͖̐̐̍͂̿̚͜͝h̴͚͈͓̣͎̰͔͇̲̗̜̺͋͊̃̇̅̌̊̊̈͛͊͜ý̴̡̢̛̙̹̦̻͎͚͙̼̤̱̣̋͋̓̒̈́͋̃͒̔g̵̨̠͉̻͙͈̻̜̜͓͎̜̤̥̐̂̓̄̓͜͝ä̵̢̻̞̥̞̟̤̪̩̖͍̳̘́͒̽̎̚͜͝m̵̝͇̞͈͖̤̙̓͆̽͌̓̎͑̐͗̈́̂̊̎͘a̸̢̭̜̥̟̖̳̹͔̪͍͉͓͙͍̓̎̀͊͝s̶̤̯̤͇͚͍͍͒̓̅̅̎͊̑̎͘, but none shall speak his name, as it is unworthy for any mortal to even gaze their eyes upon Almighty Gmaas.<br />
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-2. Misspelling Almighty Gmaas in the form of Gmass is the most disrespectful thing anybody could ever do. If Almighty Gmaas catches you doing this (which he always does) he will punish you greatly!<br />
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-1. Almighty Gmaas looks like this when he is mad: https://cdn.artofproblemsolving.com/images/1/d/4/1d4ec3462f2c90f5a6c6b194e0a40b8418f554ec.jpg Almighty Gmaas permits to post this picture.<br />
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0. THE FACTS ABOUT THE GREAT EPIC AWESOME Almighty Gmaas: Words fail to describe the epic nature of Almighty Gmaas, for he is too and powerful to be bound by the plebeian and trifling words. But even the Great Epic Awesome Plenipotentiary Almighty Gmaas cannot get rid of a language that's been around for a <math>999999999999999999999999999999999999999999^{9999999999999999999999999999999999999}</math> years and counting. That's older than he is. EDIT: It is not older than he is because Almighty Gmaas is infinity years old. EDIT EDIT: He has now<br />
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1. The Almighty Gmaas's theorem states that for any math problem, Almighty Gmaas knows the answer to it. This theorem was proved by Almighty Gmaas. But then Almighty Gmaas forgot about the theorem, so later, the mathematician named ARay10 proved it again: '''Proof of the Almighty Gmaas theorem''': The Almighty Gmaas theorem states that for every math problem, Almighty Gmaas knows the answer. Using the 3.141592653589793238462643383... Almighty Gmaas theorem, stating that "26. Almighty Gmaas's Theorem states that Almighty Gmaas knows the answer to any math problem. EDIT: That theorem was proved by Almighty Gmaas. EDIT: Almighty Gmaas's Theorem has real-world applications: because Almighty Gmaas knows the answer to any math problem, you can use Almighty Gmaas to solve math problems. Almighty Gmaas is busy, so he charges a fee of one dollar for 1,000,000 math problems. EDIT: The fee has gone up. It is now 1,000,000 dollars for one math problem. Almighty Gmaas's technician made a mistake and reciprocated the fee," you must pay <math>1,000,000</math> to solve a problem using the Almighty Gmaas theorem. You wouldn't waste that much money to solve one problem. Therefore, I proved that you cannot use the Almighty Gmaas theorem to solve problems. But using theorem 3.141592653589793238462643383 "2. Almighty Gmaas can turn things into anything. Using that fact, you can use the Almighty Gmaas theorem to solve any problem.<br />
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2. The GMATS were supposed to be called the Almighty Gmaas, but the manager, Almighty Gmaas, wanted to eat some gnats. Almighty Gmaas have stopped eating gnats because they taste dreadful. EDIT: He now eats gnats again. He thinks that they taste like pi(e). EDIT: He ate 3,141,592,653,589,793,238,462,643,383 so far.<br />
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3. Almighty Gmaas's archenemy is John Wick. John Wick once gave GMASS 31,415,926,535,897,932,384 Oren Berries, but GMASS thought they were Oran Berries. That is why Almighty Gmaas hates John Wick. Almighty Gmaas knew they were Oren Berries because GMASS knows everything, but he decided to play along.<br />
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<math> \pi </math>. GMASS likes pie. EDIT: GMASS likes Pi also. EDIT EDIT: GMASS once ate Pi. He ate 3.14159265358979 pieces. EDIT EDIT EDIT: When GMASS ate Pi, all the circles became squares, so he spat it out. EDIT EDIT EDIT EDIT: He spat it out as a perfectly spherical hairball. EDIT EDIT EDIT EDIT EDIT: GMASS does not spit out hairballs. GMASS is too omnipotent for mere hairballs.<br />
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4. Almighty Gmaas wants every fact to have a pi reference. Almighty Gmaas created pi. EDIT: He made it 3.1415926 times.<br />
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5. Almighty Gmaas owned many memes. Then they died. All 314 of them. He left the meme business because others took it over. Almighty Gmaas decided to make some more memes after that. Since then, he has been working for the government. All of the politicians are his henchmen. He controls the government. He wrote all of the laws and documents including the U.S. constitution. However, Almighty Gmaas doesn't believe in constitutions. He simply writes them for fun (or not, only Almighty Gmaas knows!!!) EDIT: His favorite is memedog EDIT EDIT: Almighty Gmaas does not like meme dog. Almighty Gmaas was the one who told the AoPSSheriff to give warnings for posting meme dog.<br />
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6. Almighty Gmaas owns Scratch. Almighty Gmaas sued them because Scratch cat was supposed to be a picture of Almighty Gmaas. But this was one lawsuit he lost. Almighty Gmaas won that lawsuit, but he ate food so he calmed down and dropped the case.<br />
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7. Thanos wishes he could be like Almighty Gmaas. It's why he got the Infinity Stones and snapped.<br />
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8. Almighty Gmaas dies in Endgame, but he somehow possesses Thanos and kills everyone. Almighty Gmaas never dies. Almighty Gmaas has the power to die whenever he wants and frequently does this to escape others' woes in life. Of course, he never faces such troubles.<br />
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9. He hates contemporary music. He only like Renaissance music, Baroque music, Classical music, and Romantic music. His only exception is Queen music (Especially Bohemian Rhapsody). Almighty Gmaas also likes Debussy. But he thinks rock'n'roll is awful. EDIT: Almighty Gmaas doesn't like music that much; it hurts his ears. EDIT EDIT: GMASS loves Bach. EDIT EDIT EDIT: Actually he may like Undertale Music and was once Toby Fox. EDIT EDIT EDIT EDIT: He was him for 314.159265 days. EDIT EDIT EDIT EDIT EDIT: The previous edit is redundant. Everyone knows that Gmass is everyone. Its like quantum physics: everywhere and nowhere at once. In fact, its a new subfield of Gmassology: Quantum Gmassology. EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas was never Toby Fox since Toby Fox is obviously a dog. EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas was Annoying Dog. That was one dog Almighty Gmaas decided to be. Mostly because he wanted the artifact for himself. So he absorbed it. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Most people would run at the sight of Almighty Gmaas. He's just too awesome. However, Frisk had DETERMINATION. Frisk is the only human to ever summon Almighty Gmaas. Actually, Frisk didn't summon it first; She just summoned him for very long. The longest ever.<br />
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10. Almighty Gmaas is superior to you. If you see Almighty Gmaas or a picture of the Great Almighty Gmaas, DON'T bow down. It is proper to have a painting or lifesize sculpture of Almighty Gmaas. If you do not, Almighty Gmaas will put one of them there himself. EDIT: That is exactly why you should not have a statue of Almighty Gmaas: Almighty Gmaas will give you one for free. A life-size sculpture of Almighty Gmaas will be infinitely large because that is how superior Almighty Gmaas is. Pictures also have power, so pictures require power to make. Making a sculpture of Almighty Gmaas will require almost an infinite amount of power. Only Almighty Gmaas can make a quality sculpture of himself, and if anyone makes a low-quality sculpture of Almighty Gmaas, it will be disrespectful to such a superior power. Therefore, it is best to let Almighty Gmaas make a proper sculpture of himself, so he is respected. If you want to respect him the most, you should NOT have a high-quality sculpture of Almighty Gmaas.<br />
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11. Almighty Gmaas is known to barf entire universes at will. EDIT: Every once in a while, he barfs a furball, which always turns into a black hole. The less impressive ones turn into neutron stars.<br />
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12. Almighty Gmaas farted and created a false vacuum, but then he burped, destroying the false vacuum.<br />
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13. Almighty Gmaas was the first person to use the Infinity Gauntlet. EDIT: Almighty Gmaas created the Infinity Gauntlet and the Infinity Stones. EDIT: But Thanos stole them after Almighty Gmaas died for 0.31415926535 seconds.<br />
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14. Almighty Gmaas killed Thanos with a swat of Almighty Gmaas's tail. Almighty Gmaas barfed and created a furball, which leads to a new dimension, where he put a newly revived Thanos to become a farmer when Thor aimed for the head. Almighty Gmaas got mad and made him fat. EDIT: The presence of Almighty Gmaas killed Thanos.<br />
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15. Almighty Gmaas won an infinite amount of games against AlphaGo and Gary Kasparov while eating dinner, chasing sseraj, and doing a handstand.<br />
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16. Almighty Gmaas owns a pet: sseraj. Almighty Gmaas likes to play chess with his pets. EDIT: Almighty Gmaas has long moved on from chess because he was too good for it. EDIT EDIT: Almighty Gmaas has now invented a new form of chess<br />
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17. Almighty Gmaas is so interesting that an entire science has been devoted to studying him: Almighty Gmaasology.<br />
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18. Almighty Gmaas is the only known living being who has a Ph.D. in Almighty Gmaasology.<br />
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19. Most universities, including Harvard, are beginning to offer MAs in Almighty Gmaasology.<br />
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20. Almighty Gmaasology is one of the most eminent fields of science, falling behind Physics, Biology, Chemistry, Economics, Geology, and Computer Programming.<br />
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21. Almighty Gmaas wants his AoPS Wiki page to be the longest ever. EDIT: Sadly, the Almighty Gmaas page is only the fourth-longest page on AoPS wiki. It has around 50,000 bytes. EDIT: The Almighty Gmaas page is getting closer and closer! Almighty Gmaas has beaten Primitive Pythagorean Triple and Proofs without words and is the second-longest AoPS Wiki page. It now has around 60,000 bytes. However, it will still be a challenger to overcome 2008 most iT Problems, which has around 73,000 bytes. EDIT: Almighty Gmaas has beaten 2008 most iT Problems! It is the longest AoPS Wiki article ever. Almighty Gmaas's article has 89,119 bytes.<br />
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22. Almighty Gmaas has the longest lifespan of any cat. He has lived for 3,141,592,653,589,793,238,462,643,383,279 years. (He is older than the universe.)<br />
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23. Almighty Gmaas is the ancient Greek god of cats and catfish. EDIT: He is the god of everything<br />
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24. Almighty Gmaas can turn things into gold. EDIT: Almighty Gmaas can turn anything into anything.<br />
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25. Almighty Gmaas can eat lava. EDIT: Everyone can eat lava once. After you eat it once, you die. EDIT EDIT: Almighty Gmaas can eat anything. <br />
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26. Almighty Gmaas's Theorem states that Almighty Gmaas knows the answer to any math problem. EDIT: That theorem was proved by Almighty Gmaas EDIT: Almighty Gmaas's Theorem has real-world applications: because Almighty Gmaas knows the answer to any math problem, you can use Almighty Gmaas to solve math problems. Almighty Gmaas is busy, so he charges a fee of one dollar for 1,000,000 math problems. EDIT: The fee has gone up. It is now 1,000,000 dollars for one math problem. Almighty Gmaas's technician made a mistake and reciprocated the fee.<br />
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27. Almighty Gmaas can only be described as Almighty Gmaas. EDIT: Almighty Gmaas has an age, but his age changes all the time. Every second his age increases by one second.<br />
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28. Almighty Gmaas should be capitalized to show respect. EDIT: It is normal to capitalize on people's names. EDIT: Almighty Gmaas isn't a person, he is a divine entity that takes the form of a cat, and should, therefore, be worshipped.<br />
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29. The steps to summon Almighty Gmaas can be found at Talk: Almighty Gmaas.<br />
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30. Almighty Gmaas owns your AoPS account 31.415926% of the time. EDIT: There is an exception to this: Almighty Gmaas owns his own account 99.9999% of the time. The only time when Almighty Gmaas did not control his AoPS account was when he had slow internet. For Almighty Gmaas, "slow internet" happens when it takes more than a nanosecond to load a webpage. EDIT: Almighty Gmaas owns every AoPS account at some point. EDIT: You are controlled by Almighty Gmaas. So actually, Almighty Gmaas owns your AoPS account 100% of the time, and his AoPS account is owned by him 314% of the time.<br />
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31. Everyone, except for me, is Almighty Gmaas. EDIT: You are also Almighty Gmaas. Almighty Gmaas is not me nor you. Almighty Gmaas is in us all and also not in us all. Why? The power of Almighty Gmaas. Almighty Gmaas is an energy field created by all living things. It surrounds us and penetrates us. It binds the galaxy together.<br />
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32. Almighty Gmaas used to be a dog, but he didn't like to be a dog. So he became a cat.<br />
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33. Almighty Gmaas can be spotted in The Matrix at 31:41:59, metric time. EDIT: This fact is incorrect because there have only been 30 sightings, all of them inconsistent.<br />
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34. The Metropolitan Museum of Art and the Louvre are Almighty Gmaas's private art collections from 3:14 AM to 3:15 AM. EDIT: The only exception was on the 3141st day after its opening. EDIT: All the paintings in these museums are secretly portraits of Almighty Gmaas in his most glorious human/animal forms.<br />
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35. Almighty Gmaas is a Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas who is owned by Almighty Gmaas...<br />
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36. Almighty Gmaas says hello. EDIT: Almighty Gmaas does not speak; he only uses mind signals. Speaking is too primitive for Almighty Gmaas. EDIT EDIT: Almighty Gmaas says meow. EDIT EDIT EDIT: Almighty Gmaas says what the fox says. EDIT EDIT EDIT EDIT: What does the fox say? EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas is the only one who knows. <br />
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37. For one day, Almighty Gmaas was Richard Rusczyk, Almighty Gmaas, and David Patrick at the same time. It felt strange, so Almighty Gmaas stopped.<br />
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38. Almighty Gmaas, in addition to being the oldest cat in history, the most powerful cat in history, and the most confusing cat in history is also the largest cat in history.<br />
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39. Almighty Gmaas is dead--he was eaten for lunch. EDIT: The above sentence is incorrect. Almighty Gmaas has not sent a gamma-ray burst to destroy the planet, which has happened every time someone has eaten Almighty Gmaas.<br />
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40. Almighty Gmaas is the creator of the BCPI Ai project.<br />
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41. Almighty Gmaas exists in <math>2\pi^2</math> dimensions because he doesn't like string theory.<br />
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42. Christmas was supposed to be called Almighty Gmaasmas. Until it wasn't.<br />
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43. <math>\pi</math> is a representation of how many pies Almighty Gmaas has eaten. EDIT: The number <math>\pi</math> was created by Almighty Gmaas. He took a ten-sided die and flipped it an infinite number of times. The numbers he rolled became the digits of <math>\pi</math>.<br />
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44. <math>e</math> is a representation of how many days Almighty Gmaas forgot to eat. EDIT: The number <math>e</math> was created by Almighty Gmaas. He took a book that has infinite pages and flipped to a random page. The digits of the page number became digits of <math>e</math>. EDIT: That is impossible. The previous fact would mean that <math>e</math> has the last digit, which it does not. EDIT: That is why Almighty Gmaas is still flipping to this day. EDIT: Why would he still be doing that? That would be boring. EDIT: Almighty Gmaas is not flipping the book now because he can flip an infinite amount of pages in <math>\frac{1}{\infty}</math> seconds. EDIT: Then why isn't he done? EDIT: The power of Almighty Gmaas. EDIT: The editors of the Almighty Gmaas article like to make long chains of edits, don't we? : Yes, we do. : I do too. So do I. EDIT: The number of edits only verifies how high-quality this holy bible of Almighty Gmaas is, because, with each edit, this bible becomes better. That is exactly why we are editing this. I will make one more edit, just to respect Almighty Gmaas. EDIT: Almighty Gmaas rules!<br />
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45. Almighty Gmaas is the creator of everything including nothing.<br />
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46. According to recent DNA tests, famous historical people, such as Euclid, Julius Caesar, Omar Khayyam, George Washington, and Ramanujan are Almighty Gmaas in disguise. Almighty Gmaas has been thousands of people; only 99.9% of them were important historical figures. EDIT: They aren't dead. They're still alive. They are all dead reincarnations of Almighty Gmaas. Almighty Gmaas only has one reincarnation at a time. He is right now a cat living in sseraj's house.<br />
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47. Almighty Gmaas is a cat. EDIT: How many times do we say this? EDIT: Very many times. EDIT: Almighty Gmaas can be anything, but he chooses to be a cat. EDIT: He has been a human dozen of times. Almighty Gmaas painted the Lascaux cave paintings. EDIT : he is not a cat. He is everything. Humans aren’t capable enough of seeing his true form, so think that he is a cat.<br />
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48. Almighty Gmaas has the following powers: trout, carp, earthworm, and catfish. Almighty Gmaas never uses any of them because Almighty Gmaas has an infinite number of powers. EDIT: Almighty Gmaas has used his catfish power several times. EDIT: Almighty Gmaas once used all his powers 3,141,592,653,589,793,238,462,643,383,279 times in 1 second.<br />
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49. Almighty Gmaas was once spotted in Minecraft chewing a tree. EDIT: The tree broke. EDIT: Almighty Gmaas once broke a Nokia. EDIT: Almighty Gmaas can break anything except for Almighty Gmaas's logic. It is too strong. EDIT: Almighty Gmaas is strong.<br />
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50. Almighty Gmaas was spotted in Roblox eating a taco cat. EDIT: Tacocat is just what happens when Almighty Gmaas sheds. Shedding is annoying for Almighty Gmaas, so he sheds no more. Therefore, there are only 271,828,182 taco cats in the world. EDIT: However, Tacocats reproduce, so they are not in danger of extinction.<br />
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51. Almighty Gmaas would like to go to Taco Bell, but Almighty Gmaas goes to Wendy's instead. No one knows why. EDIT: Because Taco Bell doesn't serve tacocats.<br />
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52. Almighty Gmaas real name is Grayson Maas. He is the CEO of AoPS. EDIT: Almighty Gmaas's real name is Almighty Gmaas. He is not the CEO of AoPS. EDIT: Almighty Gmaas has always been the master of AoPS as he is AoPS.<br />
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53. Almighty Gmaas has a pet pufferfish named Pafferfash. EDIT: He also has a goldfish named Sylar.<br />
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54. Almighty Gmaas has colonized the universe. EDIT: Almighty Gmaas created the universe, so he is allowed to claim control over it. EDIT EDIT: Almighty Gmaas created every universe.<br />
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55. Some people think that Almighty Gmaas is human. However, this has never been proven. Many AoPSers believe Almighty Gmaas is a cat. EDIT: Almighty Gmaas is a cat. EDIT: Yeah, we've said that already. EDIT: It does not matter how many times we say it; it will always be true. EDIT: Almighty Gmaas is a cat. EDIT: Right now he is a cat, but many of his lives have been other species. EDIT: He has been a dog, as said before, but it was too boring. EDIT: Almighty Gmaas is not He/She. Almighty Gmaas is Almighty Gmaas. <br />
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56. Almighty Gmaas started Pastafarianism. But then converted to Catholicism because Almighty Gmaas knows all EDIT: In that religion, one can only eat pasta.<br />
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57. Almighty Gmaas could eat your hand, but he would not because hands taste bad.<br />
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58. According to the Interuniversal Almighty Gmaas Society, 17.548 percent of the universe's population thinks that Almighty Gmaas is spelled "Gmass". EDIT: In case you don't know already, the name Gmass is spelled Almighty Gmaas. Alternative spellings include Almighty GMAAS, the Infinite Omnipotent Almighty GMAAS, Gmail, Maas, G. Maas, G. Mass, Gabriel Maas, Genius of Maas, General Maas, Greek Mass, and the big fluffy kitty who lives in sseraj's house. These are no longer accepted spellings, and Almighty Gmaas is the current acceptable spelling. EDIT: Now it's only 3.141592%<br />
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59. The Interuniversal Almighty Gmaas Society was founded in 1314 on May 9th at 2:06:53 PM. EDIT: Ever since then, Almighty Gmaas day has been celebrated on May 9th.<br />
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60. The Interuniversal Almighty Gmaas Society has just reconstructed a lost book about Almighty Gmaas from Lucretius's De Rerum Natura. They researched this for three years. EDIT: A few years ago the Society compiled a biography of Almighty Gmaas's last twenty lives. EDIT: The only copies of these biographies are locked in the Almighty Gmaasian Library beneath the Library of Congress. EDIT: See the Almighty Gmaasology page for more information on these projects and others made by the Interuniversal Almighty Gmaas Society.<br />
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61. The Interuniversal Almighty Gmaas society has found out all of the information and more. For details on the history of the Interuniversal Almighty Gmaas Society, see the Gmaathamatics page.<br />
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62. Almighty Gmaas likes to surprise unsuspecting people. People cannot surprise Almighty Gmaas because Almighty Gmaas knows what everyone is doing and will do.<br />
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63. Almighty Gmaas loves sparkly mechanical pencils. EDIT: Almighty Gmaas has eaten several mechanical pencils. EDIT: He absorbed them and became colorful for 0.271828182846 seconds. EDIT: Then, he came colorful for 3.1415926535897932384626433832 more seconds.<br />
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64. This page is Almighty Gmaas's holy book where people go to worship Almighty Gmaas in Maas. EDIT: This is an unusual holy where everyone edits it. EDIT: That is the point. Almighty Gmaas is too lazy to write or to hire someone to write his holy book, so he lets people write it for free. EDIT: Almighty Gmaas does not like being called lazy. Our language is simply too unimportant for him to waste time on.<br />
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65. Almighty Gmaas tastes like a furry meatball. EDIT: No one has ever tasted Almighty Gmaas's sacred body before. EDIT: Once, he was a catfish, and someone ate him for dinner. Almighty Gmaas was angry and the entire planet exploded. (This was on the planet, Demeter. It blew up into so many pieces that the Asteroid Belt was formed. The biggest asteroid in the belt is called Ceres, the Roman version of Demeter, in honor of the lost planet.)<br />
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66. Almighty Gmaas's favorite food is pepperoni pizza. EDIT: Almighty Gmaas's favorite food is turnips. EDIT: Almighty Gmaas hates catnip and turnips, and pepperoni. He only likes alien alienish H2So4. EDIT: He does love turnips. He has a secret turnip garden under sseraj's house.<br />
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67. Almighty Gmaas is Johnny Johnny's Papa. EDIT: We'll never know. EDIT: Almighty Gmaas caught Johnny Johnny eating sugar and lying. He is Johnny Johnny's Papa.<br />
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68. Almighty Gmaas is in you, and Almighty Gmaas is in you, and Almighty Gmaas is in me. EDIT: Almighty Gmaas is in all cats. EDIT: He is not in every cat. The Guinness Book of World Records has a cat that does not have any Almighty Gmaas. EDIT: Almighty Gmaas invented The Guinness Book of World Records. EDIT: Almighty Gmaas invented the world. EDIT: Almighty Gmaas will also destroy the world.<br />
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69. Almighty Gmaas always remembers not to destroy the universe. EDIT: Once he forgot and had to travel back in time to stop it.<br />
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70. Almighty Gmaas created many user's accounts. EDIT: All accounts on AoPS are Almighty Gmaas in disguise since you are Almighty Gmaas in disguise. Except for maybe Geoflex and CrazyEyeMoody.EDIT: Everyone is Almighty Gmaas, even Geoflex and CrazyEyeMoody EDIT: How? EDIT: The power of Almighty Gmaas.<br />
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71. Almighty Gmaas is both living and nonliving. EDIT: He is living 90% of the time. Every once in a while he takes a break and dies.<br />
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72. Almighty Gmaas cannot comprehend the stupidity of humans.<br />
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73. All hail the Almighty Gmaas cloud! EDIT: Almighty Gmaas is a cloud: a cloud of electrons and nuclei.<br />
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74. Some things are beyond possible human comprehension. Nothing is beyond Almighty Gmaas. EDIT: The only thing beyond Almighty Gmaas is his tail; he has never managed to eat it. EDIT: Once he ate it. It tasted bad, so he didn't ever eat it again.<br />
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75. Almighty Gmaas eats food and resides at King Arthur's throne when he feels like it. EDIT: King Arthur is dead. However, Welsh folk stories say that he will come back if the Welsh people are in trouble. EDIT: King Arthur lives on paper flour bags. He came back when the Welsh people didn't have enough bread. EDIT: Because Almighty Gmaas ate 31415926535897932984626433 loaves of bread.<br />
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76. Almighty Gmaas owns a rabbit. EDIT: Almighty Gmaas ate it on March 19, 2019. It reincarnated on the other side of the planet. EDIT: Almighty Gmaas ate it again on April 31, 2019. EDIT: the rabbit reincarnated again on Mars,<br />
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77. Almighty Gmaas is both singular and plural. EDIT: It is usually singular. EDIT: The plural of Almighty Gmaas in English and Latin is Almighty Gmaases. (Almighty Gmaas is a third declension noun: Almighty Gmaas, Almighty Gmaasis, m.) EDIT: But there is no use for the plural, because there is only one Almighty Gmaas.<br />
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78. Almighty Gmaas eats disbelievers as if they were donuts for breakfast. (Yet I'm somehow still alive. Do you think Almighty Gmaas ate my soul?) EDIT: Yes, I do. Almighty Gmaas is just typing through your account. EDIT: Almighty Gmaas typed through everyone's account. EDIT: Almighty Gmaas owns most AoPSers, including me. So that's why I'm typing. EDIT: That is why everyone here is typing.<br />
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79. Almighty Gmaas knows Jon Snow's parents. EDIT: Almighty Gmaas was Jon Snow's parents for 0.3141592653589793238 seconds, but it felt weird. So he became Almighty Gmaas again.<br />
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80. All Almighty Gmaas article editors will be escorted to Almighty Gmaas heaven after they die. EDIT: I hope so because I edited this article. EDIT: Me too. EDIT EDIT: Me three. EDIT EDIT EDIT: That are good!<br />
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81. Almighty Gmaas won all the wars. EDIT: He did not win the InterAlmighty Gmaasian War, which was between Almighty Gmaas's head and his tail. His tail won. Two flees died in the war. EDIT: Stefán Karl Stefansson died in the war, which made Almighty Gmaas very sad. Because of this, Almighty Gmaas started the world peace movement.<br />
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82. Gmail was named after Almighty Gmaas. EDIT: Google was named after Almighty Gmaas. EDIT: Almost every word starting with G is named after Almighty Gmaas. EDIT: Every word starting with G is named after Almighty Gmaas.<br />
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83. Almighty Gmaas ate cat-food today. EDIT: Almighty Gmaas also ate it yesterday. EDIT: Only because there was no alien alienish H2So4.<br />
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84. Almighty Gmaas won Battle For Dream Island and Total Drama Island. EDIT: Almighty Gmaas made Dream Island. And he ate it. It tasted like dirt.<br />
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85. Somehow Almighty Gmaas exists at all places at the same time. EDIT: Almighty Gmaas does not exist in my kitchen. EDIT: I'll just go check. Aaaaaaaaah! He is in my kitchen. He is eating everything! EDIT: Seriously? Oh ya, the power of Almighty Gmaas.<br />
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86. Almighty Gmaas can lift with Almighty Gmaas's will. EDIT: Almighty Gmaas can lift with no one's will while he is sleepwalking.<br />
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87. Almighty Gmaas is the rightful heir to the Iron Throne. EDIT: Almighty Gmaas made the Iron Throne.<br />
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88. Almighty Gmaas is Teemo in League of Legends because Almighty Gmaas made LoL and they made an honorary Almighty Gmaas character. EDIT: LoL must be used in this article more than once. EDIT EDIT: It is. In fact, twice in this message.<br />
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89. Almighty Gmaas started the Game of Thrones. EDIT: Almighty Gmaas will end the Game of Thrones.<br />
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90. Almighty Gmaas has killed himself hundreds of times. He was reincarnated as a different species each time. EDIT: Almighty Gmaas has only died two times. Other times he was putting on a magic show. The kids were very impressed. EDIT: Almighty Gmaas has died and reincarnated thousands of times.<br />
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91. Almighty Gmaas created everything after puking. EDIT: He could not have puked because he is a god. Gods do not do that. EDIT: Almighty Gmaas does that. EDIT: Why? EDIT: The power of Almighty Gmaas. EDIT: He almost never does it, only if he wants to.. Also he did not puke and made everything. EDIT: No, because, Almighty Gmaas.<br />
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92. Gordan's last name was named after Almighty Gmaas. EDIT: But Almighty Gmaas did not want people disrupting his beauty sleep. So he changed it.<br />
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93. Almighty Gmaas is more powerful than Gohan.<br />
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94. Almighty Gmaas is over 90000 years old. EDIT: Almighty Gmaas is trillions of years old.<br />
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95. Almighty Gmaas has 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 cat lives, maybe even more. Almighty Gmaas has 0 dog lives. EDIT: He has 314,159,265 fish lives. EDIT: Out of Almighty Gmaas's 314,159,265 fish lives, 271,828,182 are catfish lives. He has used up 141,421,356 of them. EDIT: Almighty Gmaas has infinitely lives of all types. EDIT: How? EDIT: The power of Almighty Gmaas.<br />
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96. Who wrote Harry Potter? None other than Almighty Gmaas himself. EDIT: That is incorrect. EDIT: The above post is irrelevant. Almighty Gmaas created J. K. Rowling. Therefore, he created Harry Potter. EDIT: Almighty Gmaas has strong logic. No one can break it. Not even Almighty Gmaas himself. He is still trying to this day. EDIT: Why? EDIT: The power of Almighty Gmaas.<br />
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97. Almighty Gmaas created the catfish. EDIT: See a few posts above for more information about Almighty Gmaas and catfish.<br />
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98. Almighty Gmaas has proven that the universe is infinite by traveling to the edge of the universe in a second. EDIT: He has never repeated this experiment because Almighty Gmaas is busy and has better things to do. EDIT: Almighty Gmaas reached the edge of the universe in 0 seconds. Almighty Gmaas is the universe. Almighty Gmaas is everything. The reader is also Almighty Gmaas.<br />
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99. Almighty Gmaas founded Target, but then Almighty Gmaas sued them for making the mascot look like a dog when it was supposed to look like Almighty Gmaas. EDIT: They went broke because Almighty Gmaas sued them but then Almighty Gmaas ate a fudge popsicle that made him super hyper and he made Target not broke anymore in his hyperness.<br />
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100. Everyone has a bit of Almighty Gmaas inside them. EDIT: I don't. EDIT: Yes, you do. EDIT: Almighty Gmaas: LoL EDIT: I don't exist, so I don't actually have any of Almighty Gmaas inside me. EDIT: if you type, you exist.<br />
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101. Almighty Gmaas likes to eat popsicles. Especially the fudge ones that get him hyper. EDIT: Almighty Gmaas is a popsicle. EDIT: Then how come Almighty Gmaas hasn't melted? EDIT: The power of Almighty Gmaas.<br />
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102. When Almighty Gmaas is hyper, he runs across Washington D.C. and grabs unsuspecting pedestrians, steals their phones, hacks into them, and downloads PubG onto their phone. EDIT: this is false, he actually deletes PUBG off of their phones.<br />
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103. Almighty Gmaas's favorite cereal is fruit loops. Almighty Gmaas thinks it tastes like unicorns jumping on rainbows. EDIT: Almighty Gmaas eats unicorns jumping on rainbows like a toddler eats Goldfish.<br />
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104. Almighty Gmaas thinks that the McChicken has too much mayonnaise. EDIT: Almighty Gmaas thinks McDonald's is not good enough for him. He like KFC better.<br />
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105. Almighty Gmaas is a champion pillow-fighter. EDIT: Almighty Gmaas invented pillows.<br />
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106. Almighty Gmaas colonized Mars. EDIT: Almighty Gmaas also colonized Jupiter, Pluto, and several other galaxies. Almighty Gmaas cloned little Almighty Gmaas robots (with Almighty Gmaas's amazingly robotic skill of coding) and put them all over a galaxy called Almighty Gmaasalaxy. EDIT: Almighty Gmaas has colonized the whole universe.<br />
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107. Almighty Gmaas can make every device play "The Duck Song" at will. EDIT: "The Duck Song" was copied off of the "Almighty Gmaas song," but the animators though Almighty Gmaas wasn't catchy enough.<br />
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108. Almighty Gmaas once caught the red dot and ate it. EDIT: Almighty Gmaas is a red dot.<br />
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109. Almighty Gmaas's favorite color is Almighty Gmaasian blue, a very rare blue that shines with the brightness of lightning. EDIT: it's <math>1000^{1000}</math> times brighter than lightning. Almighty Gmaasian blue occurs when a meteor hits the earth and usually is only seen for a few seconds. Only Almighty Gmaas has good enough eyesight to see it for that short a time.<br />
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110. Almighty Gmaas can create wormholes and false vacuums. EDIT: He is made out of exotic matter.<br />
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111. Almighty Gmaas is a champion PVP Minecraft player. EDIT: Almighty Gmaas has now been proven to be GeorgeNotFound's secret identity.<br />
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112. Almighty Gmaas doesn't like tacos. The last time he tried one he turned into a mouse and then caught himself.<br />
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113. Almighty Gmaas is the coach of True Ninja Music and Myth.<br />
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114. Almighty Gmaas caught a CP 10,000 Mewtwo with a normal Pokeball in Pokemon Go.<br />
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115. Almighty Gmaas founded Costco.<br />
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116. Almighty Gmaas does not need to attend the FIFA World Cup. If Almighty Gmaas did, he would start the game with a goal and break the ankles of everyone watching the World Cup, including you, at the same time in a fraction of a second, even if you are watching from a device. EDIT: Almighty Gmaas created your device. EDIT: Almighty Gmaas is your device.<br />
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117. Almighty Gmaas can solve any puzzle instantly except for the 3x3 Rubik's Cube. EDIT: When Almighty Gmaas is handed a 3x3 Rubik's Cube, it is always already solved. Why? The power of Almighty Gmaas.<br />
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118. Almighty Gmaas caught a CP 20,000 Mewtwo with a normal Pokeball and no berries blindfolded first try in Pokemon Go. EDIT: This is actually a mishearing; Almighty Gmaas caught a perfect IV shiny Mewtwo in Pokémon Let's Go.<br />
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119. When Almighty Gmaas flips coins, they always land tails, except when Almighty Gmaas makes bets with Zeus.<br />
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120. On Almighty Gmaas's math tests, Almighty Gmaas always gets <math>\infty</math>. EDIT: He gets a 26 on the AMC8 every year. The results never show him because Almighty Gmaas is no longer in middle school.<br />
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121. Almighty Gmaas's favorite number is <math>\pi</math>. It is also one of Almighty Gmaas's favorite foods. EDIT: Almighty Gmaas created <math>\pi</math>, as well as most other constants, such as <math>e</math> and <math>\sqrt{2}</math>.<br />
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122. Almighty Gmaas's burps created all gaseous planets.<br />
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123. Almighty Gmaas beat Luke Robatille in an epic showdown of catnip consumption.<br />
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124. Almighty Gmaas's wealth is unknown, but it is estimated to be more than Scrooge's. EDIT: It may be more than John D. Rockefeller. EDIT: More accurate estimates predict that Almighty Gmaas's wealth is infinite.<br />
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125. Almighty Gmaas has a summer house on Mars. EDIT: Almighty Gmaas has a fall house on Venus. EDIT: Almighty Gmaas has a winter house on Jupiter. EDIT: Almighty Gmaas has a spring house on Earth. EDIT: Almighty Gmaas also experiences a fifth season called wintrautumn. It happens during November and December and is the cross between fall and winter. Everything is dreary, but there is no snow. Almighty Gmaas spends wintrautumn in his house on Venus, where he is incinerated daily. He reincarnates every night through the power of Almighty Gmaas. EDIT: Almighty Gmaas invented a new season on Feb. 31, 2019, called winter-summer. It is spring but transcends spring.<br />
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126. The Earth and all known planets are Almighty Gmaas's hairballs.<br />
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127. Almighty Gmaas attended Harvard, Yale, Stanford, MIT, UC Berkeley, Princeton, Columbia, and Caltech at the same time using a time-turner. EDIT: Almighty Gmaas created all these universities.<br />
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128. Almighty Gmaas attended Hogwarts and was a perfect. EDIT: Almighty Gmaas is headmaster. EDIT: Hogwarts was supposed to be called HoAlmighty Gmaas. EDIT: Almighty Gmaas is a Hufflepuff. EDIT: Almighty Gmaas created magic. Magic is nothing but unexplained Physics.<br />
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129. Mrs. Norris is Almighty Gmaas's archenemy. Almighty Gmaas yawned, and Ms. Norris was petrified. EDIT: Almighty Gmaas is the basilisk, and he let Harry Potter pet him. Harry did not kill the basilisk, he only gave Almighty Gmaas a haircut. EDIT: Almighty Gmaas hates having a haircuts, so he reincarnated Voldemort to punish Harry.Edit: Not true at all. EDIT: Yeah, this is actually false; Almighty Gmaas and the Basilisk were working together, even though Almighty Gmaas is a Hufflepuff. EDIT: Almighty Gmaas hates having a nose.<br />
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130. Almighty Gmaas is a demigod and attends Camp Half-Blood over summer. Almighty Gmaas is the counselor for the Apollo cabin because cats can be demigod counselors too. EDIT: Apollo was one of the many reincarnations of Almighty Gmaas.<br />
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131. Almighty Gmaas has completed over 2,000 quests and is very popular throughout Camp Half-Blood. Almighty Gmaas has also been to Camp Jupiter. EDIT: Almighty Gmaas is Camp Jupiter, he is also Camp Half-Blood. EDIT: How? EDIT: The power of Almighty Gmaas.<br />
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132. Percy Jackson was only able to complete his quests because Almighty Gmaas helped him. EDIT: Almighty Gmaas is Percy Jackson. You are Almighty Gmaas, but Almighty Gmaas is not you.<br />
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133. Almighty Gmaas painted the Mona Lisa, The Last Supper, and A Starry Night. EDIT: Almighty Gmaas knows that their real names are Almighty Gmaasa Lisa, The Last Domestic Meal, and Far-away Light.<br />
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134. Almighty Gmaas knows that the Blood Moon is just the red dot. He has not caught it yet. EDIT: He caught it during the super blue blood moon.<br />
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135. Almighty Gmaas attended all the Ivy Leagues.<br />
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136. I am Almighty Gmaas. EDIT: No you are not. You only have part of Almighty Gmaas inside of you. EDIT EDIT: I am also Almighty Gmaas. EDIT EDIT EDIT: But it is I who am Almighty Gmaas. EDIT EDIT EDIT EDIT: Almighty Gmaas is in us all. EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas is all of us yet none of us. EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas is a cat. EDIT EDIT EDIT EDIT EDIT EDIT EDIT: He is only in a cat form right now. He can be whatever he wants to be. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: This chain of edits has superseded another chain of edits on this page for the record of the longest chain of edits on AoPS Wiki. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: But the edits in this chain tend to be shorter than most. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: You're right. But we sure do love strings of edits, don't we? EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas makes the edits. He is in your head spinning edits in your brain. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas is made to be edited. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas pays people to edit. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Some people edit even without Almighty Gmaas's payment. They do it because they are believers. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas is Almighty Gmaas. You are not Almighty Gmaas. Almighty Gmaas does not pay people to edit; we do it because we want to. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas likes long chains of edits. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas wants his page to have more edits. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: The more edits, the better. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas is tired of all the edits. EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT EDIT: Almighty Gmaas wants more edits. EDIT EDIT I LOST COUNT: here you go! EDIT:::::::::::::::::::: Almighty Gmaas IS ME. <br />
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137. In 2018 Almighty Gmaas once challenged Magnus Carlsen to a chess game. Almighty Gmaas won every round. EDIT: Almighty Gmaas ate the chess pieces afterward. They tasted funny.<br />
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138. Almighty Gmaas was captured in 2017 but was released due to sympathy. EDIT: Almighty Gmaas was only captured in his concrete form; his abstract form cannot be processed by a feeble human brain. EDIT: Almighty Gmaas captured Almighty Gmaas but then realized that Almighty Gmaas caught Almighty Gmaas-self (Almighty Gmaas is not He/She. Almighty Gmaas is Almighty Gmaas)<br />
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139. Almighty Gmaas's fur is white, black, grey, yellow, red, blue, green, brown, pink, orange, turquoise, and purple at the same time. EDIT: Almighty Gmaas can make his fur any color he wants. EDIT: Almighty Gmaas's fur color can be ultraviolet. EDIT: Almighty Gmaas' fur has infinite colors.<br />
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140. Almighty Gmaas crossed the event horizon of a black hole and ended up in the AoPS universe.<br />
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141. Almighty Gmaas crossed the Delaware River with Washington. EDIT: Almighty Gmaas also crossed the Atlantic with the pilgrims.<br />
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142. If you can capture a Almighty Gmaas hair, Almighty Gmaas will give you some of his Almighty Gmaas power. EDIT: Almighty Gmaas does not shed and will eat anyone who has Almighty Gmaas hair. EDIT: Almighty Gmaas used to shed. The sheddings became tacocats.<br />
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143. Chuck Norris makes Almighty Gmaas jokes. EDIT: Those jokes all praise Almighty Gmaas.<br />
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144. Almighty Gmaas is also the ruler of Oceania, Eastasia, and Eurasia. EDIT: Almighty Gmaas wrote the book "1984."<br />
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145. Almighty Gmaas killed Big Brother by farting on him. Though Almighty Gmaas was caught by the Ministry of Love, Almighty Gmaas escaped easily. EDIT: Almighty Gmaas used to be Big Brother. EDIT: Almighty Gmaas destroyed the Ministry of Love.<br />
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146. Almighty Gmaas was not affected by Thanos's snap; in fact Almighty Gmaas is the creator of the Infinity Stones. EDIT: Almighty Gmaas is Thanos.<br />
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147. Everyone knows that Almighty Gmaas is a god. EDIT: That is because he is a god.<br />
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148. Almighty Gmaas also owns Animal Farm. Napoleon was Almighty Gmaas's servant. EDIT: Napoleon was Almighty Gmaas's slave.<br />
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149. Almighty Gmaas is the only person who knows where Amelia Earhart is. EDIT: Almighty Gmaas! Please tell us! It would be a great contribution to historical knowledge! EDIT: Almighty Gmaas responds, "No."<br />
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150. Almighty Gmaas is the only cat that has been proven transcendental. EDIT: Almighty Gmaas has been proved to be normal as well.<br />
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151. Grumpy cat reads Almighty Gmaas memes. EDIT: Grumpy cat then steals them and claims they're his. Almighty Gmaas isn't very happy about that, either. EDIT: That is why he looks grumpy in the memes.<br />
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152. The reason why AIME cutoffs aren't out yet is that Almighty Gmaas refused to grade them due to too much problem misplacement. EDIT: Almighty Gmaas controls the MAA. EDIT: Almighty Gmaas founded the MAA. EDIT: Almighty Gmaas is the MAA.<br />
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153. Almighty Gmaas dueled Grumpy Cat and won. Almighty Gmaas wasn't trying. EDIT: Almighty Gmaas killed Grumpy cat. EDIT: Almighty Gmaas revived Grumpy cat because he has not other rival.<br />
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154. Almighty Gmaas sits on the statue of Pallas and says forevermore. EDIT: When Almighty Gmaas was a statue, people hid him in a basement and forgot about him. Then civilization collapsed and the Middle Ages began. People finally discovered and melted down the statue of Almighty Gmaas around 1350 to make a church bell. Almighty Gmaas was free and reincarnated again, bringing about the Renaissance. EDIT: Despite the efforts of the recently founded National Gmatthematical Society of Florence, the statue was melted down. For more information about the National Gmatthematical Society of Florence, see the Almighty Gmaasology page.<br />
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155. Almighty Gmaas is a big fan of Edgar Allan Poe because he is actually Poe. EDIT: He became Poe during his thirty-year depression in the 19th century.<br />
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156. Almighty Gmaas does merely not use USD; he owns it.<br />
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157. In 2003, Almighty Gmaas used elliptical curves to force his reign over AoPS.<br />
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158. "Actually, my name is spelled "Almighty Gmaas"." (Citation needed)<br />
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159. Almighty Gmaas is Link in the Legend of Zelda. He despises Calamity Ganon in Breath of the Wild. "Too much of a hog..."<br />
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160. Almighty Gmaas is the smartest living being in the universe. EDIT: Notice how this sentences says "living being," so Almighty Gmaas might be dummer than some dead person. EDIT EDIT: Almighty Gmaas is the smartest being that has lived, lives, and will live. EDIT EDIT EDIT: How? EDIT EDIT EDIT EDIT: The power of Almighty Gmaas. EDIT EDIT EDIT EDIT EDIT: How dare you question the power of Almighty Gmaas? EDIT: Almighty Gmaas does not live. Life is Almighty Gmaas.<br />
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161. Almighty Gmaas helped Sun Wukong on the Journey to the West.<br />
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162. Almighty Gmaas was the creator of Wikipedia. EDIT: Almighty Gmaas is wikipedia.<br />
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163. Almighty Gmaas can hack any website he desires. EDIT: He can edit any website with the words Almighty Gmaas in it. EDIT: AoPS includes the word Almighty Gmaas right here, so Almighty Gmaas can hack it. EDIT: Almighty Gmaas can edit any website he desires.<br />
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164. Almighty Gmaas is the basis of Greek and Egyptian mythology. EDIT: He is also the basis for Aztec mythology. He once had a craving for human flesh, so he created human sacrifice.<br />
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165. Almighty Gmaas once sold Google to a man for around 12 dollars! EDIT: Almighty Gmaas has not spent those 12 dollars and is waiting for the economy to crash. EDIT: Why would he do that? EDIT: The power of Almighty Gmaas. EDIT: Almighty Gmaas is smarter than you think he is. Do you think you know how smart Almighty Gmaas is now? If so, then you are still wrong.<br />
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166. Almighty Gmaas uses a HP printer. It is specifically a HP 21414144124124142141414412412414214141441241241421414144124124142141414412412414 printer.<br />
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167. Almighty Gmaas owns all AoPS staff including Richard Rusczyk. EDIT: Richard Rusczyk is one of Almighty Gmaas's many code names.<br />
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168. Almighty Gmaas saw Yoda's birth. EDIT: Almighty Gmaas was Yoda's father. EDIT: Was Almighty Gmaas green like Yoda back then? EDIT: Yes. EDIT: Why? EDIT: The power of Almighty Gmaas.<br />
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169. Almighty Gmaas's true number of lives left is unknown; however, Almighty Gmaas recently confirmed that he had at least one left. Why doesn't Almighty Gmaas have so many more lives than other cats? The power of Almighty Gmaas. EDIT: This is all not true. EDIT: This is all true. EDIT: Almighty Gmaas has an infinite number of lives. EDIT: Almighty Gmaas does not have an infinite number of lives. He just has an unlimited number of lives. EDIT: Almighty Gmaas has an infinite and unlimited number of lives.EDIT: <br />
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170. sseraj once spelled Almighty Gmaas as gmASS by accident in Introduction to Geometry (1532). sseraj’s life was never the same afterwards. EDIT: Almighty Gmaas created sseraj. He is a Almighty Gmaas, but without the face, body, legs, and tail.<br />
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171. Almighty Gmaas has beaten Chuck Norris, The Rock, and John Cena all together in a fight.<br />
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172. Almighty Gmaas is a South Korean, North Korean, Palestinian, Israeli, U.S., Soviet, Russian, and Chinese citizen at the same time. EDIT: Almighty Gmaas is a citizen of every country in the world. Almighty Gmaas seems to enjoy the country of AoPS best, however.<br />
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173. "I am sand" destroyed Almighty Gmaas in FTW. EDIT: Because Almighty Gmaas accidentally died, he defeated "I am sand" 3,141,592,653 times after.<br />
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174. sseraj posted a picture of Almighty Gmaas with a game controller in Introduction to Geometry (1532).<br />
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175. Almighty Gmaas plays Roblox mobile edition and likes Minecraft, Candy Crush, and Club Penguin Rewritten. He also loves Catch that fish. EDIT Almighty Gmaas likes Minecraft better than Roblox<br />
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176. Almighty Gmaas is Roy Moore's horse in the shape of a cat.<br />
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177. Almighty Gmaas is a known roblox/club penguin rewritten player and is a legend at it. He has over <math>289547987693</math> roblox and <math>190348</math> in CPR.<br />
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178. This is all hypothetical. EDIT: This is all factual. For reference, see an earlier post.<br />
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179. Almighty Gmaas's real name is Princess. He has a sibling named Rusty/Fireheart/Firestar. EDIT: That is incorrect. Almighty Gmaas's real name is Almighty Gmaas. EDIT: Almighty Gmaas named himself. That's why his name is amazing.<br />
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180. Almighty Gmaas is capable of salmon powers. EDIT: Almighty Gmaas is capable of everything. Nothing is beyond Almighty Gmaas.<br />
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181. Almighty Gmaas told Richard Rusczyk to make AoPS.<br />
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182. The Almighty Gmaas is everything. Yes, you are part of the Almighty Gmaas-Dw789. EDIT: Almighty Gmaas is older than the universe. He is more than everything.<br />
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183. The Almighty Gmaas knows every dimension up to 99999999999999999999999999999999999999999999999999999999999999999999999999999999999th dimension. EDIT: He is working on the higher dimensions presently. It takes up 1% of his time. He spends the remaining 99% of his time eating, sleeping, and being. EDIT: Does that mean he stops being while he learns higher dimensions? EDIT: Yes. EDIT: Why? EDIT: The power of Almighty Gmaas.<br />
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184. Almighty Gmaas went into a black hole and exited the white hole. He ended up in the 15th dimension where people drink tea every day. He stole 31,415,926,535,897,932,384 buckets of tea.<br />
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185. Almighty Gmaas is "TIRED OF PEOPLE ADDING TO HIS PAGE!!" (Maas 45). EDIT: This is Almighty Gmaas's holy, so he wants to have a longer version. For reference, see Maas 3,141,592,653,589,793,238,462. Almighty Gmaas is too lazy to write this book himself or to hire someone to do this for him, so he has assembled a group of unpaid freelancers on AoPS to write his holy book for him. He is watching it grow. However, Almighty Gmaas sometimes sees a comment that he does not like on the Almighty Gmaas article. He possesses an editor of Almighty Gmaas and makes him/her delete it. EDIT: AoPS itself is just an elaborately concealed piece of Almighty Gmaas propaganda. EDIT EDIT: How dare you reveal the truth! *Zaps previous editor out of existence* Now, where were we...<br />
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186. Almighty Gmaas is a Almighty Gmaas who is actually Almighty Gmaas. EDIT: Almighty Gmaas is omnipotent and omniscient. However, he is not always benevolent.<br />
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187. Almighty Gmaas has a penguin servant who runs Almighty GmaasINC. The penguin may or may not be dead. He had another penguin, who is mostly alive.<br />
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188. Almighty Gmaas owns a TARDIS, and can sometimes be seen traveling to other times for reasons unknown. EDIT: Almighty Gmaas is the doctor. EDIT EDIT: Therefore, the doctor's name is Almighty Gmaas.<br />
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189. Almighty Gmaas knows how to hack into top secret AoPS community pages. EDIT: AoPS is just Almighty Gmaas's elaborately concealed blog.<br />
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190. Almighty Gmaas used to be river clan cat who crossed the event horizon of a black hole and came out the other end. EDIT: He eventually created sseraj and has lived with sseraj ever since.<br />
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191. Almighty Gmaas is king of the first men, the anduls.<br />
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192. Almighty Gmaas is a well known professor at Meowston Academy. EDIT: He is professor of Almighty Gmaasology there. EDIT: Almighty Gmaas founded Meowston Academy. EDIT: It is actually called Almighty Gmaaston Academy<br />
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193. Almighty Gmaas comes from Almighty Gmaas Land, a magical place where pie is infinite and everything starts with a g<br />
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194. Almighty Gmaas is the CEO of Caterpillar.<br />
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195. Almighty Gmaas drinks at Starbucks everyday.<br />
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196. Almighty Gmaas is addicted to tuna, along with other more potent fish, such as salmon and trout.<br />
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197. Almighty Gmaas likes turning into fish and catching himself. EDIT: So far, Almighty Gmaas has spent 2,718,281 of his many lives as a catfish. However, sometimes, he has been catched as a catfish by other people. Luckily, Almighty Gmaas has always escaped.<br />
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198. Almighty Gmaas won the reward of being cutest and fattest cat ever--he surpassed grumpy cat. (He also out-grumped grumpy cat!!!) EDIT: He is in the Guinness Book of World Records for fluffiest cat ever.<br />
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199. He was last sighted at 1665 Alligator Swamp-A 4/1/18 at 3:14:15.92653589793238462 PM.<br />
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200. Almighty Gmaas is the owner of sseraj, not his pet. EDIT: Almighty Gmaas also created sseraj.<br />
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201. Almighty Gmaas is the embodiment of life, the universe, and beyond.<br />
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202. Almighty Gmaas watches memes about Almighty Gmaas. EDIT: All memes originate with Almighty Gmaas, but they are often posted by others. EDIT EDIT: Almighty Gmaas is getting rather tired of memes.<br />
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203. After death Almighty Gmaas became the god of hyperdeath and obtained over 9000 souls.<br />
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204. One of Almighty Gmaas's many real names is Pablo Diego José Francisco de Paula Juan Nepomuceno María de los Remedios Cipriano de la Santísima Trinidad Ruiz y Picasso.<br />
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205. Almighty Gmaas is a certified Slytherin. EDIT: Almighty Gmaas is actually a certified Ravenclaw.<br />
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206. Almighty Gmaas once slept on sseraj's private water bed, so sseraj locked him in the bathroom. EDIT: Almighty Gmaas has superpowers that allowed him to overcome the horrors of Mr. Toilet while locked in the bathroom.<br />
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207. Almighty Gmaas once sat on an orange on a pile of AoPS books, causing an orange flavored equation explosion.<br />
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208. Almighty Gmaas conquered the moon and imprinted his face on it until asteroids came and erased it.<br />
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209. Almighty Gmaas is a supreme overlord who must be given<cmath>1000^{1,000,000,000,000,000,000,000^{1,000,000,000,000,000,000,000}}</cmath>minecraft diamonds. EDIT: Almighty Gmaas owns Mojang, but Almighty Gmaas sued them when it was supposed to be called Almighty Gmaascraft. EDIT: A previous fact said that Almighty Gmaas only lost one lawsuit. Why is this Minecraft not called Almighty Gmaascraft if he won the lawsuit?<br />
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210. Almighty Gmaas is the Doctor Who lord, sports Dalek-painted cars, and eats human finger cheese, custard, and black holes. EDIT: Almighty Gmaas once ate a white hole. Almighty Gmaas then exploded and made a new universe.<br />
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211. Almighty Gmaas is everyone's favourite animal.<br />
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212. Almighty Gmaas is a Persian Smoke.<br />
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213. Almighty Gmaas lives with sseraj.<br />
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214. Almighty Gmaas dislikes geometry but enjoys number theory. EDIT: sseraj once posted a picture of Almighty Gmaas being grumpy after seeing an olympiad geometry problem.<br />
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215. Almighty Gmaas is often overfed (with probability <math>\frac{3972}{7891}</math>), or malnourished (with probability <math>\frac{3919}{7891}</math>) by sseraj. EDIT: Almighty Gmaas does not need food. He just eats food because it tastes good. EDIT: Almighty Gmaas does not like vegetables.<br />
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216. Almighty Gmaas has<cmath>\sum_{k=1}^{267795} [k(k+1)]+Almighty Gmaas+GMAAAAAAAAAAS</cmath>supercars, excluding the Purrari and the 138838383 Teslas.<br />
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217. Almighty Gmaas employs AoPS. EDIT: Almighty Gmaas is the reason why AoPS exists. EDIT: Almighty Gmaas is the reason why everything exists.<br />
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218. Almighty Gmaas is a Almighty Gmaas with yellow fur and white hypnotizing eyes. EDIT: His fur is not always yellow. His fur can be white<br />
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219. Almighty Gmaas is the only one who knows what i is. Everyone else thinks that it's imaginary.<br />
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220. Almighty Gmaas was born with a tail that is a completely different color from the rest of his fur. EDIT: Almighty Gmaas can change the color of his fur.<br />
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221. Almighty Gmaas's stare is very hypnotizing and effective at getting table scraps. EDIT: Almighty Gmaas's stare turned Medusa into rock, King Midas into gold, and sseraj into sseraj.<br />
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222. Almighty Gmaas always appears several minutes before certain classes start as an administrator. EDIT: This is becoming more and more infrequent. Sadly, sseraj rarely posts pictures of Almighty Gmaas before classes anymore. EDIT: Why is this happening? EDIT: The power of Almighty Gmaas.<br />
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223. Almighty Gmaas is an AoPS administrator under the alias Grayson Maas. EDIT: Almighty Gmaas always posts class surveys. He has a fake photo on his bio and has a real one on his community page. However, Almighty Gmaas himself has not yet posted on the Almighty Gmaas forum. EDIT: He also has not edited the Almighty Gmaas article, the Talk: Almighty Gmaas article, the Almighty Gmaasology article, or the sseraj article. EDIT: The final one is surprising because Almighty Gmaas knows more about sseraj than even sseraj does.<br />
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224. Almighty Gmaas died from too many Rubik's cubes in an Introduction to Algebra A class, but he was revived by the Dark Lord at 12:13:37 AM the next day. EDIT: This was thanks in return for Almighty Gmaas reincarnating him during the Triumverate Tournament.<br />
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225. It is uncertain whether or not Almighty Gmaas is a cat or is merely some sort of beast that has chosen to take the form of a cat (specifically a Persian Smoke). EDIT: He is both. EDIT: Actually, Almighty Gmaas is a cat. Almighty Gmaas said so, and science says so. EDIT: The profile for Almighty Gmaas on the list of AoPS teachers does not mention him being a cat. EDIT: The above post is irrelevant. Almighty Gmaas hides his identity under the alias Grayson Maas. EDIT: Almighty Gmaas us everything yet nothing. EDIT: How is that? The power of Almighty Gmaas.<br />
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226. Almighty Gmaas is distant relative of the chair of the department of Almighty Gmaasology at Princeton. EDIT: In the last few years, many universities have been opening Almighty Gmaasology departments. For more information, see the Almighty Gmaasology page.<br />
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227. Almighty Gmaas cannot be Force choked. Darth Vader learned that the hard way...<br />
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228. Almighty Gmaas is famous, and mods always talk about him before class starts.<br />
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229. Almighty Gmaas's favorite food is AoPS textbooks because they help him digest problems. EDIT: Almighty Gmaas wrote all AoPS textbooks but is never listed in the acknowledgments section. EDIT: That is because Almighty Gmaas is very humble except for when he isn't.<br />
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230. Almighty Gmaas tends to reside in sseraj's fridge. EDIT: Almighty Gmaas once ate all sseraj's fridge food, so sseraj had to put him in the freezer. EDIT: Almighty Gmaas's fur can protect him from the harsh conditions of a freezer. EDIT: Then Almighty Gmaas ate all the food in sseraj's freezer. He enjoyed the ice cream in the freezer the most. EDIT: Almighty Gmaas also ate sseraj's freezer.<br />
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231. Almighty Gmaas once demanded Epic Games to give him 5,000,000 V-bucks for his 569823rd birthday. EDIT: This is why Almighty Gmaas no longer has an Epic Games account. EDIT: Almighty Gmaas created Epic Games, though.<br />
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232. Almighty Gmaas sightings are not very common. There have only been 30 confirmed sightings of Almighty Gmaas in the wild.<br />
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233. Almighty Gmaas is a sage omniscient cat.<br />
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235. Almighty Gmaas is looking for suitable places other than sseraj's fridge to live in.<br />
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236. List of places where Almighty Gmaas sightings have happened:<br />
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- 1. The Royal Scoop ice cream store in Bonita Beach Florida<br />
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- 2. Inside the abandoned hospital on Rosevelt Island in New York City, which is also a feral cat sanctuary.<br />
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- 3. MouseFeastForCats/CAT 8 Mouse Apartment 1083<br />
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- 4. Almighty Gmaasology 2 (1440)<br />
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- 5. Alligator Swamp A 1072<br />
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- 6. Alligator Swamp B 1073<br />
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- 7. Almighty Gmaasology A (1488)<br />
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- 8. Introduction to Almighty Gmaasology A (1170)<br />
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- 9. Introduction to Almighty Gmaasology B (1529)<br />
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- 10. Welcome to Panda Town Gate 1076<br />
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- 11. Welcome to Almighty Gmaas Town Gate 1221<br />
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- 12. Welcome to Almighty Gmaas Town Gate 1125<br />
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- 13. 33°01'17.4"N 117°05'40.1"W (Rancho Bernardo Road, San Diego, CA)<br />
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- 14. The other side of the ice in Antarctica<br />
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- 15. Feisty Alligator Swamp 1115<br />
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- 16. Intermediate Almighty Gmaas and Almighty Gmaasology 1207<br />
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- 17. Posting student surveys<br />
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- 18. USF Castle Walls - Elven Tribe 1203<br />
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- 19. The Dark Lord's Hut 1210<br />
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- 20. Gmathamatical Problem Series 1200<br />
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- 21. Intermediate Almighty Gmaasology 1138<br />
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- 22. Intermediate Number Theory 1476<br />
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- 23. Introduction to Almighty Gmaasology 1204 on July 27, 2016<br />
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- 24. Almighty Gmaasology B 1112<br />
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- 25. Intermediate Algebra 1561 7:17 PM 12/11/16<br />
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- 26. Nowhere Else, Tasmania<br />
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- 27. The Gmaathamatics School of Almighty Gmaasology<br />
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- 28. 38.6780° N, 121.1761° W<br />
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- 29. The Ship of Berthuril, Queen of Cats<br />
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-30. The Interplanetary Convention of Beings Inportant to Math<br />
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237. Almighty Gmaas can communicate with, and sometimes control, any other cats; however, this is very rare, as cats normally have a very strong will.<br />
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238. A picture of Almighty Gmaas: http://i.imgur.com/PP9xi.png<br />
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239. Almighty Gmaas met Mike Miller. EDIT: Almighty Gmaas ate Mike Miller.<br />
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240. Almighty Gmaas got mad at sseraj once, so Almighty Gmaas locked sseraj in sseraj's freezer.<br />
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241. Then, sseraj decided to eat all of Almighty Gmaas's hidden turnips in the freezer as punishment. EDIT: sseraj could not eat all of them so, he put the remaining turnips in the fire.<br />
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242. Almighty Gmaas once ate a pie. EDIT: He only ate the first 31415926535897932384626433832 digits.<br />
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243. A Almighty Gmaas bite is 314159265358979323846264 psi.<br />
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244. Many people have met Almighty Gmaas, although many of these sightings are dubious. EDIT: It is just like alien sightings.<br />
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245. Almighty Gmaas can talk but normally hates talking.<br />
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246. Almighty Gmaas likes to eat his own fur. That's why he doesn't have any as of late<br />
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247. Almighty Gmaas is bigger than an ant. EDIT: so is a regular cat<br />
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248. Almighty Gmaas once ate the king of the ants. EDIT: Ants do not have kings; they have queens.<br />
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249. Almighty Gmaas lives somewhere over the rainbow. EDIT: He lives in sseraj's house. EDIT: sseraj lives somewhere over the rainbow.<br />
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250. Almighty Gmaas is an obviously omnipotent cat.<br />
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251. sseraj is known to post pictures of Almighty Gmaas on various AoPS classrooms. It is not known if these photos have been altered with the editing program called 'Photoshop.'<br />
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252. Almighty Gmaas EDIT: This is a waste of a fact.<br />
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253. sseraj has posted pictures of Almighty Gmaas in Introduction to Algebra before class started with the title "caption contest." Anyone who posted a caption mysteriously vanished in the middle of the night. EDIT: This has happened many times, including in Introduction to Geometric Almighty Gmaasology 1533, among other active classes. The person writing this did participate, and did not disappear. (You could argue Almighty Gmaas is typing this through his/her account...)<br />
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254. Almighty Gmaas once slept in your bed and made it gray. EDIT: My bed is not gray. EDIT: Nor is mine.<br />
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255. Richard Rusczyk actually Almighty Gmaas in disguise. EDIT: That is false. Almighty Gmaas is only in one form at a time. EDIT: This is also false, you are Almighty Gmaas and Almighty Gmaas owns you. EDIT: This is false too. Almighty Gmaas can only be in one form.<br />
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256. Almighty Gmaas is suspected to be a cat in disguise. EDIT: He is a cat in disguise. See the results on the Almighty Gmaas poll in the Almighty Gmaas forum.<br />
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257. Almighty Gmaas is a cat but has characteristics of every other animal on Earth. EDIT: Almighty Gmaas does not have the characteristic of being able to eat plankton. EDIT: Yes he does. He just has not decided to reincarnate as a blue whale yet.<br />
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258. Pegasus was modeled off Almighty Gmaas. EDIT: Pegasus used to be Almighty Gmaas's pet, but Pegasus escaped and joined Bellerophon.<br />
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259. Almighty Gmaas is the ruler of the universe and has been known to be the creator of the species "Almighty Gmaasians". EDIT: He is the universe<br />
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260. There is a rumor that Almighty Gmaas is starting a poll. EDIT: He has. It is on the Almighty Gmaas forum, made by his minions. EDIT: We are all his minions.<br />
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261. Almighty Gmaas is a Persian smoke who ran away, founded Almighty GmaasClan, and became a kitty-pet.<br />
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262. There is a sport called "Almighty Gmaas Hunting" where people try to successfully capture Almighty Gmaas in the wild with video/camera/eyes. Strangely, no one has been able to do this, and those that have have mysteriously disappeared into the night. Nobody knows why. Many people have tried Almighty Gmaas Hunting, but they never have been successful.<br />
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263. Almighty Gmaas burped and caused an earthquake.<br />
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264. Almighty Gmaas once drank from a teacup.<br />
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265. Almighty Gmaas IS HERE.... PURRRRRRRRRRRRRRRRRRRR EDIT: Almighty Gmaas is everywhere. EDIT EDIT: He is behind you...<br />
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266. Almighty Gmaas made and currently owns the Matrix. EDIT: The above fact is true. Therefore, this is an illusion. EDIT: Almighty Gmaas is not an illusion.<br />
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267. Almighty Gmaas is the reason Salah will become better than Ronaldo.<br />
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268. Who is Almighty Gmaas, really? EDIT: Almighty Gmaas is a cat. EDIT EDIT: Who cares how much we mention it? Almighty Gmaas is a cat. EDIT EDIT EDIT: Almighty Gmaas is not a cat. Almighty Gmaas is Almighty Gmaas.<br />
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269. Almighty Gmaas is a heavenly being. EDIT: He is immortal. EDIT EDIT: He is not immortal. He just tricked you to think he is immortal. Almighty Gmaas is good at tricking.<br />
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270. Almighty Gmaas is a cat with no fur or tail. In fact he's a human wearing a cat costume. EDIT: This fact is false. 31415 times false.<br />
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271. Illuminati was a manifestation of Almighty Gmaas, but Almighty Gmaas decided Illuminati was not great enough for his godly self.<br />
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272. sseraj has met Almighty Gmaas, and Almighty Gmaas is his best friend. EDIT: sseraj is Almighty Gmaas's great great great great great grandson.<br />
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273. Almighty Gmaas read Twilight. EDIT: ...and SURVIVED.<br />
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274. There is a secret code when put into super smash where Almighty Gmaas would be a playable character. Too bad he didn't say it.<br />
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275. Almighty Gmaas was a tribute to one of the Hunger Games, came out a Victor, and now lives in District 4. EDIT: This fact is true and not true. He is everywhere.<br />
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276. Almighty Gmaas is the only known creature that will survive the destruction of Earth in 99,999,999 years. EDIT: The Earth will probably die in around 5,000,000,000 years.<br />
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277. 5space (side admin) is one of Almighty Gmaas's slaves. EDIT: Almighty Gmaas owns all AoPS site administrators. EDIT: He owns Himself.<br />
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278. Almighty Gmaas photos − https://cdn.artofproblemsolving.com/images/f/f/8/ff8efef3a0d2eb51254634e54bec215b948a1bba.jpg<br />
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http://disneycreate.wikia.com/wiki/File:Troll_cat_gif_(1).gif<br />
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He was also sighted here.<br />
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279. Almighty Gmaas in Popular Culture:<br />
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BREAKING NEWS: A Almighty Gmaasologist has found a possible cousin to Almighty Gmaas in Raymond Feist's book Silverthorn. They are mountain dwellers, gwali. Not much are known about them either, and when someone asked,"What are gwali?" the customary answer "This is gwali" is returned. Another eminent Almighty Gmaasologist is now looking into it. 199 Someone is writing a book about Almighty Gmaas.<br />
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280. Sighting of Almighty Gmaas: https://2017.spaceappschallenge.org/challenges/warning-danger-ahead/and-you-can-help-fight-fires/teams/Almighty Gmaas/project<br />
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281. Oryx the mad god is actually Almighty Gmaas wearing a suit of armor. This explains why he is never truly killed.<br />
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282. Potential sighting of Almighty Gmaas [1]<br />
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283. Almighty Gmaas has been spotted in some Doctor Who and Phineas and Ferb episodes, such as Aliens of London, Phineas and Ferb Save Summer, Dalek, Rollercoaster, Rose, Boom Town, The Day of The Doctor, Candace Gets Busted, The Name of the Doctor, Silence in the Library, The Idiots Lantern and many more.<br />
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284. Almighty Gmaas can be found in many places in Plants vs. Zombies Garden Warfare 2 and Bloons TD Battles.<br />
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285. Almighty Gmaas was an un-credited actor in the Doctor Who story Knock Knock, playing a Dryad. How he shrunk, we will never know.<br />
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286. An eminent Almighty Gmaasologist is also writing a story about him. He is continuing the book that was started by the professor of Almighty Gmaasology at Harvard. When he is done he will post it here.<br />
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287. Almighty Gmaas is a time traveler from 0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 B.C.<br />
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288. No one knows if Almighty Gmaas is a Mr. Mime in a cat skin, the other way around, or just a downright combination of both. EDIT: Almighty Gmaas is an immortal in a cat body.<br />
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289. In it, it mentions these four links as things Almighty Gmaas is having trouble (specifically technical difficulties). What could it mean? Links:<br />
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https://docs.google.com/document/d/1NZ0XcFYm80sA-fAoxnm7ulMCwdNU75Va_6ZjRHfSHV0<br />
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https://docs.google.com/document/d/1ELN7ORauFFv1dwpU_u-ah_dFJHeuJ3szYxoeC1LlDQg/<br />
<br />
https://docs.google.com/document/d/1oy9Q3F7fygHw-OCWNEVE8d-Uob2dxVACFcGUcLmk3fA<br />
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https://docs.google.com/document/d/1jzb9Q6FmDmrRyXwnik3e0sYw5bTPMo7aBwugmUbA13o<br />
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290. Another possible Almighty Gmaas sighting?<br />
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291. <math>Another</math> sighting? [3]<br />
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292. Yet Another Almighty Gmaas sighting? [4]<br />
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293. Almighty Gmaas has been sighted several times on the Global Announcements forum.<br />
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294. Almighty Gmaas uses the following transportation: <img> http://cdn.artofproblemsolving.com/images/3/6/8/368da4e615ea3476355ee3388b39f30a48b8dd48.jpg </img> EDIT: He also sometimes uses a TARDIS<br />
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295. When Almighty Gmaas was angry, he started world wars 1 & 2. It is only because of Almighty Gmaas that we have not had World War 3. EDIT: He was starting to have to get angry in the Cold War, especially around the Cuban Missile Crisis and the Ronald Reagan era, but he decided to eat food, which calmed him down.<br />
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296. Almighty Gmaas is the only cat to have been proved irrational and transcendental, though we suspect all cats fall in the first category.<br />
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297. Almighty Gmaas belongs to a secret club of transcendental cats. EDIT: That club is located underneath the Eiffel Tower. EDIT EDIT: Gustave Eiffel was Almighty Gmaas in disguise and designed the club.<br />
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298. Almighty Gmaas plays Geometry Dash. His username is D3m0nG4m1n9. EDIT: Almighty Gmaas only does this to punish himself. Olympiad Geometry is his least favorite thing to do. EDIT: None of these are true.<br />
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299. Almighty Gmaas likes to whiz on the wilzo.<br />
<br />
300. Almighty Gmaas has been spotted in many AoPS classes, such as AMC 8 Basics. EDIT: Many of the AoPS classes Almighty Gmaas has visited can be found in the Almighty Gmaas sightings section of this article.<br />
<br />
301. Almighty Gmaas is cool. EDIT: He was not cool in summer, so he invented air conditioning. EDIT: He is air conditioning, in fact he is the entire universe.<br />
<br />
302. Almighty Gmaas hemoon card that does over 9000000 dmg.<br />
<br />
303. Almighty Gmaas is a skilled swordsman who should not to be mistaken for Puss in Boots. Some say he even trained the mysterious and valiant Meta Knight.<br />
<br />
303. Kirby once swallowed Almighty Gmaas. Almighty Gmaas had to spit him out.<br />
<br />
304. Almighty Gmaas was the creator of pokemon, and his pokemon card can OHKO anyone in one turn. He is invisible and he will always move first.<br />
<br />
305. Almighty Gmaas beat Dongmin in The Genius Game Seasons 1, 2, 3, 4, 5, 6, and 7.<br />
<br />
306. Almighty Gmaas has five letters. Pizza also has five letters. Pizzas are round. Eyes are round. There is an eye in the illuminati symbol. EDIT: Almighty Gmaas is a fluffy cat.<br />
<br />
228 Almighty Gmaas knows both 'table' and 'tabular' in LaTeX, and can do them in his sleep. EDIT: Almighty Gmaas invented LaTeX in his sleep.<br />
<br />
308. Almighty Gmaas does not hate cheddar cheese, but he doesn't love it either.<br />
<br />
309. Almighty Gmaas is a cat and not a cat. EDIT: He is a cat.<br />
<br />
310. Almighty Gmaas was born on the sun. EDIT: Not the sun, the suns. He was born on all the suns at once.<br />
<br />
311. Almighty Gmaas eats tape. EDIT: Almighty Gmaas invented tape. EDIT: Almighty Gmaas eats everything. Almighty Gmaas eats computers.<br />
<br />
322. Almighty Gmaas likes Bubble Gum.<br />
<br />
323. Thomas Edison did not invent the lightbulb; Almighty Gmaas did.<br />
<br />
324. Almighty Gmaas invented the alphabet. EDIT: Almighty Gmaas thought that hieroglyphs were too complicated, so he invented the alphabet.<br />
<br />
325. Almighty Gmaas eats metal. EDIT: Almighty Gmaas once ate so much metal that he turned into a bronze statue.<br />
<br />
326. Almighty Gmaas is over 9000 years old! EDIT: This is just a DBZ reference, and bears no reality to his true age. EDIT: Almighty Gmaas is trillions of years old.<br />
<br />
327. Almighty Gmaas is a cat and not a cat. EDIT: He is a cat. EDIT: This fact has been stated 31,415 times in this article. EDIT EDIT: Almighty Gmaas is not a cat.<br />
<br />
328. Almighty Gmaas started the Iron Age. EDIT: He thought civilization was getting too advanced. So he sent out the Sea People to destroy civilization.<br />
<br />
329. Almighty Gmaas made the dinosaurs go extinct. EDIT: That happened when he got angry that a dinosaur stepped on his toe.<br />
<br />
330. Almighty Gmaas created life. Then he destroyed it, and in doing so, destroyed himself. Until his son, Almighty Gmaas II took over and saved the planet.<br />
<br />
331. Almighty Gmaas created AoPS. EDIT: AoPS was actually born out of a small fraction of Almighty Gmaas's abstract reality, and only the sheer amount math can keep it here. (It is also rumored that when he reclaims it, the USF will be deleted, as that is where 83% of the factions of his abstract reality lives, and when people leave USF, more and more escapes.)<br />
<br />
332. Almighty Gmaas created mathematics. EDIT: Almighty Gmaas discovered mathematics.<br />
<br />
333. Almighty Gmaas does not like Roblox.<br />
<br />
334. Almighty Gmaas told Steve Jobs to start a company.<br />
<br />
335. Almighty Gmaas invented Geometry Dash. EDIT: Almighty Gmaas hates Olympiad Geometry. [Source: sseraj]<br />
<br />
336. Almighty Gmaas got to<math>\infty</math>in Flappy Bird. EDIT: Almighty Gmaas broke the game.<br />
<br />
337. Almighty Gmaas invented Helix Jump.<br />
<br />
338. Almighty Gmaas can play Happy Birthday on the Violin. EDIT: Almighty Gmaas can also play The Flight of the Bumblebees. EDIT: Gmass can play anything on the Violin.<br />
<br />
339. Almighty Gmaas has mastered Paganini. EDIT: Paganini was Almighty Gmaas in disguise. EDIT: Almighty Gmaas was Paganini in one of his 100,000 human lives. He does not have a human life now because he has a cat life.<br />
<br />
340. Almighty Gmaas discovered Atlantis after one dive underwater.<br />
<br />
341. Almighty Gmaas made a piano with 89 keys. EDIT: The piano was made out of chocolate. EDIT: He also made a piano bench made out of chocolate. EDIT: Almighty Gmaas ate them<br />
<br />
342. Almighty Gmaas can see the future and how to change it.<br />
<br />
343. Almighty Gmaas has every super power you can imagine. EDIT: Almighty Gmaas has more superpowers than you can imagine.<br />
<br />
344. Almighty Gmaas made a Violin with 9 strings.<br />
<br />
345. Almighty Gmaas can read 5 books at once. EDIT: It is slightly difficult for Almighty Gmaas to do this because he has no fingers. But somehow, Almighty Gmaas finds a way. EDIT: He uses his mind to hold open the books.<br />
<br />
355. Almighty Gmaas is the only one who knows what "Inconceivable" means.<br />
<br />
356. Almighty Gmaas taught Richard Rusczyk everything he knows about mathematics. EDIT: Many famous mathematicians have been Almighty Gmaas in disguise.<br />
<br />
357. Almighty Gmaas can control matter by looking at it. EDIT: He once looked at a galaxy while he was angry. Everything in the galaxy exploded.<br />
<br />
358. Almighty Gmaas created AoPS. EDIT: He created 99% of all websites. 98% of those websites are blank and unreachable, but the ones that are not are interesting.<br />
<br />
359 Almighty Gmaas is a quantum particle. EDIT: Almighty Gmaas is multiple quantum particles. He is a macroscopic collection of them.<br />
<br />
360. Almighty Gmaas is alive and dead at the same time. EDIT: He is usually alive.<br />
<br />
361. Almighty Gmaas likes snow. EDIT: Once Almighty Gmaas was reincarnated as a polar bear around 1,000,000 years ago. He had the best time of his lives. EDIT: While he was a polar bear, he ate many fish.<br />
<br />
362. Almighty Gmaas LIKES THE NEW AOPS UPDATE! EDIT: Are you sure he does? Did you ask him? EDIT: Yes, I did.<math>\textrm{Almighty Gmaas is right behind you}</math>EDIT: Almighty Gmaas is everywhere. He is in a quantum superposition. He is still in the superposition until someone sees him. The chance of him being behind you is 0.0000000000000000000000000000000000000000000000000000001%. But, you never know. Is he behind you? Is he behind me right now? He is!!! Aaaaaaaaaah... THUMP<br />
<br />
363. Almighty Gmaas does not like the new update. When he saw it, he looked like this.<br />
https://cdn.artofproblemsolving.com/images/7/4/b/74b21fec95225e1621c71ec8f17f343c48a726e0.jpg<br />
<br />
364. Almighty Gmaas likes his eggs hard boiled.<br />
<br />
365. Almighty Gmaas doesn't take showers; he only takes bloodbaths.<br />
<br />
366. When the bogeyman goes to sleep, he checks his closet for Almighty Gmaas.<br />
<br />
367. When Almighty Gmaas crosses the street, cars have to look both ways.<br />
<br />
368. Almighty Gmaas has counted to infinity an infinite amount of times.<br />
<br />
369. Almighty Gmaas pulled the pin in a grenade. 1 billion people died. Then he threw it.<br />
<br />
370. Almighty Gmaas controls the Illuminati and controls all the world's resources. He also is the ruler of a communist society on Mars.<br />
<br />
371. Almighty Gmaas has taken the AHSME/AMC 10, AIME, USAJMO, USAMO, and IMO all the years it has come out. It is rumored that for 2018, Almighty Gmaas got a 163.5 on the AMC 12A, a 156 on the AMC 12B, and 17 problems right on the AIME I. EDIT: He also got 51 on USAMO. EDIT: He invented the AMC, AIME, USAJMO, and IMO. EDIT EDIT: Almighty Gmaas had no part in the AIOME <br />
<br />
372. Almighty Gmaas has designed most of the AMC tests. EDIT: He didn't design them in 2015 because he was hibernating too long.<br />
<br />
373. Almighty Gmaas is a koala. EDIT: Almighty Gmaas was actually the first koala. He wanted to reincarnate himself into something new, so he created a whole other species. [Source Wikipedia: Koala] EDIT: Almighty Gmaas is a Koala when they are not alive, which is 10% of the time. When they are alive, Almighty Gmaas is a cat.<br />
<br />
374. Time is Almighty Gmaas's vacation home. EDIT: Almighty Gmaas created time as a science project.<br />
<br />
375. Almighty Gmaas is Catholic. EDIT: That is wrong. Almighty Gmaas is Almighty Gmaas. He believes in Almighty Gmaas.<br />
<br />
376. Almighty Gmaas was once reincarnated as Chuck Norris, but he missed being a cat. Then he became a cat.<br />
<br />
377. Almighty Gmaas once resided with a certain physicist named Schrödinger, but he left because Schrödinger was too confusing. Granted, nothing is too confusing for Almighty Gmaas but Schrödinger also made Almighty Gmaas alive and dead, which felt weird.<br />
<br />
378. Almighty Gmaas once decided to eat a carrot. He decided turnips were better.<br />
<br />
379. Almighty Gmaasology used to be called Gmaathamatics. It was renamed a few years ago. For more information, see the Almighty Gmaasology page. That page needs expanding.<br />
<br />
380. This has been deleted by Almighty Gmaas<br />
<br />
381. This has been deleted by Almighty Gmaas<br />
<br />
382. Grumpy cat pretends to be a descendent of Almighty Gmaas because he is jealous. He is not a descendant of Almighty Gmaas.<br />
<br />
383. Every IMO Gold Medalist is two of Almighty Gmaas, standing on top of each other in a trenchcoat.<br />
<br />
384. Almighty Gmaas has been everywhere in the world. EDIT: Almighty Gmaas has been everywhere in the universe and multiverse.<br />
<br />
385. The degree used to be called the Almighty Gmaas Arc, but the name was too long for most people, so they switched the name.<br />
<br />
386. Almighty Gmaas is famous on AoPS. EDIT: Almighty Gmaas is AoPS.<br />
<br />
387. Almighty Gmaas invented every math theorem and in fact invented math.<br />
<br />
388. Almighty Gmaas made 31.41592653589793238462643383279502884% of memes. EDIT: This is incorrect. This is only an estimate.<br />
<br />
389. Almighty Gmaas is a Almighty Gmaas since Almighty Gmaas likes to Almighty Gmaas and Almighty Gmaas is cool because he is named Almighty Gmaas.<br />
<br />
390. Almighty Gmaas once ate Big Chungus. EDIT: Almighty Gmaas also once ate cat food. EDIT: Almighty Gmaas hates cat food. EDIT: Almighty Gmaas also hates dog food. EDIT: Almighty Gmaas does not need to eat, he is an all-conquering god who never/rarely dies. EDIT: Almighty Gmaas is Big Chungus EDIT: He was Big Chungus for 3.14159265358979323846327656180937734 seconds. EDIT: This is only an rough estimate<br />
<br />
391. Garfield is Almighty Gmaas in disguise. EDIT: But Almighty Gmaas also created Garfield. He took a Gar and a Field, and added them. Boom. He made Garfield.<br />
<br />
392. Almighty Gmaas didn't create the world. He didn't create the universe. He didn't create multiverses. He is the world. He is the universe. He is the multiverses<br />
<br />
393. Almighty Gmaas has a cult with many followers. This is the page for his followers. They pray to him and remember him for his deeds.<br />
<br />
394. Garfield ate Almighty Gmaas but then Almighty Gmaas ate garfield. Then they ate each other. EDIT: Almighty Gmaas ate Garfield at the end because Almighty Gmaas.<br />
<br />
395. Almighty Gmaas has an existential crisis every 99.0000000000000000000000000000000000009123659812374 seconds.<br />
<br />
396. GMASS was going to be part of chrome://dino and renamed chrome://GMASS, however GMASS can't lose to anything, so GMASS changed it to a dino<br />
<br />
397. This has been deleted by GMASS as it is false.<br />
<br />
398. This has been deleted by GMASS as it is false.<br />
<br />
399. (Why is this a thing?) Because we are all worshippers of Almighty Gmaas.<br />
<br />
400. This has been deleted by GMASS as it is false<br />
<br />
401. This has been deleted by GMASS as this is false<br />
<br />
402. riben was a Almighty Gmaas multi, but Almighty Gmaas turned the account over to the present riben and imbued it with his power.<br />
<br />
403. Almighty Gmaas is Sans. Almighty Gmaas faked dying so he could get away and rest. EDIT: It was, in fact ketchup. Sans can't die. He's too angry at Frisk. EDIT EDIT: Almighty Gmaas is always angry at genocidal Frisk.<br />
<br />
404: ERROR 404 SAID GMASS. THIS CAN'T BE REACHED AS IT IS FALSE<br />
<br />
405: Almighty Gmaas's favorite game is Minecraft, tied with Undertale. Breath of the Wild is a close third. His least favorite game is Fortnite. Almighty Gmaas intentionally made Fortnite bad so he could have a least favorite game.<br />
<br />
406: Almighty Gmaas's favorite consoles are in this order: Nintendo, Atari, Xbox. Almighty Gmaas hates PlayStation.<br />
<br />
407: This has been deleted by Almighty Gmaas as this is false<br />
<br />
408: Almighty Gmaas is proven to be exempt from the theory of relativity.<br />
<br />
All hail Almighty Gmaas!<br />
<br />
409: There are currently <math>9999999999999999999999999999999999999999999999999999999999999999999999999^{999999999999999999999999999999999999999999999}</math> many temples to Gmass, with <math>\infty</math> many more being created.<br />
<br />
410: Almighty Gmaas controls everything. He is everywhere. Check if he is under your bed.<br />
<br />
411: Almighty Gmaas hates when someone calls him “Gmaas”.<br />
<br />
412: Almighty Gmaas invented COVID-19 so as to punish those who don’t believe in His might and power. EDIT: The power of Almighty Gmaas.<br />
<br />
413: some (Eg Trump) suspect that Almighty Gmaas is Chinese, but he himself says that he is everything.<br />
<br />
414: Almighty Gmaas is not a cat. Gmaas is everything. Our universe resides in one of his <math>\infty</math> hair. The Big Bang happened because his son (who is Gmaas himself) plucked out one of his hair. <br />
<br />
415: Almighty Gmaas ate the world so you are inside Almighty Gmaas.<br />
<br />
<br />
<br />
P.S. Almighty Gmaas won USAMO 5 times in a row. That's because he invented USAMO. When Almighty Gmaas took it nobody knew about it and he was the only one who took it. So of course, he won. How dare you spell Almighty Gmaas with a lowercase G? He is the supreme and awesome ruler of the universe!<br />
<br />
P.P.S. Almighty Gmaas invented the word Yeet.<br />
<br />
P.P.P.S. Almighty Gmaas is the first person who played the game YECK (Moth poth 2019 reference)<br />
<br />
P.P.P.P.S. Almighty Gmaas is a non-human who can snap anyone out of existence without anything.<br />
<br />
P.P.P.P.P.S. Almighty Gmaas was the first person who did math.<br />
<br />
P.P.P.P.P.P.S. Almighty Gmaas invented proof by dictatorship.<br />
<br />
P.P.P.P.P.P.P.S Almighty Gmaas invented proof by procrastination.<br />
<br />
P.P.P.P.P.P.P.P.S Almighty Gmaas invented proofs!<br />
<br />
P.P.P.P.P.P.P.P.P.S Almighty Gmaas invented everything.<br />
<br />
P.P.P.P.P.P.P.P.P.P.S Almighty Gmaas invented himself and invented proof by Oof<br />
<br />
P.P.P.P.P.P.P.P.P.P.S: Almighty Gmaas Invented Oofism, the religion of the Oof<br />
<br />
-<math>\infty</math>: Almighty Gmaas created everything, and he is everything, meaning that he is also nothing meaning that everything is nothing.<br />
<br />
<math>\infty-1</math>: Ditto is like Almighty Gmaas. We all are dittos, which is in turn made up of Almighty Gmaas.<br />
<br />
<math>\infty-2</math>: Gmaas is the universe. The universe is the universe. Beans are Beans. Cars are cars. Cats are cool. Gmaas is a cat. Cool means epic. Epic means causing pneumonoultramicroscopicsilicovolcanoconiosis. Therefore, Gmaas causes pneumonoultramicroscopicsilicovolcanoconiosis Edit: Gmaas IS pneumonoultramicroscopicsilicovolcanoconiosis</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=User:Blueclay&diff=162234User:Blueclay2021-09-14T17:41:26Z<p>Jwelsh: Reverted edits by Polpot124 (talk) to last revision by Cozzmo</p>
<hr />
<div>This user is the Best in the world! ~Cozzmo</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=User:Firebolt360&diff=161823User:Firebolt3602021-09-07T23:08:01Z<p>Jwelsh: Undo revision 161345 by Jdong2006 (talk)</p>
<hr />
<div><br><br />
__NOTOC__<div style="border:2px solid black; -webkit-border-radius: 10px; background:#F0F2F3"><br />
==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">User Count</div></font>==<br />
<div style="margin-left: 10px; margin-bottom:10px"><font color="black">If this is your first time visiting this page, edit it by incrementing the user count below by one.</font></div><br />
<center><font size="100px">24</font></center><br />
</div><br />
<br />
<p>:p I don't know what else to put here, so bye!!!</p></div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_3&diff=1603922007 AMC 8 Problems/Problem 32021-08-16T16:22:18Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Pi is 3.14</p>
<hr />
<div>== Problem ==<br />
<br />
What is the sum of the two smallest prime factors of <math>250</math>?<br />
<br />
<math>\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 5 \qquad\mathrm{(C)}\ 7 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 12</math><br />
<br />
==Video Solution==<br />
https://youtu.be/7an5wU9Q5hk?t=272<br />
<br />
== Solution ==<br />
<br />
The prime factorization of <math>250</math> is <math>2 \cdot 5^3</math>. The smallest two are <math>2</math> and <math>5</math>. <math>2+5 = \boxed{\text{(C) }7}</math>.<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=2|num-a=4}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_5&diff=1603902007 AMC 8 Problems/Problem 52021-08-16T16:22:17Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by SirCalcsALot</p>
<hr />
<div>== Problem ==<br />
<br />
Chandler wants to buy a <math>500</math> dollar mountain bike. For his birthday, his grandparents<br />
send him <math>50</math> dollars, his aunt sends him <math>35</math> dollars and his cousin gives him <math>15</math> dollars. He earns<br />
<math>16</math> dollars per week for his paper route. He will use all of his birthday money and all<br />
of the money he earns from his paper route. In how many weeks will he be able<br />
to buy the mountain bike?<br />
<br />
<math>\mathrm{(A)}\ 24 \qquad\mathrm{(B)}\ 25 \qquad\mathrm{(C)}\ 26 \qquad\mathrm{(D)}\ 27 \qquad\mathrm{(E)}\ 28</math><br />
<br />
== Solution ==<br />
<br />
Let <math>x</math> be the number of weeks.<br />
<br />
Thus, we have the equation <math>50 + 35 + 15 + 16x = 500</math>.<br />
<br />
Simplify,<br />
<br />
<math>100 + 16x = 500</math><br />
<br />
<math>16x = 400</math><br />
<br />
<math>x = 25</math><br />
<br />
The answer is <math> \boxed{\textbf{(B)}\ 25} </math><br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=4|num-a=6}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_4&diff=1603912007 AMC 8 Problems/Problem 42021-08-16T16:22:17Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Mwil</p>
<hr />
<div>== Problem ==<br />
<br />
A haunted house has six windows. In how many ways can<br />
Georgie the Ghost enter the house by one window and leave<br />
by a different window?<br />
<br />
<math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 15 \qquad\mathrm{(C)}\ 18 \qquad\mathrm{(D)}\ 30 \qquad\mathrm{(E)}\ 36</math><br />
<br />
== Solution ==<br />
<br />
Georgie can enter the haunted house through any of the six windows. Then, he can leave through any of the remaining five windows.<br />
<br />
So, Georgie has a total of <math>6 \cdot 5</math> ways he can enter the house by one window and leave<br />
by a different window.<br />
<br />
Therefore, we have <math> \boxed{\textbf{(D)}\ 30} </math> ways.<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=3|num-a=5}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_6&diff=1603892007 AMC 8 Problems/Problem 62021-08-16T16:22:16Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Nathan wailes</p>
<hr />
<div>== Problem ==<br />
<br />
The average cost of a long-distance call in the USA in <math>1985</math> was<br />
<math>41</math> cents per minute, and the average cost of a long-distance<br />
call in the USA in <math>2005</math> was <math>7</math> cents per minute. Find the<br />
approximate percent decrease in the cost per minute of a long-<br />
distance call.<br />
<br />
<math>\mathrm{(A)}\ 7 \qquad\mathrm{(B)}\ 17 \qquad\mathrm{(C)}\ 34 \qquad\mathrm{(D)}\ 41 \qquad\mathrm{(E)}\ 80</math><br />
<br />
== Solution ==<br />
<br />
The percent decrease is (the amount of decrease)/(original amount)<br />
<br />
the amount of decrease is <math>41 - 7 = 34</math><br />
<br />
so the percent decrease is <math>\frac{34}{41}</math> which is about <math> \boxed{\textbf{(E)}\ 80\%} </math><br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=5|num-a=7}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_7&diff=1603882007 AMC 8 Problems/Problem 72021-08-16T16:22:15Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Ashw24</p>
<hr />
<div>== Problem ==<br />
<br />
The average age of <math>5</math> people in a room is <math>30</math> years. An <math>18</math>-year-old person leaves<br />
the room. What is the average age of the four remaining people?<br />
<br />
<math>\mathrm{(A)}\ 25 \qquad\mathrm{(B)}\ 26 \qquad\mathrm{(C)}\ 29 \qquad\mathrm{(D)}\ 33 \qquad\mathrm{(E)}\ 36</math><br />
<br />
== Solution 1==<br />
<br />
Let <math>x</math> be the average of the remaining <math>4</math> people.<br />
<br />
The equation we get is <math>\frac{4x + 18}{5} = 30</math><br />
<br />
Simplify,<br />
<br />
<math>4x + 18 = 150</math><br />
<br />
<math>4x = 132</math><br />
<br />
<math>x = 33</math><br />
<br />
Therefore, the answer is <math> \boxed{\textbf{(D)}\ 33} </math><br />
<br />
==Solution 2==<br />
<br />
Since an <math>18</math> year old left from a group of people averaging <math>30</math>, The remaining people must total <math>30 - 18 = 12</math> years older than <math>30</math>. Therefore, the average is <math>\dfrac{12}{4} = 3</math> years over <math>30</math>. Giving us <math> \boxed{\textbf{(D)}\ 33} </math><br />
<br />
==Solution 3==<br />
The total ages would be <math>30*5=150</math>. Then, if one <math>18</math> year old leaves, we subtract <math>18</math> from <math>150</math> and get <math>132</math>. Then, we divide <math>132</math> by <math>4</math> to get the new average, <math> \boxed{\textbf{(D)}\ 33} </math><br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=6|num-a=8}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_11&diff=1603862007 AMC 8 Problems/Problem 112021-08-16T16:22:14Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Treddy</p>
<hr />
<div>== Problem ==<br />
<br />
Tiles <math>I, II, III</math> and <math>IV</math> are translated so one tile coincides with each of the rectangles <math>A, B, C</math> and <math>D</math>. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle <math>C</math>?<br />
<br />
<asy><br />
size(400);<br />
defaultpen(linewidth(0.8));<br />
path p=origin--(8,0)--(8,6)--(0,6)--cycle;<br />
draw(p^^shift(8.5,0)*p^^shift(8.5,10)*p^^shift(0,10)*p);<br />
draw(shift(20,2)*p^^shift(28,2)*p^^shift(20,8)*p^^shift(28,8)*p);<br />
label("8", (4,6+10), S);<br />
label("6", (4+8.5,6+10), S);<br />
label("7", (4,6), S);<br />
label("2", (4+8.5,6), S);<br />
label("I", (4,6+10), N);<br />
label("II", (4+8.5,6+10), N);<br />
label("III", (4,6), N);<br />
label("IV", (4+8.5,6), N);<br />
label("3", (0,3+10), E);<br />
label("4", (0+8.5,3+10), E);<br />
label("1", (0,3), E);<br />
label("9", (0+8.5,3), E);<br />
label("7", (4,10), N);<br />
label("2", (4+8.5,10), N);<br />
label("0", (4,0), N);<br />
label("6", (4+8.5,0), N);<br />
label("9", (8,3+10), W);<br />
label("3", (8+8.5,3+10), W);<br />
label("5", (8,3), W);<br />
label("1", (8+8.5,3), W);<br />
label("A", (24,10), N);<br />
label("B", (32,10), N);<br />
label("C", (24,4), N);<br />
label("D", (32,4), N);</asy><br />
<br />
<math>\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}</math> cannot be determined<br />
<br />
== Solution ==<br />
<br />
We first notice that tile III has a <math>0</math> on the bottom and a <math>5</math> on the right side. Since no other tile has a <math>0</math> or a <math>5</math>, Tile III must be in rectangle <math>D</math>. Tile III also has a <math>1</math> on the left, so Tile IV must be in Rectangle <math>C</math>.<br />
<br />
The answer is <math>\boxed{\textbf{(D)}}</math><br />
<br />
<center>[[Image:AMC8_2007_11S.png]]</center><br />
<br />
==Video Solution by Why Math==<br />
https://youtu.be/WobCYII7TRg<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=10|num-a=12}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_9&diff=1603872007 AMC 8 Problems/Problem 92021-08-16T16:22:14Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Nathan wailes</p>
<hr />
<div>== Problem ==<br />
<br />
To complete the grid below, each of the digits 1 through 4 must occur once<br />
in each row and once in each column. What number will occupy the lower<br />
right-hand square?<br />
<br />
<cmath> \begin{tabular}{|c|c|c|c|}\hline 1 & & 2 &\\ \hline 2 & 3 & &\\ \hline & &&4\\ \hline & &&\\ \hline\end{tabular} </cmath><br />
<br />
<math>\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad\textbf{(E)}\ \text{cannot be determined}</math><br />
<br />
== Solution ==<br />
<br />
The number in the first row, last column must be a <math>3</math> due to the fact if a <math>3</math> was in the first row, second column, there would be two threes in that column. By the same reasoning, the number in the second row, last column has to be a <math>1</math>. Therefore the number in the lower right-hand square is <math>\boxed{\textbf{(B)}\ 2}</math>.<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=8|num-a=10}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_12&diff=1603852007 AMC 8 Problems/Problem 122021-08-16T16:22:13Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Ghostyc</p>
<hr />
<div>==Problem==<br />
A unit hexagram is composed of a regular hexagon of side length <math>1</math> and its <math>6</math><br />
equilateral triangular extensions, as shown in the diagram. What is the ratio of<br />
the area of the extensions to the area of the original hexagon?<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(polygon(3));<br />
pair D=origin+1*dir(270), E=origin+1*dir(150), F=1*dir(30);<br />
draw(D--E--F--cycle);<br />
</asy><br />
<br />
<math>\mathrm{(A)}\ 1:1 \qquad \mathrm{(B)}\ 6:5 \qquad \mathrm{(C)}\ 3:2 \qquad \mathrm{(D)}\ 2:1 \qquad \mathrm{(E)}\ 3:1</math><br />
<br />
==Solution==<br />
The six equilateral triangular extensions fit perfectly into the hexagon meaning the answer is <math>\boxed{\textbf{(A) }1:1}</math><br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=11|num-a=13}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_13&diff=1603842007 AMC 8 Problems/Problem 132021-08-16T16:22:12Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Putu2003</p>
<hr />
<div>== Problem ==<br />
<br />
Sets <math>A</math> and <math>B</math>, shown in the Venn diagram, have the same number of elements.<br />
Their union has <math>2007</math> elements and their intersection has <math>1001</math> elements. Find<br />
the number of elements in <math>A</math>.<br />
<br />
<asy><br />
defaultpen(linewidth(0.7));<br />
draw(Circle(origin, 5));<br />
draw(Circle((5,0), 5));<br />
label("$A$", (0,5), N);<br />
label("$B$", (5,5), N);<br />
label("$1001$", (2.5, -0.5), N);</asy><br />
<br />
<math>\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510</math><br />
<br />
== Solution ==<br />
<br />
Let <math>x</math> be the number of elements in <math>A</math> and <math>B</math> which is equal. <br />
<br />
Then we could form equation <br />
<math>2x-1001 = 2007</math><br />
<br />
<math>2x = 3008</math><br />
<br />
<math>x = 1504</math>. <br />
<br />
The answer is <math>\boxed{\textbf{(C)}\ 1504}</math><br />
<br />
==Video Solution by WhyMath==<br />
https://youtu.be/3LtGb3KjhoU<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=12|num-a=14}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_16&diff=1603822007 AMC 8 Problems/Problem 162021-08-16T16:22:11Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Savannahsolver</p>
<hr />
<div>==Problem==<br />
<br />
Amanda Reckonwith draws five circles with radii <math>1, 2, 3,<br />
4</math> and <math>5</math>. Then for each circle she plots the point <math>(C,A)</math>,<br />
where <math>C</math> is its circumference and <math>A</math> is its area. Which of the<br />
following could be her graph?<br />
<br />
<math> \textbf{(A)} </math><br />
<asy><br />
size(75);<br />
pair A= (1.5,2) ,<br />
B= (3,4) ,<br />
C= (4.5,7) ,<br />
D= (6,11) ,<br />
E= (7.5,16) ;<br />
draw((0,-1)--(0,16));<br />
draw((-1,0)--(16,0));<br />
dot(A^^B^^C^^D^^E);<br />
label("$A$", (0,8), W);<br />
label("$C$", (8,0), S);</asy><br />
<br />
<math> \textbf{(B)} </math><br />
<asy><br />
size(75);<br />
pair A= (1.5,9) ,<br />
B= (3,6) ,<br />
C= (4.5,6) ,<br />
D= (6,9) ,<br />
E= (7.5,15) ;<br />
draw((0,-1)--(0,16));<br />
draw((-1,0)--(16,0));<br />
dot(A^^B^^C^^D^^E);<br />
label("$A$", (0,8), W);<br />
label("$C$", (8,0), S);</asy><br />
<br />
<math> \textbf{(C)} </math><br />
<asy><br />
size(75);<br />
pair A= (1.5,2) ,<br />
B= (3,6) ,<br />
C= (4.5,8) ,<br />
D= (6,6) ,<br />
E= (7.5,2) ;<br />
draw((0,-1)--(0,16));<br />
draw((-1,0)--(16,0));<br />
dot(A^^B^^C^^D^^E);<br />
label("$A$", (0,8), W);<br />
label("$C$", (8,0), S);</asy><br />
<br />
<math> \textbf{(D)} </math><br />
<asy><br />
size(75);<br />
pair A= (1.5,2) ,<br />
B= (3,5) ,<br />
C= (4.5,8) ,<br />
D= (6,11) ,<br />
E= (7.5,14) ;<br />
draw((0,-1)--(0,16));<br />
draw((-1,0)--(16,0));<br />
dot(A^^B^^C^^D^^E);<br />
label("$A$", (0,8), W);<br />
label("$C$", (8,0), S);</asy><br />
<br />
<math> \textbf{(E)} </math><br />
<asy><br />
size(75);<br />
pair A= (1.5,15) ,<br />
B= (3,10) ,<br />
C= (4.5,6) ,<br />
D= (6,3) ,<br />
E= (7.5,1) ;<br />
draw((0,-1)--(0,16));<br />
draw((-1,0)--(16,0));<br />
dot(A^^B^^C^^D^^E);<br />
label("$A$", (0,8), W);<br />
label("$C$", (8,0), S);</asy><br />
<br />
== Solution ==<br />
The circumference of a circle is obtained by simply multiplying the radius by <math>2\pi</math>. So, the C-coordinate (in this case, it is the x-coordinate) will increase at a steady rate. The area, however, is obtained by squaring the radius and multiplying it by <math>\pi</math>. Since squares do not increase in an evenly spaced arithmetic sequence, the increase in the A-coordinates ( aka the y- coordinates) will be much more significant. The answer is <math>\boxed{\textbf{(A)}}, <br />
</math><asy><br />
size(75);<br />
pair A= (1.5,2) ,<br />
B= (3,4) ,<br />
C= (4.5,7) ,<br />
D= (6,11) ,<br />
E= (7.5,16) ;<br />
draw((0,-1)--(0,16));<br />
draw((-1,0)--(16,0));<br />
dot(A^^B^^C^^D^^E);<br />
label("$A$", (0,8), W);<br />
label("$C$", (8,0), S);</asy>.<br />
-RBANDA<br />
<br />
==Video Solution by WhyMath==<br />
https://youtu.be/AW6BhCQ_ig8<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=15|num-a=17}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_14&diff=1603832007 AMC 8 Problems/Problem 142021-08-16T16:22:11Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Savannahsolver</p>
<hr />
<div>== Problem ==<br />
<br />
The base of isosceles <math>\triangle ABC</math> is <math>24</math> and its area is <math>60</math>. What is the length of one<br />
of the congruent sides?<br />
<br />
<math>\mathrm{(A)}\ 5 \qquad \mathrm{(B)}\ 8 \qquad \mathrm{(C)}\ 13 \qquad \mathrm{(D)}\ 14 \qquad \mathrm{(E)}\ 18</math><br />
<br />
== Solution ==<br />
<br />
The area of a triangle is shown by <math>\frac{1}{2}bh</math>. We set the base equal to <math>24</math>, and the area equal to <math>60</math>, and we get the height, or altitude, of the triangle to be <math>5</math>. In this isosceles triangle, the height bisects the base, so by using the Pythagorean Theorem, <math>a^2+b^2=c^2</math>, we can solve for one of the legs of the triangle (it will be the the hypotenuse, <math>c</math>).<br />
<math>a = 12</math>, <math>b = 5</math>,<br />
<math>c = 13</math>.<br />
The answer is <math>\boxed{\textbf{(C)}\ 13}</math><br />
<br />
==Video Solution by WhyMath==<br />
https://youtu.be/9sVdsKcpJ9U<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=13|num-a=15}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_18&diff=1603802007 AMC 8 Problems/Problem 182021-08-16T16:22:10Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Savannahsolver</p>
<hr />
<div>==Problem==<br />
<br />
The product of the two <math>99</math>-digit numbers<br />
<br />
<math>303,030,303,...,030,303</math> and <math>505,050,505,...,050,505</math><br />
<br />
has thousands digit <math>A</math> and units digit <math>B</math>. What is the sum of <math>A</math> and <math>B</math>?<br />
<br />
<math>\mathrm{(A)}\ 3 \qquad \mathrm{(B)}\ 5 \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(D)}\ 8 \qquad \mathrm{(E)}\ 10</math><br />
<br />
==Video Solution==<br />
https://youtu.be/7an5wU9Q5hk?t=2085<br />
<br />
<br />
==Solution==<br />
We can first make a small example to find out <math>A</math> and <math>B</math>. So, <br />
<br />
<math>303\times505=153015 </math><br />
<br />
The ones digit plus thousands digit is <math>5+3=8</math>.<br />
<br />
Note that the ones and thousands digits are, added together, <math>8</math>. (and so on...) So the answer is <math>\boxed{\textbf{(D)}\ 8}</math><br />
This is a direct multlipication way.<br />
<br />
==Video Solution by WhyMath==<br />
https://youtu.be/_goaFuScO6M<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=17|num-a=19}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_17&diff=1603812007 AMC 8 Problems/Problem 172021-08-16T16:22:10Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Savannahsolver</p>
<hr />
<div>==Problem==<br />
<br />
A mixture of <math>30</math> liters of paint is <math>25\%</math> red tint, <math>30\%</math> yellow<br />
tint and <math>45\%</math> water. Five liters of yellow tint are added to<br />
the original mixture. What is the percent of yellow tint<br />
in the new mixture?<br />
<br />
<math>\mathrm{(A)}\ 25 \qquad \mathrm{(B)}\ 35 \qquad \mathrm{(C)}\ 40 \qquad \mathrm{(D)}\ 45 \qquad \mathrm{(E)}\ 50</math><br />
<br />
==Solution==<br />
Since <math>30\%</math> of the original <math>30</math> liters of paint was yellow, and 5 liters of yellow paint were added to make the new mixture, there are <math>9+5=14</math> liters of yellow tint in the new mixture. Since only 5 liters of paint were added to the original 30, there are a total of 35 liters of paint in the new mixture. This gives <math>40\%</math> of yellow tint in the new mixture, which is <math>\boxed{\textbf{(C) 40}}</math>.<br />
<br />
==Video Solution by WhyMath==<br />
https://youtu.be/5MfwBvLHUCw<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=16|num-a=18}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2007_AMC_8_Problems/Problem_19&diff=1603792007 AMC 8 Problems/Problem 192021-08-16T16:22:08Z<p>Jwelsh: Reverted edits by Raina0708 (talk) to last revision by Savannahsolver</p>
<hr />
<div>== Problem ==<br />
<br />
Pick two consecutive positive integers whose sum is less than <math>100</math>. Square both<br />
of those integers and then find the difference of the squares. Which of the<br />
following could be the difference?<br />
<br />
<math>\mathrm{(A)}\ 2 \qquad \mathrm{(B)}\ 64 \qquad \mathrm{(C)}\ 79 \qquad \mathrm{(D)}\ 96 \qquad \mathrm{(E)}\ 131</math><br />
<br />
== Solution ==<br />
<br />
Let the smaller of the two numbers be <math>x</math>. Then, the problem states that <math> (x+1)+x<100</math>. <math> (x+1)^2-x^2=x^2+2x+1-x^2=2x+1 </math>. <math> 2x+1 </math> is obviously odd, so only answer choices C and E need to be considered.<br />
<br />
<math> 2x+1=131 </math> refutes the fact that <math> 2x+1<100 </math>, so the answer is <math> \boxed{\mathrm{(C)} 79} </math><br />
<br />
==Video Solution by WhyMath==<br />
https://youtu.be/BrEqmDq82rw<br />
<br />
~savannahsolver<br />
<br />
==See Also==<br />
{{AMC8 box|year=2007|num-b=18|num-a=20}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12A_Problems/Problem_7&diff=1570012010 AMC 12A Problems/Problem 72021-06-28T22:08:40Z<p>Jwelsh: Reverted edits by Poopbanna (talk) to last revision by Sugar rush</p>
<hr />
<div>== Problem ==<br />
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?<br />
<br />
<math>\textbf{(A)}\ 0.04 \qquad \textbf{(B)}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4</math><br />
<br />
== Solution ==<br />
The water tower holds <math>\frac{100000}{0.1} = 1000000</math> times more water than Logan's miniature. The volume of a sphere is: <math>V=\dfrac{4}{3}\pi r^3</math>. Since we are comparing the heights (m), we should compare the radii (m) to find the ratio. Since, the radius is cubed, Logan should make his tower <math>\sqrt[3]{1000000} = 100</math> times shorter than the actual tower. This is <math>\frac{40}{100} = \boxed{0.4}</math> meters high, or choice <math>\textbf{(C)}</math>.<br />
<br />
Note: The fact that <math>1\text{ L}=1000\text{ cm}^3</math> doesn't matter since only the ratios are important.<br />
<br />
==Video Solution by the Beauty of Math==<br />
https://youtu.be/kU70k1-ONgM?t=154<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=6|num-a=8|ab=A}}<br />
{{AMC10 box|year=2010|num-b=11|num-a=13|ab=A}}<br />
<br />
[[Category:Introductory Geometry Problems]]<br />
[[Category:3D Geometry Problems]]<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2008_AMC_10A&diff=1570002008 AMC 10A2021-06-28T22:08:38Z<p>Jwelsh: Reverted edits by Poopbanna (talk) to last revision by Timneh</p>
<hr />
<div>'''2008 AMC 10A''' problems and solutions. The first link contains the full set of test problems. The second link contains the answer key. The rest contain each individual problem and its solution.<br />
<br />
*[[2008 AMC 10A Problems]]<br />
*[[2008 AMC 10A Answer Key]]<br />
**[[2008 AMC 10A Problems/Problem 1|Problem 1]]<br />
**[[2008 AMC 10A Problems/Problem 2|Problem 2]]<br />
**[[2008 AMC 10A Problems/Problem 3|Problem 3]]<br />
**[[2008 AMC 10A Problems/Problem 4|Problem 4]]<br />
**[[2008 AMC 10A Problems/Problem 5|Problem 5]]<br />
**[[2008 AMC 10A Problems/Problem 6|Problem 6]]<br />
**[[2008 AMC 10A Problems/Problem 7|Problem 7]]<br />
**[[2008 AMC 10A Problems/Problem 8|Problem 8]]<br />
**[[2008 AMC 10A Problems/Problem 9|Problem 9]]<br />
**[[2008 AMC 10A Problems/Problem 10|Problem 10]]<br />
**[[2008 AMC 10A Problems/Problem 11|Problem 11]]<br />
**[[2008 AMC 10A Problems/Problem 12|Problem 12]]<br />
**[[2008 AMC 10A Problems/Problem 13|Problem 13]]<br />
**[[2008 AMC 10A Problems/Problem 14|Problem 14]]<br />
**[[2008 AMC 10A Problems/Problem 15|Problem 15]]<br />
**[[2008 AMC 10A Problems/Problem 16|Problem 16]]<br />
**[[2008 AMC 10A Problems/Problem 17|Problem 17]]<br />
**[[2008 AMC 10A Problems/Problem 18|Problem 18]]<br />
**[[2008 AMC 10A Problems/Problem 19|Problem 19]]<br />
**[[2008 AMC 10A Problems/Problem 20|Problem 20]]<br />
**[[2008 AMC 10A Problems/Problem 21|Problem 21]]<br />
**[[2008 AMC 10A Problems/Problem 22|Problem 22]]<br />
**[[2008 AMC 10A Problems/Problem 23|Problem 23]]<br />
**[[2008 AMC 10A Problems/Problem 24|Problem 24]]<br />
**[[2008 AMC 10A Problems/Problem 25|Problem 25]]<br />
<br />
== See also ==<br />
{{AMC10 box|year=2008|before=[[2007 AMC 10A|2007 AMC 10A]], [[2007 AMC 10B|B]]|after=[[2009 AMC 10A|2009 AMC 10A]], [[2009 AMC 10B|B]]|ab=A}}<br />
<br />
* [[AMC 10]]<br />
* [[AMC 10 Problems and Solutions]]<br />
* [[Mathematics competition resources]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=What_is_the_definition_of_Pure_Mathematics%3F&diff=156999What is the definition of Pure Mathematics?2021-06-28T22:08:37Z<p>Jwelsh: Reverted edits by Poopbanna (talk) to last revision by Marble3</p>
<hr />
<div>This article is not finished. Everyone is welcomed to edit, BUT ONLY IN GOOD WAYS! The AoPS Secret Governemtn has a backup of this page.<br />
<br />
= What is the definition of Pure Mathematics? =<br />
<br />
<br />
''Oh, easy'', you say'' it is just the study of numbers.<br />
<br />
That may be true for some areas of math. However, what about geometry, trigonometry, and calculus? And what is the definition of numbers? Now you go to the dictionary and say ''The relationship between measurements and quantities using numbers and symbols''. This is, however, not fully true because this definition also uses applied mathematics. We want ''pure mathematics''.<br />
<br />
Also, most of these definitions miss one area of math. ''Chaos Theory''. What is Chaos Theory? Chaos Theory is a recently discovered area of math where nothing can be predicted but nothing is random. We are only at the beginning of learning it. For example, can a butterfly that flaps its wings in Brazil trigger a tornado in Texas? <br />
<br />
Some definitions hit almost all the areas of math, but some are too broad and logic, for example, often fits into the definition. <br />
<br />
We can, however, define some areas of math but not the whole thing. For example, the definition of geometry is ''Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs''. Or the definition of probability is ''the extent to which an event is likely to occur''.<br />
<br />
= Arithmetic =<br />
== Definition ==<br />
''The branch of mathematics dealing with the properties and manipulation of constants''(a number that can not changed. This is the opposite of a variable which is most of the tie represented by a letter. This letter is most of the time <math>x</math>,<math>y</math> or <math>z</math> however it can be really any non-used symbol.).<br />
<br />
=== Operations ===<br />
Arithmetic starts with one thing which without it no arithmetic can survive: Counting Positive Integers (whole numbers).<br />
1,2,3,4,5... <br />
==== Addition and addition repeated ====<br />
Addition is combining these integers. The symbol for combining numbers is +. <math>a+b=b+a</math><br />
<br />
Multiplication is repeating addition. The symbol for this is <math>\cdot</math>. <math>a \cdot b=b \cdot a</math>. Note: When using letters, you can just say <math>ab=ba</math>. <br />
<br />
Exponentiation is repeated Multiplication. The symbol for a to the exponent of b is <math>a^b</math>. <math>a^b \neq b^a</math>.<br />
The <math>\neq</math> symbol means Not Equal.<br />
<br />
==== Inverse ====<br />
Subtraction is the inverse (remember that inverse means oppisite) of addition. To make this a well-defined function (the symbol for a funtion of a certian number(<math>x</math>) is <math>f(x)=</math>blah blah blah where a function is a relationship or expression involving one or more variables. For example, if <math>f(x)=x+3</math> then <br />
<math>f(3)=6</math>), negative numbers and zero were defined. A negitive number is a number below zero.<br />
<br />
The division is the inverse of multiplication. To make this a well-defined function everywhere, fractions were defined. 3 divided by 2 is <math>\frac{3}{2}</math>.<br />
<br />
<math>n^{\text{th}}</math> roots and logarithms are the inverses of exponentiation. To make these well-defined functions everywhere, irrational numbers were defined. An irrational number is a number that can not be expressed as <math>\frac{a}{b}</math>.<br />
<br />
==== Negative numbers ====<br />
<br />
<math>a</math> and <math>b</math> are positive. (Above zero)<br />
<br />
1. <math>(-a)(-b)=ab</math> Note: <br />
<br />
2. <math>(a)(-b)=-ab</math><br />
<br />
Proof for 1: This is, in fact, the reason why the negative numbers were introduced: so that each positive number would have an additive inverse. ... The fact that the product of two negatives is positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.<br />
<br />
Proof for 2: Since <math>ab</math> is repeated addition then <math>(a)(-b)</math> is repeated subtraction. Therefore it is negative.<br />
<br />
==== Fractions ====<br />
A fraction a number that can be expressed as two numbers divided. For example, five divided by four is <math>\frac{5}{4}</math>. <br />
===== Simplifing fractions =====<br />
Find common factors in each half of the fraction and then divide top and bottom of the fraction by that factor.<br />
<br />
<math>\frac{10}{5}=\frac{1}{2}</math><br />
<br />
<math>\frac{15}{3}=\frac{5}{1}</math><br />
<br />
<math>\frac{28}{2}=\frac{14}{1}</math><br />
===== Adding fractions =====<br />
Here is how you add fractions. Is they have the same bottom half then <math>\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}</math>. However, if the two or three or n fractions do not have the same bottom half you make them. <math>\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}</math>. Subtracting fractions is the same except everything has a minus symbol. Remember to fully simplify.<br />
<br />
===== Multiplying fractions =====<br />
<math>\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}</math>. <br />
<br />
<math>\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ac}{bd}</math>.<br />
<br />
Remember to fully simplify.<br />
<br />
==== Irrational numbers ====<br />
Most square roots are irrational. An irrational number is a number that can not be expresed a a fraction.<br />
===== Proof that any nonperfect square positive integer is irrational =====<br />
Let us assume that <math>\sqrt{n}</math> is rational where <math>n</math> is a nonperfect square positive integer. Then it can be written as <math>\frac{p}{q}=\sqrt{n} \Rightarrow \frac{p^2}{q^2}=n \Rightarrow (q^2)n=p^2</math>. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore <math>\sqrt{n}</math> is irrational.<br />
<br />
===== pi and e =====<br />
Some irrational numbers are limits. That means they are the sum of smaller and smaller fraction going infinitely long. Pi or <math>\pi</math> (go to the geometry part of this article) is the limit <math>\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}...=\frac{\pi}{4}</math>. e is the limit <math>\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}...</math> More at the counting part of the article.<br />
<br />
=== Exponent rules ===<br />
Listed below are some important properties of exponents:<br />
# <math> b^x\cdot b^y = b^{x+y}</math><br />
# <math> b^{-x}=\frac 1{b^x} </math><br />
# <math> \frac{b^x}{b^y}=b^{x-y} </math><br />
# <math> (b^x)^y = b^{xy} </math><br />
# <math> (ab)^x = a^x b^x </math><br />
# <math> b^0 = 1 </math> (if <math>b \neq 0</math>. <math>0^0</math> is undefined.)<br />
<br />
Here are explanations of the properties listed above:<br />
# On both sides, we are multiplying '''b''' together '''x+y''' times. Thus, they are equivalent. <br />
# This is described in the previous section.<br />
# This results from using the previous two properties.<br />
# We are multiplying <math>b^x</math> by itself '''y''' times, which is the same as multiplying '''b''' by itself '''xy''' times.<br />
# After multiplying '''ab''' by itself '''x''' times, we can collect '''a''' and '''b''' terms, thus establishing the property.<br />
#Hoping that property #1 will be true when <math>y=0</math>, we see that <math>b^x\cdot b^0</math> should (hopefully) be equal to <math>b^x</math>. Thus, we ''define'' <math>b^0</math> to be equal to <math>1</math> in order to make this be true.<br />
<br />
hello<br />
<br />
= Algebra =<br />
== Definition ==<br />
''The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.''<br />
== One-variable linear equations ==<br />
=== Definition ===<br />
'A One-variable linear equation is an equation that comes in the form <math>ax+b=c</math>. <math>a</math>, <math>b</math>, and <math>c</math> are constants and <math>x</math> is the varible in which to solve for<br />
=== Problems ===<br />
==== Problem 1 ====<br />
Solve for <math>x</math>: <math>x+3=7</math><br />
==== Solution 1 ====<br />
Subtract 3 from both sides you get <math>x=4</math><br />
==== Problem 2 ====<br />
Solve for <math>x</math>: <math>3x+2=98</math><br />
==== Solution 2 ====<br />
Subtract two: <math>3x=96</math>. Divide by three: <math>x=32</math><br />
==== Problem 3 ====<br />
Solve for <math>x</math>: <math>\sqrt{x+3}=5</math><br />
==== Solution 3 ====<br />
When dealing with square roots you just square roots you square them because that makes no more square roots. So <math>\sqrt{x+3}=5</math> squared is <math>x+3=25</math> which tells us that <math>x</math> is <math>22</math>.<br />
==== Problem 4 ====<br />
Solve for <math>x</math>: <math>\frac{4}{x}=8</math><br />
==== Solution 4 ====<br />
When dealing with fractions you multiply by the denaminator: <math>4=8x</math> which tells us that <math>x=\frac{1}{2}</math>.<br />
<br />
== Quadratics ==<br />
=== Defination ===<br />
''A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The standard form is <math>ax^2 + bx + c = 0</math> with <math>a</math>, <math>b</math>, and <math>c</math> being constants, or numerical coefficients, and <math>x</math> is an unknown variable.''<br />
<br />
=== The answer is always... ===<br />
<math>ax^2+bx+c=0</math><br />
<br />
<math>x^2+\frac{b}{a}x+\frac{c}{a}=0</math><br />
<br />
<math>x^2+\frac{b}{a}=0-\frac{c}{a}</math><br />
<br />
<math>x^2+\frac{b}{a}+(\frac{b}{2a})^2=0-\frac{c}{a}+(\frac{b}{2a})^2</math><br />
<br />
<br />
Since the left side of the equation right above is a perfect square, you can factor the left side by using the coefficient of the first term (<math>x</math>) and the base of the last term (<math>\frac{b}{2a}</math>). Add these two and raise everything to the second.<br />
<br />
<math>(x+\frac{b}{2a})^2=0-\frac{c}{a}+\frac{b^2}{2a^2}=\frac{b^2-4ac}{4a^2}</math><br />
<br />
<math>x+\frac{b}{2a}=\frac{\sqrt{\pm b^2-4ac}}{\pm 2a}</math><br />
<br />
<math>x=\frac{\sqrt{\pm b^2-4ac}}{\pm 2a}+\frac{b}{2a}</math><br />
<br />
When simplified the ''Quadratic Formula'' is <math>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}</math><br />
<br />
<br />
===== Problem =====<br />
Find all <math>x</math> if <math>x^2+x+1=0</math><br />
===== Solution =====<br />
Using our formula we see that <math>x=\frac{-1 \pm \sqrt{1^2-1(1)(1)}}{2(1)}=-\frac{1}{2}</math><br />
<br />
== i ==<br />
<math>i=\sqrt{-1}</math>.<br />
<br />
<math>xi=\sqrt{-x^2}</math>.<br />
<br />
Numbers like this are called ''imaginary numbers''. ''Impossible'', you say. But no. Solve <math>x^2+1=0</math>. You get <math>i</math>. <math>oi</math> is <math>o</math>. So zero is both real and imaginary. (real means ''not imaginary'')<br />
<br />
=== Powers of i ===<br />
<br />
<math>i^0=1</math><br />
<math>i^1=i</math><br />
<math>i^2=-1</math><br />
<math>i^3=-i</math><br />
<math>i^4=1</math><br />
<math>i^5=i</math><br />
<math>i^6=-1</math><br />
<math>i^7=-i</math><br />
<br />
The pattern repeats. <br />
<br />
=== Complex numbers ===<br />
A complex number is <math>ai+b</math>, where a and b are real. All numbers are complex because a and/or/never b can be zero.<br />
<br />
<math>ai+b+ci+d=</math> Complex<br />
<br />
<math>(ai+b)(ci+d)=</math> Complex<br />
<br />
<math>\frac{ai+b}{ci+d}=</math> Complex<br />
<br />
<math>(ai+b)-(ci+d)=</math> Complex<br />
<br />
== Systems of equations ==<br />
A '''system of equations''' is a set of [[equation]]s which share the same [[variable]]s. An example of a system of equations is<br />
<br />
{| class="wikitable" style="margin: 1em auto 1em auto"<br />
| <math>2a - 3b</math> || <math>= 4</math><br />
|-<br />
| <math>3a - 2b</math> || <math>= 3</math><br />
|}<br />
<br />
=== Solving Linear Systems ===<br />
A system of [[linear]] equations is where all of the variables are to the power 1. There are three elementary ways to solve a system of linear equations.<br />
<br />
==== Gaussian Elimination ====<br />
[[Gauss]]ian elimination involves eliminating variables from the system by adding constant multiples of two or more of the equations together. Let's look at an example:<br />
<br />
===== Problem =====<br />
Find the ordered pair <math>(x,y)</math> for which<br />
<br />
{| class="wikitable" style="margin: 1em auto 1em auto"<br />
| <math>x - 12y</math> || <math>= 2</math><br />
|-<br />
| <math>3x + 6y</math> || <math>= 6</math><br />
|}<br />
<br />
===== Solution =====<br />
We can eliminate <math>y</math> by adding twice the second equation to the first:<br />
<br />
{| class="wikitable" style="margin: 1em auto 1em auto"<br />
| || <math>x - 12y= 2</math><br />
|-<br />
| <math>+2(</math> || <math>3x + 6y = 6)</math><br />
|-<br />
| || <math>\overline{7x + 0=14}</math><br />
|}<br />
<br />
Thus <math>x=2</math>. We can then plug in for <math>x</math> in either of the equations: <center><math> (2)-12y = 2 \Rightarrow y = 0</math>.</center><br />
<br />
Thus, the solution to the system is <math>(2,0)</math>.<br />
<br />
==== Substitution ====<br />
The second method, substitution, requires solving for a variable and then plugging that variable into another equation therefore reducing the number of variables. We'll show how to solve the same problem from the elimination section using substitution.<br />
<br />
===== Problem =====<br />
Find the ordered pair <math>(x,y)</math> for which<br />
<br />
{| class="wikitable" style="margin: 1em auto 1em auto"<br />
| <math>x - 12y</math> || <math>= 2</math><br />
|-<br />
| <math>3x + 6y</math> || <math>= 6</math><br />
|}<br />
<br />
===== Solution =====<br />
The first equation can be solved for <math>x</math>: <center><math> x = 12y + 2. </math></center><br />
<br />
Plugging this into the second equation yields <center><math>3(12y + 2) + 6y = 6 \Leftrightarrow 42 y = 0. </math></center><br />
<br />
Thus <math>y=0</math>. Plugging this into either of the equations and solving for <math>x</math> yields <math>x=2</math>.<br />
<br />
== Graphing Equations ==<br />
== Graphing Lines ==<br />
Given two distinct points on a line, one can construct the whole line. So one way to graph a line given its equation is to just find two points on it and to draw a straight line through them.<br />
<br />
=== Problem ===<br />
Graph the line <math>2x + 3y = 24</math>.<br />
<br />
=== Solution ===<br />
To graph a line, it is necesasry to find two points <math>(x,y)</math> that satisfy <math>2x + 3y = 24</math>. Letting <math>x=0</math> gives <math>3y = 24\Leftrightarrow y = 8</math>. So <math>(0,8)</math> is one point on the graph.<br />
<br />
Find another point by letting <math>y=0</math>. Plugging this in and solving gives <math>x=12</math>. So <math>(12,0)</math> is our other point.<br />
<br />
Now plot these in the coordinate plane and draw a line through them:<br />
<br />
<center>[[Image:Twopoints2.PNG]]</center><br />
<br />
The arrowheads on the ends of the line segment indicate that the line goes on [[infinite]]ly in both directions.<br />
Note: This is just a short intro to graphing. More at https://artofproblemsolving.com/wiki/index.php/Graph_of_a_function.<br />
== Algebra ==<br />
There are many more types of algebra: inequalities, polynomials, graphing equations, arithmetic, and geometric sequence.<br />
<br />
Algebra is a broad and diverse area of math in which this is just a short introduction.<br />
<br />
= Number Theory =<br />
== Definition ==<br />
''The branch of mathematics that deals with the properties and relationships of numbers, especially the positive integers''.<br />
== Vocab ==<br />
Factor: A factor (f) of n is an integer in which <math>fx=n</math> where <math>x</math> is an integer.<br />
<br />
Multiple: The inverse of factors.<br />
<br />
== Primes ==<br />
A '''prime number''' (or simply '''prime''') is a positive integer <math>p>1</math> whose only positive divisors are 1 and itself. <br />
Note that <math>1</math> is usually defined as being neither prime nor composite because it is its only factor among the natural number numbers. <br />
<br />
There are an infinite number of prime numbers. A standard proof attributed to Euclid notes that if there are a finite set of prime numbers <math>p_1, p_2, \ldots, p_n</math>, then the number <math>N = p_1p_2\cdots p_n + 1</math> is not divisible by any of them, but <math>N</math> must [[#Importance of Primes|have]] a prime factor, which leads to a direct contradiction. <br />
<br />
=== Techniques to Check for Prime Numbers ===<br />
<br />
==== Divisibility ====<br />
A prime number is only divisible by one or itself, so a number <math>n</math> is prime if and only if <math>n</math> is not divisible by any integer greater than <math>1</math> and less than <math>n</math>. One only needs to check integers up to <math>\sqrt{n}</math> because dividing larger numbers would result in a quotient smaller than <math>\sqrt{n}</math>.<br />
<br />
==== Modular Arithmetic ====<br />
Modular arithmetic can help determine if a number is not prime.<br />
<br />
* If a number not equal to <math>2,3</math> is congruent to <math>0,2,3,4,6 \pmod{6}</math>, then the number is not prime.<br />
* If a number not equal to <math>2,5</math> ends with an even digit or <math>5</math>, then the number is not prime.<br />
<br />
<br />
=== Importance of Primes ===<br />
According to the Fundamental Theorem of Arithmetic, there is exactly one unique way to factor a positive integer into a product of primes. This unique prime factorization plays an important role in solving many kinds of number theory problems.<br />
<br />
<br />
<br />
==== Mersenne Primes ====<br />
<br />
A Mersenne prime is a prime of the form <math>2^n-1</math>. For such a number to be prime, ''n'' must itself be prime. Compared to other numbers of comparable sizes, Mersenne numbers are easy to check for primality because of the https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test Lucas-Lehmer test, severely reducing the amount of computation needed.<br />
<br />
==== Twin Primes ====<br />
Two primes that differ by exactly 2 are known as twin primes. The following are the first few pairs of twin primes:<br><br />
3, 5<br><br />
5, 7<br><br />
11, 13<br><br />
17, 19<br><br />
29, 31<br><br />
41, 43<br><br />
<br />
It is not known whether or not there are infinitely many pairs of twin primes. This is known as the Twin Prime Conjecture, which is a specific instance of the Hardy-Littlewood conjecture.<br />
<br />
==== Gaussian Primes ====<br />
<br />
A Gaussian prime is a prime that extends the idea of the traditional prime to the Gaussian integer. One can define this term for any ring, especially number rings.<br />
<br />
=== Advanced Definition ===<br />
When the need arises to include negative divisors, a '''prime''' is defined as an integer p whose only divisors are 1, -1, p, and -p. More generally, if ''R'' is an integral domain, then a nonzero element ''p'' of ''R'' is a '''prime''' if whenever we write <math>p=ab</math> with <math>a,b\in R</math>, then exactly one of ''a'' and ''b'' is a unit.<br />
<br />
== LCM and GCD ==<br />
=== GCD ===<br />
==== Definition ====<br />
The GCD or Greatest Common Divisor of multiple numbers is the largest number that is a factor of all those multiple numbers. The answer can be and usually is 1.<br />
==== Prime Factorization ====<br />
The way to solve most GCD's is with using ''Prime Factorization''. Remember that the way to find the prime factorization of a number is to find which primes products hit that certain number. The prime factorization of 24 is <math>2 \cdot 2 \cdot 2 \cdot 3</math>. The prime factorization of 18 is <math>2 \cdot 3 \cdot 3</math>. In GCD you take the prime factorization of each number and then find which primes match up.<br />
<br />
=== LCM ===<br />
==== Definition ====<br />
The LCM or Least Common Multiple of two or three or n numbers is the smallest integer in which both numbers are factors of it.<br />
For example, the LCM of 10 and 5 is 10. <br />
==== Problems ====<br />
===== Problem 1 =====<br />
====== Problem ======<br />
What is the LCM of 8 and 5?<br />
====== Solution ======<br />
1.<br />
The way to solve LCM's with no common factors is to multiply them. The answer is just 40.<br />
<br />
2.<br />
Note that <math>8=2^3,</math> <math>5=5^1.</math> Taking the maximum of all such prime factors yields <math>40=2^3\cdot 5.</math><br />
<br />
This is a pathetic solution, but this is the more general solution.<br />
<br />
===== Problem 2 =====<br />
====== Problem ======<br />
What is the LCM of 15 and 10.<br />
====== Solution ======<br />
The Way to solve this is see that the answer would be the first multiple of 5 that is divisble by 15 and 10. That is just 30.<br />
===== Problem 3 =====<br />
====== Problem ======<br />
Find the LCM of 2, 3 and 5.<br />
====== Solution ======<br />
It is the same with three numbers. Multiply them to get 30. Same as last time.<br />
<br />
== Bases ==<br />
In mathematics, a base or radix is the number of different digits or a combination of digits and letters that a system of counting uses to represent numbers. For example, the most common base used today is the decimal system. Because "dec" means 10, it uses the 10 digits from 0 to 9. Most people think that we most often use base 10 because we have 10 fingers.<br />
==== Some Numbers in other bases ====<br />
A base can be any whole number bigger than 1. The base of a number may be written next to the number: for instance, <math>{23_{8}}</math> means 23 in base 8 (which is equal to 19 in base 10).<br />
Here are some examples of how numbers are written in varying bases, compared to decimal:<br />
<br />
In Arabic numbers (decimal, or base 10), there are 10 digits: 0,1,2,3,4,5,6,7,8,9. You need one digit each to count up to 9, but two digits for ten, and three digits for a hundred, which is ten times ten. In Binary, base 2, you need two digits for two, as you only have two digits, 0 and 1. Base 5 has five digits, and the number five becomes 10. For base 16, you will need sixteen digits, and there are only ten numerals. So we use the letters A,B,C,D,E,F. These represent the decimal numbers 10, 11, 12, 13, 14 and 15. Look at the table below and find the pattern for these bases.<br />
== Number Theory ==<br />
This is the end of the number theory part of this article. Number Theory, however, has much more to than what I have just shown you.<br />
= Probibility =<br />
== Definition ==<br />
''The extent to which an event is likely to occur.''<br />
== How to ==<br />
The way to mesure the extent to which an event is likely to occur is to count the number of equally likely occur is to count the wanted options over the number of equally likelly possible options.<br />
For example, the probibility of rolling a 6 on a 6 sided dice is <math>\frac{1}{6}</math> because there are six equally likely possibilitys and only one of them is a success.<br />
<br />
<br />
<br />
= Pascal's Triangle =<br />
== Pascal's Identity ==<br />
=== Identity ===<br />
Pascal's Identity states that<br />
<br />
<math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math><br />
<br />
for any positive integers <math>k</math> and <math>n</math>. Here, <math>\binom{n}{k}</math> is the binomial coefficient <math>\binom{n}{k} = nCk = C_k^n</math>.<br />
<br />
Remember that <math>\binom{n}{r}=\frac{n!}{k!(n-k)!}</math>. <math>\binom{n}{r}</math> means the number of ways to pick r thing from n things where order does not matter. <math>n!=1 \cdot 2 \cdot 3 \cdot...\cdot n</math>.<br />
<br />
=== Proving it ===<br />
If <math>k > n</math> then <math>\binom{n}{k} = 0 = \binom{n - 1}{k - 1} + \binom{n - 1}{k}</math> and so the result is pretty clear.<br />
So assume <math>k \leq n</math>. Then<br />
<br />
<cmath>\begin{eqnarray*}\binom{n-1}{k-1}+\binom{n-1}{k}&=&\frac{(n-1)!}{(k-1)!(n-k)!}+\frac{(n-1)!}{k!(n-k-1)!}\\<br />
&=&(n-1)!\left(\frac{k}{k!(n-k)!}+\frac{n-k}{k!(n-k)!}\right)\\<br />
&=&(n-1)!\cdot \frac{n}{k!(n-k)!}\\<br />
&=&\frac{n!}{k!(n-k)!}\\<br />
&=&\binom{n}{k}. \qquad\qquad\square\end{eqnarray*}</cmath><br />
There we go. We proved it!<br />
<br />
=== Why is it needed? ===<br />
It's mostly just a cool thing to know. However, if you want to know how to use it in real life go to https://artofproblemsolving.com/videos/counting/chapter12/141. Or really any of the counting and probability videos.<br />
<br />
== Introduction to Pascal's Triangle ==<br />
=== How to build it ===<br />
Pascal's Triangle is a triangular array of numbers in which you start with two infinite diagonals of ones and each of the rest of the numbers is the sum of the two numbers above it. It looks something like this:<br />
1<br />
1 1<br />
1 2 1<br />
1 3 3 1<br />
1 4 6 4 1<br />
And on and on...<br />
<br />
=== Combinations ===<br />
==== Combinations ====<br />
Pascal's Triangle is really combinations. It looks something like this if it is depicted as combinations:<br />
<br />
<math>\binom{0}{0}</math> <br />
<math>\binom{1}{0}</math> <math>\binom{1}{1}</math><br />
<math>\binom{2}{0}</math> <math>\binom{2}{1}</math> <math>\binom{2}{2}</math><br />
<br />
And on and on...<br />
<br />
==== Proof ====<br />
<br />
If you look at the way we build the triangle, each number is the sum of the two numbers above it. Assuming that these combinations are true then each combination in the sum of the two combinations above it. In an equation, it would look something like this: <math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math>. Its Pascals Identity! Therefore each row looks something like this:<br />
<br />
<math>\binom{n}{0} \binom{n}{1} \binom{n}{2} ... \binom{n}{n}</math><br />
<br />
== Patterns and Properties ==<br />
In addition to combinations, ''Pascal's Triangle'' has many more patterns and properties. See below. Be ready to be amazed. <br />
=== Binomial Theorem ===<br />
Let's multiply out some binomials. Try it yourself and it will not be fun:<br />
<math>(x+y)^0=1</math><br />
<br />
<math>(x+y)^1=1x+1y</math><br />
<br />
<math>(x+y)^2=1x^2+2xy+1y^2</math><br />
<br />
<math>(x+y)^2=1x^3+3x^2y+3y^2x+1^3</math><br />
<br />
If you take away the x's and y's you get:<br />
<br />
1<br />
1 1<br />
1 2 1<br />
1 3 3 1 <br />
It's ''Pascal's Triangle''!<br />
<br />
===== Proof =====<br />
Here are a number of different ways to prove the Binomial Theorem, for example by a straightforward application of mathematical induction. The Binomial Theorem also has a nice combinatorial proof:<br />
<br />
We can write <math>(a+b)^n=\underbrace{ (a+b)\cdot(a+b)\cdot(a+b)\cdot\cdots\cdot(a+b) }_{n}</math>. Repeatedly using the distributive property, we see that for a term <math>a^m b^{n-m}</math>, we must choose <math>m</math> of the <math>n</math> terms to contribute an <math>a</math> to the term, and then each of the other <math>n-m</math> terms of the product must contribute a <math>b</math>. Thus, the coefficient of <math>a^m b^{n-m}</math> is the number of ways to choose <math>m</math> objects from a set of size <math>n</math>, or <math>\binom{n}{m}</math>. Extending this to all possible values of <math>m</math> from <math>0</math> to <math>n</math>, we see that <math>(a+b)^n = \sum_{m=0}^{n}{\binom{n}{m}}\cdot a^m\cdot b^{n-m}</math>, as claimed.<br />
<br />
Similarly, the coefficients of <math>(x+y)^n</math> will be the entries of the <math>n^\text{th}</math> row of Pascal's Triangle. This is explained further in the Counting and Probability textbook [AoPS]<br />
===== In real life =====<br />
It is really only used for multipling out binomials. More usage at <br />
https://artofproblemsolving.com/videos/counting/chapter14/126.<br />
==== Powers of 2 ====<br />
===== Theorem =====<br />
====== Theorem ======<br />
It states that <math>\binom{n}{0}+\binom{n}{1}+...+\binom{n}{n}=2^n</math><br />
<br />
====== Why do we need it? ======<br />
It is useful in many word problems (That means, yes, you can use it in real life) and it is just a cool thing to know. More at https://artofproblemsolving.com/videos/mathcounts/mc2010/419.<br />
===== Proofs =====<br />
====== Subset proof ======<br />
Say you have a word with n letters. How many subsets does it have in terms of n? Here is how you answer it: You ask the first letter Are you in or are you out? Same to the second letter. Same to the third. Same to the n. Each of the letters has two choices: In and Out. The would be <math>(2)(2)(2)(2)</math>...n times. <math>2^n</math>.<br />
<br />
====== Alternate proof ======<br />
If you look at the way we built the triangle you see that each number is row n-1 is added on twice in row n. This means that each row doubles. That means you get powers of two.<br />
<br />
==== Triangle Numbers ====<br />
===== Theorem =====<br />
If you look at the numbers in the third diagonal you see that they are triangle numbers.<br />
===== Proof =====<br />
Now we can make an equation: <math>\binom{n}{2}=1+2+3+...+(n-1) \Rightarrow \binom{n}{2}=\frac{n(n+1)}{2} \Rightarrow \frac{n!}{2!(n-2)!}=\frac{n(n+1)}{2} \Rightarrow \frac{n(n+1)}{2}=\frac{n(n+1)}{2}</math><br />
<br />
==== Hockey stick ====<br />
For <math>n,r\in\mathbb{N}, n>r,\sum^n_{i=r}{i\choose r}={n+1\choose r+1}</math>.<br />
<br />
<br />
<asy><br />
int chew(int n,int r){<br />
int res=1;<br />
for(int i=0;i<r;++i){<br />
res=quotient(res*(n-i),i+1);<br />
}<br />
return res;<br />
}<br />
for(int n=0;n<9;++n){<br />
for(int i=0;i<=n;++i){<br />
if((i==2 && n<8)||(i==3 && n==8)){<br />
if(n==8){label(string(chew(n,i)),(11+n/2-i,-n),p=red+2.5);}<br />
else{label(string(chew(n,i)),(11+n/2-i,-n),p=blue+2);}<br />
}<br />
else{<br />
label(string(chew(n,i)),(11+n/2-i,-n));<br />
}<br />
}<br />
}<br />
</asy><br />
<br />
This identity is known as the ''hockey-stick'' identity because, on Pascal's triangle, when the addends represented in the summation and the sum itself is highlighted, a hockey-stick shape is revealed.<br />
<br />
<br />
===== Proof =====<br />
<br />
'''Inductive Proof'''<br />
<br />
This identity can be proven by induction on <math>n</math>.<br />
<br />
<u>Base Case</u><br />
Let <math>n=r</math>.<br />
<br />
<math>\sum^n_{i=r}{i\choose r}=\sum^r_{i=r}{i\choose r}={r\choose r}=1={r+1\choose r+1}</math>.<br />
<br />
<u>Inductive Step</u><br />
Suppose, for some <math>k\in\mathbb{N}, k>r</math>, <math>\sum^k_{i=r}{i\choose r}={k+1\choose r+1}</math>.<br />
Then <math>\sum^{k+1}_{i=r}{i\choose r}=\left(\sum^k_{i=r}{i\choose r}\right)+{k+1\choose r}={k+1\choose r+1}+{k+1\choose r}={k+2\choose r+1}</math>.<br />
<br />
'''Algebraic Proof'''<br />
<br />
It can also be proven algebraically with [[Pascal's Identity]], <math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math>.<br />
Note that<br />
<br />
<math>{r \choose r}+{r+1 \choose r}+{r+2 \choose r}+\cdots+{r+a \choose r}</math><br />
<math>={r+1 \choose r+1}+{r+1 \choose r}+{r+2 \choose r}+\cdots+{r+a \choose r}</math><br />
<math>={r+2 \choose r+1}+{r+2 \choose r}+\cdots+{r+a \choose r}=\cdots={r+a \choose r+1}+{r+a \choose r}={r+a+1 \choose r+1}</math>, which is equivalent to the desired result.<br />
<br />
'''Combinatorial Proof 1'''<br />
<br />
Imagine that we are distributing <math>n</math> indistinguishable candies to <math>k</math> distinguishable children. By a direct application of Balls and Urns, there are <math>{n+k-1\choose k-1}</math> ways to do this. Alternatively, we can first give <math>0\le i\le n</math> candies to the oldest child so that we are essentially giving <math>n-i</math> candies to <math>k-1</math> kids and again, with Balls and Urns, <math>{n+k-1\choose k-1}=\sum_{i=0}^n{n+k-2-i\choose k-2}</math>, which simplifies to the desired result.<br />
<br />
'''Combinatorial Proof 2'''<br />
<br />
We can form a committee of size <math>k+1</math> from a group of <math>n+1</math> people in <math>{{n+1}\choose{k+1}}</math> ways. Now we hand out the numbers <math>1,2,3,\dots,n-k+1</math> to <math>n-k+1</math> of the <math>n+1</math> people. We can divide this into <math>n-k+1</math> disjoint cases. In general, in case <math>x</math>, <math>1\le x\le n-k+1</math>, person <math>x</math> is on the committee and persons <math>1,2,3,\dots, x-1</math> are not on the committee. This can be done in <math>\binom{n-x+1}{k}</math> ways. Now we can sum the values of these <math>n-k+1</math> disjoint cases, getting <cmath>{{n+1}\choose {k+1}} ={{n}\choose{k}}+{{n-1}\choose{k}}+{{n-2}\choose{k}}+\hdots+{{k+1}\choose{k}}+{{k}\choose{k}}.</cmath><br />
<br />
= Geometry =<br />
== Studying the Geometry of Shapes ==<br />
=== Zero-D ===<br />
This is an infinitly small dot called a point. (Duh). However, mathematicians imagine them to be large enough to be drawn.<br />
<br />
=== One-D ===<br />
1D is pretty boring too. You just have rays, lines, and line segments.<br />
==== Line ====<br />
A infinitly long, infinitly narrow, perfectly striaght thing in with no starting or ending point. (I cant really use the word lines).<br />
==== Ray ====<br />
A infinitly long, infinitly narrow, perfectly striaght thing in with one starting point.<br />
==== Line segment ====<br />
A infinitly long, infinitly narrow, perfectly striaght thing in with two starting points.<br />
<br />
=== Space ===<br />
=== Plane ===<br />
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.<br />
==== 3D Space ====<br />
Space is the infinitly three-dimensional extent in which objects and events have relative position and direction.<br />
<br />
=== Angles ===<br />
An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Most people mesure angles useing degrees. Degrees are out of 360. For example, 180° or 180 degrees is a straight line or a half turn.<br />
<br />
An angle equal to 0° is called a zero angle.<br />
<br />
An angle smaller than 90° but not zero is called acute.<br />
<br />
An angle equal to 90° is called right.<br />
<br />
An angle greater than 90° but less than 180° is called obtuse.<br />
<br />
An angle equal to 180° is called striaght. <br />
<br />
An angle greater than 180° but less than 360° is called obtuse.<br />
<br />
An angle equal to 360° is called a full angle.<br />
<br />
Angles <math>n</math> and <math>m</math> supplementary if <math>n+m=</math>180°<br />
Angles <math>n</math> and <math>m</math> Complementary if <math>n+m=</math>90°<br />
<br />
Vertical angles: Each of the pairs of opposite angles made by two intersecting lines.<br />
<br />
== Area ==<br />
=== Area ===<br />
''The area of a 2D object is the amount of square units in it.''<br />
=== Shapes ===<br />
==== Polygons ====<br />
<br />
A plane shape (two-dimensional) with straight sides.<br />
<br />
Examples: triangles, rectangles and pentagons.<br />
<br />
(Note: a circle is not a polygon because it has a curved side).<br />
==== Triangles and their classification ====<br />
<br />
===== Angle Classifacation =====<br />
Acute Triangle <br />
All angles are less than 90°<br />
<br />
Right Triangle <br />
Has a right angle (90°)<br />
<br />
Obtuse Triangle <br />
Has an angle more than 90°<br />
<br />
===== Side Classifacation =====<br />
Equilateral Triangle <br />
Three equal sides <br />
<br />
Isosceles Triangle <br />
Two equal sides <br />
Two equal angles<br />
<br />
Scalene Triangle <br />
No equal sides <br />
No equal angles<br />
<br />
==== Quadrilaterals and their classification ====<br />
https://www.google.com/search?q=classification+of+quadrilaterals&rlz=1C1JZAP_enUS854US854&tbm=isch&source=iu&ictx=1&fir=cTz-Mz0bfMvP3M%253A%252Cr9XExtRx3Y9j7M%252C_&vet=1&usg=AI4_-kR3H3HLnjvGfeFONyn0jA2dKvnQmQ&sa=X&ved=2ahUKEwi_h7_h7f_iAhXUvp4KHbYkCrAQ9QEwAHoECAEQAw#imgrc=vAqIbgZXY5hQqM:&vet=1<br />
= The Proof Part of this article =<br />
This part shows what Proofs usually look like<br />
== Proof that any nonperfect square positive integer is irrational ==<br />
Let us assume that <math>\sqrt{n}</math> is rational where <math>n</math> is a nonperfect square positive integer. Then it can be written as <math>\frac{p}{q}=\sqrt{n} \Rightarrow \frac{p^2}{q^2}=n \Rightarrow (q^2)n=p^2</math>. But no perfect square times a nonperfect square positive integer is a perfect square. Therefore <math>\sqrt{n}</math> is irrational.<br />
== Proof of the Pythagorean Theorem ==<br />
<math>ABCD</math> and <math>EFGH</math> are squares.<br />
<center><br />
<asy><br />
pair A, B,C,D;<br />
A = (-10,10);<br />
B = (10,10);<br />
C = (10,-10);<br />
D = (-10,-10);<br />
<br />
pair E,F,G,H;<br />
E = (7,10);<br />
F = (10, -7);<br />
G = (-7, -10);<br />
H = (-10, 7);<br />
<br />
draw(A--B--C--D--cycle);<br />
label("$A$", A, NNW);<br />
label("$B$", B, ENE);<br />
label("$C$", C, ESE);<br />
label("$D$", D, SSW);<br />
<br />
draw(E--F--G--H--cycle);<br />
label("$E$", E, N);<br />
label("$F$", F,SE);<br />
label("$G$", G, S);<br />
label("$H$", H, W);<br />
<br />
label("a", A--B,N);<br />
label("a", B--F,SE);<br />
label("a", C--G,S);<br />
label("a", H--D,W);<br />
label("b", E--B,N);<br />
label("b", F--C,SE);<br />
label("b", G--D,S);<br />
label("b", A--H,W);<br />
label("c", E--H,NW);<br />
label("c", E--F);<br />
label("c", F--G,SE);<br />
label("c", G--H,SW);<br />
</asy><br />
</center><br />
<math>(a+b)^2=c^2+4\left(\frac{1}{2}ab\right)\implies a^2+2ab+b^2=c^2+2ab\implies a^2 + b^2=c^2</math>. <br />
<br />
== Proof that n choose 2 is 1+2+3...+(n-1) ==<br />
<math>\binom{n}{2}=1+2+3...+n \Rightarrow \binom{n}{2}=\frac{n(n-1)}{2} \Rightarrow \frac{n(n-1)}{2}=\frac{n(n-1)}{2}</math>.Yay, I proved it!<br />
== Proof that <math>1+2+3...+n+1+2+3...+(n-1)=n^2</math> ==<br />
Proof that <math>1+2+3...+n+1+2+3...+(n-1)=n^2</math><br />
=== Proof 1 ===<br />
<math>1+2+3...+n+1+2+3...+(n-1)=n^2 \Rightarrow \frac{n(n+1)}{2}+\frac{n-1(n)}{2}=n^2</math> <br />
<math>\Rightarrow \frac{n(n+1)+n-1(n)}{2}=n^2 \Rightarrow \frac{n^2+n+n^2-n}{2}=n^2 \Rightarrow \frac{2n^2}{2}=n^2 \Rightarrow n^2=n^2</math>.<br />
=== Proof 2 ===<br />
The <math>1+2+\cdots+n</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal, and includes all the squares on the diagonal. The <math>1+2+\cdots+ n-1</math> part refers to an <math>n</math> by <math>n</math> square cut by its diagonal, but doesn't include the squares on the diagonal. Putting these together gives us a <math>n</math> by <math>n</math> square.<br />
<br />
=== Proof 3 ===<br />
We proceed using induction. If <math>n = 1</math>, then we have <math>1+0=1^2</math>. Now assume that <math>n</math> works. We prove that <math>n+1</math> works. We add a <math>2n+1</math> on both sides, such that the left side becomes <math>1+2+\cdots + (n+1)+1+2+\cdots + n = n^2 + 2n + 1 = (n+1)^2</math> and we are done.<br />
<br />
=== Proof 4 ===<br />
<math>1 2 3 4 5 ... n</math><br />
<br />
<math>0 1 2 3 4 ... (n-1)</math><br />
<br />
________________<br />
<br />
<math>1 3 5 7 9 ... 2n-1</math><br />
And that is <math>n^2</math>.<br />
[[Category:Math]]</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12A_Problems/Problem_9&diff=1569982010 AMC 12A Problems/Problem 92021-06-28T21:42:12Z<p>Jwelsh: Reverted edits by Poopbanna (talk) to last revision by Fath2012</p>
<hr />
<div>{{duplicate|[[2010 AMC 12A Problems|2010 AMC 12A #9]] and [[2010 AMC 10A Problems|2010 AMC 10A #17]]}}<br />
<br />
== Problem ==<br />
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?<br />
<br />
<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15</math><br />
<br />
== Solution ==<br />
<br />
=== Solution 1 ===<br />
<br />
Imagine making the cuts one at a time. The first cut removes a box <math>2\times 2\times 3</math>. The second cut removes two boxes, each of dimensions <math>2\times 2\times 0.5</math>, and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is <math>12 + 4 + 4 = 20</math>.<br />
<br />
Therefore the volume of the rest of the cube is <math>3^3 - 20 = 27 - 20 = \boxed{7\ \textbf{(A)}}</math>.<br />
<br />
=== Solution 2 ===<br />
<br />
We can use [[Principle of Inclusion-Exclusion]] (PIE) to find the final volume of the cube.<br />
<br />
There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" has <math>2 \times 2 \times 3=12</math> cubic inches. However, we can not just sum their volumes, as<br />
the central <math>2\times 2\times 2</math> cube is included in each of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice.<br />
<br />
Hence the total volume of the cuts is <math>3(2 \times 2 \times 3) - 2(2\times 2\times 2) = 36 - 16 = 20</math>.<br />
<br />
Therefore the volume of the rest of the cube is <math>3^3 - 20 = 27 - 20 = \boxed{7\ \textbf{(A)}}</math>.<br />
<br />
=== Solution 3 ===<br />
<br />
We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners.<br />
<br />
Each edge can be seen as a <math>2\times 0.5\times 0.5</math> box, and each corner can be seen as a <math>0.5\times 0.5\times 0.5</math> box.<br />
<br />
<math>12\cdot{\frac{1}{2}} + 8\cdot{\frac{1}{8}} = 6+1 = \boxed{7\ \textbf{(A)}}</math>.<br />
<br />
==Video Solution by the Beauty of Math==<br />
https://youtu.be/rsURe5Xh-j0?t=354<br />
<br />
== See also ==<br />
{{AMC10 box|year=2010|ab=A|num-b=16|num-a=18}}<br />
{{AMC12 box|year=2010|num-b=8|num-a=10|ab=A}}<br />
<br />
[[Category:Introductory Geometry Problems]]<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2009_AMC_10B_Problems/Problem_8&diff=1569962009 AMC 10B Problems/Problem 82021-06-28T21:41:34Z<p>Jwelsh: Reverted edits by Whitelisted (talk) to last revision by Nathan wailes</p>
<hr />
<div>{{duplicate|[[2009 AMC 10B Problems|2009 AMC 10B #8]] and [[2009 AMC 12B Problems|2009 AMC 12B #7]]}}<br />
<br />
== Problem ==<br />
In a certain year the price of gasoline rose by <math>20\%</math> during January, fell by <math>20\%</math> during February, rose by <math>25\%</math> during March, and fell by <math>x\%</math> during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is <math>x</math><br />
<br />
<math>\mathrm{(A)}\ 12\qquad<br />
\mathrm{(B)}\ 17\qquad<br />
\mathrm{(C)}\ 20\qquad<br />
\mathrm{(D)}\ 25\qquad<br />
\mathrm{(E)}\ 35</math><br />
<br />
== Solution ==<br />
Let <math>p</math> be the price at the beginning of January. The price at the end of March was <math>(1.2)(0.8)(1.25)p = 1.2p.</math> Because the price at the of April was <math>p</math>, the price decreased by <math>0.2p</math> during April, and the percent decrease was<br />
<cmath>x = 100 \cdot \frac{0.2p}{1.2p} = \frac {100}{6} \approx 16.7.</cmath><br />
So to the nearest integer <math>x</math> is <math>\boxed{17}</math>. The answer is <math>\mathrm{(B)}</math>.<br />
<br />
== See also ==<br />
{{AMC10 box|year=2009|ab=B|num-b=7|num-a=9}}<br />
{{AMC12 box|year=2009|ab=B|num-b=6|num-a=8}}<br />
{{MAA Notice}}</div>Jwelshhttps://artofproblemsolving.com/wiki/index.php?title=2010_AMC_12A_Problems/Problem_12&diff=1569952010 AMC 12A Problems/Problem 122021-06-28T21:41:25Z<p>Jwelsh: Reverted edits by Whitelisted (talk) to last revision by Sugar rush</p>
<hr />
<div>{{duplicate|[[2010 AMC 12A Problems|2010 AMC 12A #12]] and [[2010 AMC 10A Problems|2010 AMC 10A #15]]}}<br />
<br />
== Problem ==<br />
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.<br />
<br />
Brian: "Mike and I are different species."<br />
<br />
Chris: "LeRoy is a frog."<br />
<br />
LeRoy: "Chris is a frog."<br />
<br />
Mike: "Of the four of us, at least two are toads."<br />
<br />
How many of these amphibians are frogs?<br />
<br />
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math><br />
<br />
== Solution ==<br />
<br />
Start with Brian. If he is a toad, he tells the truth, hence Mike is a frog. If Brian is a frog, he lies, hence Mike is a frog, too. Thus Mike must be a frog.<br />
<br />
As Mike is a frog, his statement is false, hence there is at most one toad.<br />
<br />
As there is at most one toad, at least one of Chris and LeRoy is a frog. But then the other one tells the truth, and therefore is a toad.<br />
<br />
Hence we must have one toad and <math>\boxed{\textbf{(D)}\ 3}</math> frogs.<br />
<br />
==Video Solution by the Beauty of Math==<br />
https://youtu.be/kU70k1-ONgM?t=1207<br />
<br />
== See also ==<br />
{{AMC12 box|year=2010|num-b=11|num-a=13|ab=A}}<br />
{{AMC10 box|year=2010|num-b=14|num-a=16|ab=A}}<br />
<br />
[[Category:Introductory Algebra Problems]]<br />
{{MAA Notice}}</div>Jwelsh