https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Justinsun&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T13:02:24ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=LaTeX:Symbols&diff=176872LaTeX:Symbols2022-08-12T22:15:46Z<p>Justinsun: /* Greek Letters */ typo fix</p>
<hr />
<div>{{Latex}}<br />
<br />
This article provides a short list of commonly used LaTeX symbols. <br />
<br />
== Finding Other Symbols ==<br />
<br />
Here are some external resources for finding less commonly used symbols:<br />
<ul><br />
<li><br />
[http://detexify.kirelabs.org/classify.html Detexify] is an online application which allows you to draw the symbol you'd like and shows you the <math>\text{\LaTeX}</math> code for it!<br />
<br/><br/></li><br />
<br />
<li><br />
MathJax (what allows us to use <math>\text{\LaTeX}</math> on the web, (technically an AJAX library simulating it.)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list of supported commands].<br />
<br/><br/></li><br />
<br />
<li><br />
[http://mirrors.ctan.org/info/symbols/comprehensive/symbols-a4.pdf The Comprehensive LaTeX Symbol List].<br />
<br/><br/></li><br />
<br />
<li><br />
[https://mathvault.ca/wp-content/uploads/Comprehensive-List-of-Mathematical-Symbols.pdf Comprehensive List of Mathematical Symbols].<br />
<br/><br/></li><br />
</ul><br />
<br />
-------------------------------------------------------------------------------------------------------------<br />
<br />
==Operators==<br />
{| class="latextable"<br />
!Symbol!!Command!!Symbol!!Command!!Symbol!!Command<br />
|-<br />
|<math>\pm</math>||\pm||<math>\mp</math>||\mp||<math>\times</math>||\times<br />
|-<br />
|<math>\div</math>||\div||<math>\cdot</math>||\cdot||<math>\ast</math>||\ast<br />
|-<br />
|<math>\star</math>||\star||<math>\dagger</math>||\dagger||<math>\ddagger</math>||\ddagger<br />
|-<br />
|<math>\amalg</math>||\amalg||<math>\cap</math>||\cap||<math>\cup</math>||\cup<br />
|-<br />
|<math>\uplus</math>||\uplus||<math>\sqcap</math>||\sqcap||<math>\sqcup</math>||\sqcup<br />
|-<br />
|<math>\vee</math>||\vee||<math>\wedge</math>||\wedge||<math>\oplus</math>||\oplus<br />
|-<br />
|<math>\ominus</math>||\ominus||<math>\otimes</math>||\otimes||<math>\circ</math>||\circ<br />
|-<br />
|<math>\bullet</math>||\bullet||<math>\diamond</math>||\diamond||<math>\lhd</math>||\lhd<br />
|-<br />
|<math>\rhd</math>||\rhd||<math>\unlhd</math>||\unlhd||<math>\unrhd</math>||\unrhd<br />
|-<br />
|<math>\oslash</math>||\oslash||<math>\odot</math>||\odot||<math>\bigcirc</math>||\bigcirc<br />
|-<br />
|<math>\triangleleft</math>||\triangleleft||<math>\Diamond</math>||\Diamond||<math>\bigtriangleup</math>||\bigtriangleup<br />
|-<br />
|<math>\bigtriangledown</math>||\bigtriangledown||<math>\Box</math>||\Box||<math>\triangleright</math>||\triangleright<br />
|-<br />
|<math>\setminus</math>||\setminus||<math>\wr</math>||\wr||<math>\sqrt{x}</math>||\sqrt{x}<br />
|-<br />
|<math>x^{\circ}</math>||x^{\circ}||<math>\triangledown</math>||\triangledown||<math>\sqrt[n]{x}</math>||\sqrt[n]{x}<br />
|-<br />
|<math>a^x</math>||a^x||<math>a^{xyz}</math>||a^{xyz}||<math>a_x</math>||a_x<br />
|}<br />
<br />
==Relations==<br />
{| class="latextable"<br />
!Symbol !! Command !!Symbol !! Command!!Symbol !! Command<br />
|-<br />
| <math>\le</math>||\le||<math>\ge</math>||\ge||<math>\neq</math>||\neq<br />
|-<br />
| <math>\sim</math>||\sim||<math>\ll</math>||\ll||<math>\gg</math>||\gg<br />
|-<br />
| <math>\doteq</math>||\doteq||<math>\simeq</math>||\simeq||<math>\subset</math>||\subset<br />
|-<br />
| <math>\supset</math>||\supset||<math>\approx</math>||\approx||<math>\asymp</math>||\asymp<br />
|-<br />
| <math>\subseteq</math>||\subseteq||<math>\supseteq</math>||\supseteq||<math>\cong</math>||\cong<br />
|-<br />
| <math>\smile</math>||\smile||<math>\sqsubset</math>||\sqsubset||<math>\sqsupset</math>||\sqsupset<br />
|-<br />
| <math>\equiv</math>||\equiv||<math>\frown</math>||\frown||<math>\sqsubseteq</math>||\sqsubseteq<br />
|-<br />
| <math>\sqsupseteq</math>||\sqsupseteq||<math>\propto</math>||\propto||<math>\bowtie</math>||\bowtie<br />
|-<br />
| <math>\in</math>||\in||<math>\ni</math>||\ni||<math>\prec</math>||\prec<br />
|-<br />
| <math>\succ</math>||\succ||<math>\vdash</math>||\vdash||<math>\dashv</math>||\dashv<br />
|-<br />
| <math>\preceq</math>||\preceq||<math>\succeq</math>||\succeq||<math>\models</math>||\models<br />
|-<br />
| <math>\perp</math>||\perp||<math>\parallel</math>||\parallel||<br />
|-<br />
| <math>\mid</math>||\mid||<math>\bumpeq</math>||\bumpeq||<br />
|}<br />
Negations of many of these relations can be formed by just putting \not before the symbol, or by slipping an "n" between the \ and the word. Here are a couple examples, plus many other negations; it works for many of the many others as well.<br />
<br />
{| class="latextable"}<br />
!Symbol!!Command!!Symbol!!Command!!Symbol!!Command<br />
|-<br />
|<math>\nmid</math>||\nmid||<math>\nleq</math>||\nleq||<math>\ngeq</math>||\ngeq<br />
|-<br />
| <math>\nsim</math>||\nsim||<math>\ncong</math>||\ncong||<math>\nparallel</math>||\nparallel<br />
|-<br />
| <math>\not<</math>||\not<||<math>\not></math>||\not>||<math>\not=</math>||\not= or \neq or \ne<br />
|-<br />
| <math>\not\le</math>||\not\le||<math>\not\ge</math>||\not\ge||<math>\not\sim</math>||\not\sim<br />
|-<br />
|<math>\not \approx</math>||\not\approx||<math>\not\cong</math>||\not\cong||<math>\not\equiv</math>||\not\equiv<br />
|-<br />
| <math>\not\parallel</math>||\not\parallel||<math>\nless</math>||\nless||<math>\ngtr</math>||\ngtr<br />
|-<br />
| <math>\lneq</math>||\lneq||<math>\gneq</math>||\gneq||<math>\lnsim</math>||\lnsim<br />
|-<br />
| <math>\lneqq</math>||\lneqq||<math>\gneqq</math>||\gneqq<br />
|}<br />
<br />
To use other relations not listed here, such as =, >, and <, in LaTeX, you must use the symbols on your keyboard, as they are not available in <math>\LaTeX</math>.<br />
<br />
==Greek Letters==<br />
{| class="latextable"<br />
|+ Lowercase Letters<br />
!Symbol!!Command!!Symbol!!Command!!Symbol!!Command!!Symbol!!Command<br />
|-<br />
|<math>\alpha</math>||\alpha||<math>\beta</math>||\beta||<math>\gamma</math>||\gamma||<math>\delta</math>||\delta<br />
|-<br />
| <math>\epsilon</math>||\epsilon||<math>\varepsilon</math>||\varepsilon||<math>\zeta</math>||\zeta||<math>\eta</math>||\eta<br />
|-<br />
| <math>\theta</math>||\theta||<math>\vartheta</math>||\vartheta||<math>\iota</math>||\iota||<math>\kappa</math>||\kappa<br />
|-<br />
| <math>\lambda</math>||\lambda||<math>\mu</math>||\mu||<math>\nu</math>||\nu||<math>\xi</math>||\xi<br />
|-<br />
|<math>\pi</math>||\pi||<math>\varpi</math>||\varpi||<math>\rho</math>||\rho||<math>\varrho</math>||\varrho<br />
|-<br />
| <math>\sigma</math>||\sigma||<math>\varsigma</math>||\varsigma||<math>\tau</math>||\tau||<math>\upsilon</math>||\upsilon<br />
|-<br />
| <math>\phi</math>||\phi||<math>\varphi</math>||\varphi||<math>\chi</math>||\chi||<math>\psi</math>||\psi<br />
|-<br />
| <math>\omega</math>||\omega<br />
|}<br />
<br />
<br />
{| class="latextable"<br />
|+ Capital Letters<br />
!Symbol!!Command!!Symbol!!Command!!Symbol!!Command!!Symbol!!Command<br />
|-<br />
|<math>\Gamma</math>||\Gamma||<math>\Delta</math>||\Delta||<math>\Theta</math>||\Theta||<math>\Lambda</math>||\Lambda<br />
|-<br />
| <math>\Xi</math>||\Xi||<math>\Pi</math>||\Pi||<math>\Sigma</math>||\Sigma||<math>\Upsilon</math>||\Upsilon<br />
|-<br />
| <math>\Phi</math>||\Phi||<math>\Psi</math>||\Psi||<math>\Omega</math>||\Omega<br />
|}<br />
<br />
== Arrows ==<br />
<br />
{| class="latextable"<br />
!Symbol !! Command!!Symbol !! Command<br />
|-<br />
|<math>\gets</math>||\gets||<math>\to</math>||\to<br />
|-<br />
|<math>\leftarrow</math>||\leftarrow||<math>\Leftarrow</math>||\Leftarrow<br />
|-<br />
|<math>\rightarrow</math>||\rightarrow||<math>\Rightarrow</math>||\Rightarrow<br />
|-<br />
|<math>\leftrightarrow</math>||\leftrightarrow||<math>\Leftrightarrow</math>||\Leftrightarrow<br />
|-<br />
|<math>\mapsto</math>||\mapsto||<math>\hookleftarrow</math>||\hookleftarrow<br />
|-<br />
|<math>\leftharpoonup</math>||\leftharpoonup||<math>\leftharpoondown</math>||\leftharpoondown<br />
|-<br />
|<math>\rightleftharpoons</math>||\rightleftharpoons||<math>\longleftarrow</math>||\longleftarrow<br />
|-<br />
|<math>\Longleftarrow</math>||\Longleftarrow||<math>\longrightarrow</math>||\longrightarrow<br />
|-<br />
|<math>\Longrightarrow</math>||\Longrightarrow||<math>\longleftrightarrow</math>||\longleftrightarrow<br />
|-<br />
|<math>\Longleftrightarrow</math>||\Longleftrightarrow||<math>\longmapsto</math>||\longmapsto<br />
|-<br />
|<math>\hookrightarrow</math>||\hookrightarrow||<math>\rightharpoonup</math>||\rightharpoonup<br />
|-<br />
|<math>\rightharpoondown</math>||\rightharpoondown||<math>\leadsto</math>||\leadsto<br />
|-<br />
|<math>\uparrow</math>||\uparrow||<math>\Uparrow</math>||\Uparrow<br />
|-<br />
|<math>\downarrow</math>||\downarrow||<math>\Downarrow</math>||\Downarrow<br />
|-<br />
|<math>\updownarrow</math>||\updownarrow||<math>\Updownarrow</math>||\Updownarrow<br />
|-<br />
|<math>\nearrow</math>||\nearrow||<math>\searrow</math>||\searrow<br />
|-<br />
|<math>\swarrow</math>||\swarrow||<math>\nwarrow</math>||\nwarrow<br />
|-<br />
|<math>\overrightarrow{AB}</math>||\overrightarrow{AB}||<math>\overleftarrow{AB}</math>||\overleftarrow{AB}<br />
|-<br />
|<math>\overleftrightarrow{AB}</math>||\overleftrightarrow{AB}|| ||<br />
|}<br />
(For those of you who hate typing long strings of letters, \iff and \implies can be used in place of \Longleftrightarrow and \Longrightarrow respectively.)<br />
<br />
==Dots==<br />
{| class="latextable"<br />
!Symbol!!Command!!Symbol!!Command!!<br />
|- <br />
|<math>\cdot</math>||\cdot|| |<math>\vdots</math>||\vdots|| <br />
|- <br />
|<math>\dots</math>||\dots|| |<math>\ddots</math>||\ddots||<br />
|-<br />
|<math>\cdots</math>||\cdots|| |<math>\iddots</math>||\iddots||<br />
<br />
|}<br />
<br />
==Accents==<br />
{| class="latextable"<br />
!Symbol !! Command!!Symbol !! Command!!Symbol !! Command<br />
|-<br />
|<math>\hat{x}</math>||\hat{x}||<math>\check{x}</math>||\check{x}||<math>\dot{x}</math>||\dot{x}<br />
|-<br />
|<math>\breve{x}</math>||\breve{x}||<math>\acute{x}</math>||\acute{x}||<math>\ddot{x}</math>||\ddot{x}<br />
|-<br />
|<math>\grave{x}</math>||\grave{x}||<math>\tilde{x}</math>||\tilde{x}||<math>\mathring{x}</math>||\mathring{x}<br />
|-<br />
|<math>\bar{x}</math>||\bar{x}||<math>\vec{x}</math>||\vec{x}<br />
|}<br />
When applying accents to i and j, you can use \imath and \jmath to keep the dots from interfering with the accents:<br />
{| class="latextable"<br />
!Symbol !! Command!!Symbol !! Command<br />
|-<br />
|<math>\vec{\jmath}</math>||\vec{\jmath}||<math>\tilde{\imath}</math>||\tilde{\imath}<br />
|}<br />
\tilde and \hat have wide versions that allow you to accent an expression:<br />
{| class="latextable"<br />
!Symbol !! Command!!Symbol !! Command<br />
|-<br />
|<math>\widehat{7+x}</math>||\widehat{7+x}||<math>\widetilde{abc}</math>||\widetilde{abc}<br />
|}<br />
<br />
==Others==<br />
{| class="latextable"<br />
!Symbol!!Command!!Symbol!!Command!!Symbol!!Command <br />
|-<br />
|<math>\infty</math>||\infty||<math>\triangle</math>||\triangle||<math>\angle</math>||\angle<br />
|-<br />
|<math>\aleph</math>||\aleph||<math>\hbar</math>||\hbar||<math>\imath</math>||\imath<br />
|-<br />
|<math>\jmath</math>||\jmath||<math>\ell</math>||\ell||<math>\wp</math>||\wp<br />
|-<br />
|<math>\Re</math>||\Re||<math>\Im</math>||\Im||<math>\mho</math>||\mho<br />
|-<br />
|<math>\prime</math>||\prime||<math>\emptyset</math>||\emptyset||<math>\nabla</math>||\nabla<br />
|-<br />
|<math>\surd</math>||\surd||<math>\partial</math>||\partial||<math>\top</math>||\top<br />
|-<br />
|<math>\bot</math>||\bot||<math>\vdash</math>||\vdash||<math>\dashv</math>||\dashv<br />
|-<br />
|<math>\forall</math>||\forall||<math>\exists</math>||\exists||<math>\neg</math>||\neg<br />
|-<br />
|<math>\flat</math>||\flat||<math>\natural</math>||\natural||<math>\sharp</math>||\sharp<br />
|-<br />
|<math>\backslash</math>||\backslash||<math>\Box</math>||\Box||<math>\Diamond</math>||\Diamond<br />
|-<br />
|<math>\clubsuit</math>||\clubsuit||<math>\diamondsuit</math>||\diamondsuit||<math>\heartsuit</math>||\heartsuit<br />
|-<br />
|<math>\spadesuit</math>||\spadesuit||<math>\Join</math>||\Join||<math>\blacksquare</math>||\blacksquare<br />
|-<br />
|<math>\diamondsuit</math>||\diamondsuit||<math>\copyright</math>||\copyright ||<math>\underarc{XYZ}</math>||\underarc{XYZ}<br />
|-<br />
|<math>\heartsuit</math>||\heartsuit||<math>\overarc{ABC}</math>||\overarc{ABC} ||<math>\cup</math>||\cup <br />
|-<br />
|<math>\S</math>||\S||<math>\P</math>||\P||<math>\Vdash</math>||\Vdash<br />
|-<br />
|<math>\pounds</math>||\pounds||<math>\in</math>||\in||<math>\vDash</math>||\vDash<br />
|-<br />
|<math>\bigstar</math>||\bigstar<br />
|-<br />
|<math>\square</math>||\square||<br />
|-<br />
|<math>\smiley</math>||\smiley||<br />
|-<br />
|<math>\mathbb{R}</math>||\mathbb{R} (represents all real numbers)||<br />
|-<br />
|<math>\checkmark</math>||\checkmark||<br />
|-<br />
|<math>\cancer</math>||\cancer||<br />
|}<br />
Note: <math>\cancer</math> and <math>\overarc{ABC}</math> do not work in the classroom.<br />
<br />
==Command Symbols==<br />
Some symbols are used in commands, so they need to be treated in a special way.<br />
{| class="latextable"<br />
!Symbol!!Command!!Symbol!!Command!!Symbol!!Command!!Symbol!!Command<br />
|-<br />
|<math>\textdollar</math>||\textdollar or \\$||<math>\&</math>||\&||<math>\%</math>||\%||<math>\#</math>||\#<br />
|-<br />
|<math>\_</math>||\_||<math>\{</math>||\{||<math>\}</math>||\}||<math>\backslash</math>||\backslash<br />
|}<br />
<br />
(Warning: Using <nowiki>$</nowiki> for <math>\textdollar</math> will result in <math>\$</math>. This is a bug as far as we know. Depending on the version of <math>\LaTeX</math> this is not always a problem.)<br />
<br />
==European Language Symbols==<br />
{| class="latextable"<br />
!Symbol!!Command!!Symbol!!Command!!Symbol!!Command!!Symbol!!Command<br />
|-<br />
|<math>{\oe}</math>||{\oe}||<math>{\ae}</math>||{\ae}||<math>{\o}</math>||{\o}<br />
|-<br />
|<math>{\OE}</math>||{\OE}||<math>{\AE}</math>||{\AE}||<math>{\AA}</math>||{\AA}||<math>{\O}</math>||{\O}<br />
|-<br />
|<math>{\l}</math>||{\l}||<math>{\ss}</math>||{\ss}||<math>\text{!`}</math>||!`<br />
|-<br />
|<math>{\L}</math>||{\L}||<math>{\SS}</math>||{\SS}||<br />
|}<br />
<br />
==Bracketing Symbols==<br />
In mathematics, sometimes we need to enclose expressions in brackets, braces or parentheses. Some of these work just as you'd imagine in LaTeX; type ( and ) for parentheses, [ and ] for brackets, and | and | for absolute value. However, other symbols have special commands:<br />
{| class="latextable"<br />
!Symbol !! Command!!Symbol !! Command!!Symbol !! Command<br />
|-<br />
|<math>\{</math>||\{||<math>\}</math>||\}||<math>\|</math>||<nowiki>\|</nowiki><br />
|-<br />
| <math>\backslash</math>||\backslash||<math>\lfloor</math>||\lfloor||<math>\rfloor</math>||\rfloor<br />
|-<br />
| <math>\lceil</math>||\lceil||<math>\rceil</math>||\rceil||<math>\langle</math>||\langle<br />
|-<br />
| <math>\rangle</math>||\rangle<br />
|}<br />
You might notice that if you use any of these to typeset an expression that is vertically large, like<br />
<br />
:<tt>(\frac{a}{x} )^2</tt><br />
<br />
the parentheses don't come out the right size:<br />
<br />
:<math>(\frac{a}{x})^2</math><br />
<br />
If we put \left and \right before the relevant parentheses, we get a prettier expression:<br />
<br />
:<tt>\left(\frac{a}{x} \right)^2</tt><br />
<br />
gives<br />
<br />
:<math>\left(\frac{a}{x} \right)^2</math><br />
<br />
For systems of equations or piecewise functions, use the cases environment:<br />
<br />
<tt>f(x) = \begin{cases} x^2 & x \ge 0 \\ x & x < 0 \end{cases}</tt><br />
<br />
which gives<br />
<br />
<math>f(x) = \begin{cases} x^2 & x \ge 0 \\ x & x < 0 \end{cases}</math><br />
<br />
In addition to the <tt>\left</tt> and <tt>\right</tt> commands, when doing floor or ceiling functions with fractions, using <br />
<br />
<tt>\left\lceil\frac{x}{y}\right\rceil</tt> <br />
<br />
and <tt>\left\lfloor\frac{x}{y}\right\rfloor </tt><br />
<br />
gives both <math>\left\lceil\frac{x}{y}\right\rceil</math> and <math>\left\lfloor\frac{x}{y}\right\rfloor</math>, respectively.<br />
<br />
<br />
And, if you type this<br />
<br />
<tt>\underbrace{a_0+a_1+a_2+\cdots+a_n}_{x}</tt><br />
<br />
Gives<br />
<br />
<math>\underbrace{a_0+a_1+a_2+\cdots+a_n}_{x}</math><br />
<br />
Or<br />
<br />
<tt>\overbrace{a_0+a_1+a_2+\cdots+a_n}^{x}</tt><br />
<br />
Gives<br />
<br />
<math>\overbrace{a_0+a_1+a_2+\cdots+a_n}^{x}</math><br />
<br />
<br />
\left and \right can also be used to resize the following symbols:<br />
{| class="latextable"<br />
!Symbol !! Command!!Symbol !! Command!!Symbol !! Command<br />
|-<br />
|<math>\uparrow</math>||\uparrow||<math>\downarrow</math>||\downarrow||<math>\updownarrow</math>||\updownarrow<br />
|-<br />
| <math>\Uparrow</math>||\Uparrow||<math>\Downarrow</math>||\Downarrow||<math>\Updownarrow</math>||\Updownarrow<br />
|}<br />
<br />
==Multi-Size Symbols==<br />
Some symbols render differently in inline math mode and in display mode. Display mode occurs when you use <nowiki>\[...\]</nowiki> or <nowiki>$$...$$</nowiki>, or environments like \begin{equation}...\end{equation}, \begin{align}...\end{align}. Read more in the [[LaTeX:Commands|commands]] section of the guide about how symbols which take arguments above and below the symbols, such as a summation symbol, behave in the two modes.<br />
<br />
In each of the following, the two images show the symbol in display mode, then in inline mode.<br />
<br />
{| class="latextable"<br />
!Symbol !! Command!!Symbol !! Command!!Symbol !! Command<br />
|-<br />
|<math>\sum \textstyle\sum</math>||\sum||<math>\int \textstyle\int</math>||\int||<math>\oint \textstyle\oint</math>||\oint<br />
|-<br />
|<math>\prod \textstyle\prod</math>||\prod||<math>\coprod \textstyle\coprod</math>||\coprod||<math>\bigcap \textstyle\bigcap</math>||\bigcap<br />
|-<br />
|<math>\bigcup \textstyle\bigcup</math>||\bigcup||<math>\bigsqcup \textstyle\bigsqcup</math>||\bigsqcup||<math>\bigvee \textstyle\bigvee</math>||\bigvee<br />
|-<br />
|<math>\bigwedge \textstyle\bigwedge</math>||\bigwedge||<math>\bigodot \textstyle\bigodot</math>||\bigodot||<math>\bigotimes \textstyle\bigotimes</math>||\bigotimes<br />
|-<br />
|<math>\bigoplus \textstyle\bigoplus</math>||\bigoplus||<math>\biguplus \textstyle\biguplus</math>||\biguplus<br />
|}<br />
<br />
==See More==<br />
*[[LaTeX:Commands | Next: Commands]]</div>Justinsunhttps://artofproblemsolving.com/wiki/index.php?title=1975_Canadian_MO_Problems/Problem_8&diff=1745661975 Canadian MO Problems/Problem 82022-06-09T23:15:42Z<p>Justinsun: /* Problem 8 */</p>
<hr />
<div>== Problem 8 ==<br />
Let <math>k</math> be a positive integer. Find all polynomials<br />
<cmath>P(x) = a_0+a_1x+\cdots+a_nx^n</cmath><br />
where the <math>a_i</math> are real, which satisfy the equation<br />
<cmath>P(P(x)) = \{P(x)\}^k</cmath>.<br />
<br />
{{Old CanadaMO box|num-b=7|after = Last Question|year=1975}}<br />
<br />
== Solution 1==<br />
<br />
Let <math>f(n)</math> be the degree of polynomial <math>n</math>. We begin by noting that <math>f(P(x)) = k</math>. This is because the degree of the LHS is <math>f(P(x))^{f(P(x))}</math> and the RHS is <math>f(P(x))^k</math>. Now we split <math>P(x)</math> into two cases.<br />
<br />
In the first case, <math>P(x)</math> is a constant. This means that <math>c = c^k \Longrightarrow c\in \{0,1\}</math> or <math>c\in \{0,1,-1\}</math> if <math>k</math> is even.<br />
<br />
In the second case, <math>P(x)</math> is nonconstant with coefficients of <math>a_1,a_2,\hdots a_{k+1}</math>. If we divide by <math>P(x)</math> on both sides, then we have that <math>a_1 + \frac{a_2}{P(x)} + \frac{a_3}{P(x)^2} \hdots \frac{a_{k+1}}{P(x)^{k+1}} = 1</math>. This can only be achieved if <math>a_2,a_3, \hdots a_{k+1} = 0</math>. This is because if we factor out a <math>\frac{1}{P(x)}</math>, then clearly these terms are not constant. Thus, <math>a_1 = 1</math> and our second solution is <math>P(x) = x^k</math>. <br />
<br />
~bigbrain123</div>Justinsunhttps://artofproblemsolving.com/wiki/index.php?title=User:Piphi&diff=148737User:Piphi2021-03-06T23:29:24Z<p>Justinsun: /* User Count */</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">486</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 is the creator of the [[User:Piphi/Games|AoPS Wiki Games by Piphi]], the future of games on AoPS.<br><br />
<br />
Piphi started the signature trend at around May 2020.<br><br />
<br />
Piphi has been very close to winning multiple [[Greed Control]] games, piphi placed 5th in game #18 and 2nd in game #19. Thanks to piphi, Greed Control games have started to be kept track of. Piphi made a spreadsheet that has all of Greed Control history [https://artofproblemsolving.com/community/c19451h2126208p15569802 here].<br><br />
<br />
Piphi also found out who won [[Reaper]] games #1 and #2 as seen [https://artofproblemsolving.com/community/c19451h1826745p15526330 here].<br><br />
<br />
Piphi created the [[AoPS Administrators]] page, added most of the AoPS Admins to it, and created the scrollable table.<br><br />
<br />
Piphi has also added a lot of the info that is in the [[Reaper Archives]].<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 published Greed Control Game 19 statistics [https://artofproblemsolving.com/community/c19451h2126212 here].<br />
<br />
Piphi has a post that was made an announcement in a official AoPS Forum [https://artofproblemsolving.com/community/c68h2175116 here].<br />
<br />
Piphi is a proud member of [https://artofproblemsolving.com/community/c562043 The Interuniversal GMAAS Society].<br />
<br />
Piphi has won 2 gold trophies at the [https://artofproblemsolving.com/community/c1124279 Asymptote Competition] and is now part of the 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 500<br />
{{User:Piphi/Template:Progress_Bar|93.8|width=100%}}<br />
<br />
200 subpages of [[User:Piphi]]<br />
{{User:Piphi/Template:Progress_Bar|98|width=100%}}<br />
<br />
200 signups for [[User:Piphi/Games|AoPS Wiki Games by Piphi]]<br />
{{User:Piphi/Template:Progress_Bar|50.5|width=100%}}<br />
<br />
Make 10,000 edits<br />
{{User:Piphi/Template:Progress_Bar|22.60|width=100%}}</font></div><br />
</div></div>Justinsunhttps://artofproblemsolving.com/wiki/index.php?title=User:Piphi&diff=148736User:Piphi2021-03-06T23:28:54Z<p>Justinsun: /* User Count */</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">586</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 is the creator of the [[User:Piphi/Games|AoPS Wiki Games by Piphi]], the future of games on AoPS.<br><br />
<br />
Piphi started the signature trend at around May 2020.<br><br />
<br />
Piphi has been very close to winning multiple [[Greed Control]] games, piphi placed 5th in game #18 and 2nd in game #19. Thanks to piphi, Greed Control games have started to be kept track of. Piphi made a spreadsheet that has all of Greed Control history [https://artofproblemsolving.com/community/c19451h2126208p15569802 here].<br><br />
<br />
Piphi also found out who won [[Reaper]] games #1 and #2 as seen [https://artofproblemsolving.com/community/c19451h1826745p15526330 here].<br><br />
<br />
Piphi created the [[AoPS Administrators]] page, added most of the AoPS Admins to it, and created the scrollable table.<br><br />
<br />
Piphi has also added a lot of the info that is in the [[Reaper Archives]].<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 published Greed Control Game 19 statistics [https://artofproblemsolving.com/community/c19451h2126212 here].<br />
<br />
Piphi has a post that was made an announcement in a official AoPS Forum [https://artofproblemsolving.com/community/c68h2175116 here].<br />
<br />
Piphi is a proud member of [https://artofproblemsolving.com/community/c562043 The Interuniversal GMAAS Society].<br />
<br />
Piphi has won 2 gold trophies at the [https://artofproblemsolving.com/community/c1124279 Asymptote Competition] and is now part of the 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 500<br />
{{User:Piphi/Template:Progress_Bar|93.8|width=100%}}<br />
<br />
200 subpages of [[User:Piphi]]<br />
{{User:Piphi/Template:Progress_Bar|98|width=100%}}<br />
<br />
200 signups for [[User:Piphi/Games|AoPS Wiki Games by Piphi]]<br />
{{User:Piphi/Template:Progress_Bar|50.5|width=100%}}<br />
<br />
Make 10,000 edits<br />
{{User:Piphi/Template:Progress_Bar|22.60|width=100%}}</font></div><br />
</div></div>Justinsunhttps://artofproblemsolving.com/wiki/index.php?title=User:Piphi/Chess&diff=148735User:Piphi/Chess2021-03-06T23:27:27Z<p>Justinsun: /* Signups */</p>
<hr />
<div>__NOTOC__<br />
{{User:Piphi/Template:Header}}<br />
<br><br />
{{User:Piphi/Chess/Template:Header}}<br />
<br />
== Signups ==<br />
Put your username below if you want to play chess through the wiki. I will be going through this list periodically and add people to games. If you have any preference on who you want to play against then put their username in parenthesis next to your username.<br />
<br />
'''Note:''' You are allowed to be in a maximum of 3 games at once. Once someone wins a game you're in or it is abandoned, it no longer counts towards your 3 games.<br />
<br />
<br />
* ALICEW123 (Want to play game 1)<br />
*justinsun (Yea, Sucky)<br />
* piphi <span style="color:green;">assigned to game 1</span><br />
* BorealBear (But I am not on often :|) <span style="color:green;">assigned to game 1</span><br />
* awesomeguy856 <span style="color:green;">assigned to game 2</span><br />
* Heavytoothpaste, I play a lot :) <span style="color:green;">assigned to game 2</span><br />
* FaerieDragon88, (I’m on frequently) <span style="color:green;">assigned to game 3</span><br />
* williamxiao (don't know if i will be any good at this) <span style="color:green;">assigned to game 3</span><br />
* Mark888 (I'll fail horribly, let's do this) <span style="color:green;">assigned to game 4</span><br />
* ARay10 (But I'm worse) <span style="color:green;">assigned to game 4</span><br />
* wuwang2002 (I will just try) <span style="color:green;">assigned to game 5</span><br />
* Mark888 (am I allowed to run 2 games at once?) <span style="color:green;">assigned to game 5</span><br />
* Juno (EPIC FAILER HERE) <span style="color:green;">assigned to game 6</span><br />
* ARay10 <span style="color:green;">assigned to game 6</span><br />
* Paul10 <span style="color:green;">assigned to game 7</span><br />
* Kmath1234 <span style="color:green;">assigned to game 7</span><br />
* ARay10 (wants to verse piphi) <span style="color:green;">assigned to game 8</span><br />
* piphi (sure, this is your last available chess game BTW, you can still play [[User:Piphi/Checkers|checkers]] or wait until one of your games end) <span style="color:green;">assigned to game 8</span><br />
* Aspiring_Mathletes <span style="color:green;">assigned to game 9</span><br />
* LJCoder619 <span style="color:green;">assigned to game 9</span><br />
* SamuraiA (not going to be too active) <span style="color:green;">assigned to game 10</span><br />
* rf20008 <span style="color:green;">assigned to game 10</span><br />
* BrightMonkey <span style="color:green;">assigned to game 11</span><br />
* Alpha_2 <span style="color:green;">assigned to game 11</span><br />
* HonestCat (I'm excited to play chess!) <span style="color:green;">assigned to game 12</span><br />
* aw2017 (I can pk fire anyone) <span style="color:green;">assigned to game 12</span><br />
* CreativeHedgehog (I'm a beginner) <span style="color:green;">assigned to game 13</span><br />
* HonestCat <span style="color:green;">assigned to game 13</span><br />
* ARay10 <span style="color:green;">assigned to game 14</span><br />
* HonestCat <span style="color:green;">assigned to game 14</span><br />
* Bluedevils(I rly suck) <span style="color:green;">assigned to game 15</span><br />
* ATGY(I KNOW I WILL LOSE) <span style="color:green;">assigned to game 15</span><br />
* aops-g5-gethsemanea2 (I am not that much good at chess and not in the wiki as often) <span style="color:green;">assigned to game 16</span><br />
* Angrybird029 <span style="color:green;">assigned to game 16</span><br />
* CreativeHedgehog (I want to play a crazyhouse variation) <span style="color:green;">assigned to game 17</span><br />
* SaphireTurtle (normal game please) <span style="color:green;">assigned to game 18</span><br />
* Angrybird029 (I can play crazyhouse against CreativeHedgeHog) <span style="color:green;">assigned to game 17</span><br />
* Angrybird029 (I also can play normal chess against SaphireTurtle. That isn't against the rules, right?) <span style="color:green;">assigned to game 18</span><br />
* HonestCat <span style="color:green;">assigned to game 19</span><br />
* HonestCat <span style="color:green;">assigned to game 20</span><br />
* CreativeHedgehog (Doing Do Show Qi, but deciding to play a chess game with HonestCat.) <span style="color:green;">assigned to game 19</span><br />
* aops-g5-gethsemanea2 (now this is my second chess game, but the other one is done... i can do bughouse variant. so now piphi signed up, we (might) be teammates :)) <span style="color:green;">assigned to game 22</span><br />
* piphi (I'll play bughouse with aops-g5-gethsemanea2) <span style="color:green;">assigned to game 22</span><br />
* RedJive (Will do whatever but I suck) <span style="color:green;">assigned to game 20</span><br />
* CreativeHedghehog (Want to play 3-check, Piphi can use the progress bar template for this) <span style="color:green;">assigned to game 21</span><br />
* HonestCat (Can play 3-check with CreativeHedgehog, I stink) <span style="color:green;">assigned to game 21</span><br />
* CreativeHedgehog (I'm going to play bughouse against aops-g5-gethsemanea2. I am currently making a bughouse template. I would like to play with Angrybird029.) <span style="color:green;">assigned to game 22</span><br />
* Angrybird029 (I'll round out the four needed to play bughouse). <span style="color:green;">assigned to game 22</span><br />
* CreativeHedgehog (I want to play KOTH)<br />
* zhuqingzhang <span style="color:green;">assigned to game 23</span><br />
* Bluedevils (signup plz?) <span style="color:green;">assigned to game 23</span><br />
* Cozzmo pm me when ready when do u expect??? plz plz plz!!! <span style="color:green;">assigned to game 24</span><br />
* Angrybird029 (I can play with Cozzmo, and CreativeHedgehog after one of my games finish). <span style="color:green;">assigned to game 24</span><br />
* Pi31 <span style="color:green;">assigned to game 25</span><br />
* EDKthepro <span style="color:green;">assigned to game 25</span><br />
* Wood2009<br />
* ARay10 tru i'm back <span style="color:green;">assigned to game 26</span><br />
* CreativeHedgehog (wants to battle ARay10) <span style="color:green;">assigned to game 26</span><br />
* CreativeHedgheog (ARay10 isn't responding, I'll play antichess) <span style="color:green;">assigned to game 27</span><br />
* piphi (antichess anyone?) <span style="color:green;">assigned to game 27</span><br />
<br />
== Rules ==<br />
[[User:Piphi/Chess/Rules]]<br />
<br />
== Help ==<br />
[[User:Piphi/Chess/Help]]<br />
<br />
== Games ==<br />
* Game #1 piphi v BorealBear. Game is [[User:Piphi/Chess/Game_1|here]]. Discussion is [[User talk:Piphi/Chess/Game_1|here]].<br />
<br />
* Game #2 awesomeguy856 v Heavytoothpaste. Game is [[User:Piphi/Chess/Game_2|here]]. Discussion is [[User talk:Piphi/Chess/Game_2|here]].<br />
<br />
* Game #3 FaerieDragon88 v williamxiao. Game is [[User:Piphi/Chess/Game_3|here]]. Discussion is [[User talk:Piphi/Chess/Game_3|here]].<br />
<br />
* Game #4 Mark888 v ARay10. Game is [[User:Piphi/Chess/Game_4|here]]. Discussion is [[User talk:Piphi/Chess/Game_4|here]].<br />
<br />
* Game #5 wuwang2002 v Mark888. Game is [[User:Piphi/Chess/Game_5|here]]. Discussion is [[User talk:Piphi/Chess/Game_5|here]].<br />
<br />
* Game #6 Juno v ARay10. Game is [[User:Piphi/Chess/Game_6|here]]. Discussion is [[User talk:Piphi/Chess/Game_6|here]].<br />
<br />
* Game #7 Paul10 v Kmath1234. Game is [[User:Piphi/Chess/Game_7|here]]. Discussion is [[User talk:Piphi/Chess/Game_7|here]].<br />
<br />
* Game #8 ARay10 v piphi. Game is [[User:Piphi/Chess/Game_8|here]]. Discussion is [[User talk:Piphi/Chess/Game_8|here]].<br />
<br />
* Game #9 Aspiring_Mathletes v LJCoder619. Game is [[User:Piphi/Chess/Game_9|here]]. Discussion is [[User talk:Piphi/Chess/Game_9|here]].<br />
<br />
* Game #10 SamuraiA v rf20008. Game is [[User:Piphi/Chess/Game_10|here]]. Discussion is [[User talk:Piphi/Chess/Game_10|here]].<br />
<br />
* Game #11 BrightMonkey v Alpha_2. Game is [[User:Piphi/Chess/Game_11|here]]. Discussion is [[User talk:Piphi/Chess/Game_11|here]].<br />
<br />
* Game #12 HonestCat v aw2017. Game is [[User:Piphi/Chess/Game_12|here]]. Discussion is [[User talk:Piphi/Chess/Game_12|here]].<br />
<br />
* Game #13 HonestCat v CreativeHedgehog. Game is [[User:Piphi/Chess/Game_13|here]]. Discussion is [[User talk:Piphi/Chess/Game_13|here]].<br />
<br />
* Game #14 ARay10 v HonestCat. Game is [[User:Piphi/Chess/Game_14|here]]. Discussion is [[User talk:Piphi/Chess/Game_14|here]].<br />
<br />
* Game #15 Bluedevils v ATGY. Game is [[User:Piphi/Chess/Game_15|here]]. Discussion is [[User talk:Piphi/Chess/Game_15|here]].<br />
<br />
* Game #16 aops-g5-gethsemanea2 v Angrybird029. Game is [[User:Piphi/Chess/Game_16|here]]. Discussion is [[User talk:Piphi/Chess/Game_16|here]].<br />
<br />
* Game #17 CreativeHedgehog v Angrybird029. Game is [[User:Piphi/Chess/Game_17|here]]. Discussion is [[User talk:Piphi/Chess/Game_17|here]].<br />
<br />
* Game #18 SaphireTurtle v Angrybird029. Game is [[User:Piphi/Chess/Game_18|here]]. Discussion is [[User talk:Piphi/Chess/Game_18|here]].<br />
<br />
* Game #19 HonestCat v CreativeHedgehog. Game is [[User:Piphi/Chess/Game_19|here]]. Discussion is [[User talk:Piphi/Chess/Game_19|here]].<br />
<br />
* Game #20 HonestCat v RedJive. Game is [[User:Piphi/Chess/Game_20|here]]. Discussion is [[User talk:Piphi/Chess/Game_20|here]].<br />
<br />
* Game #21 CreativeHedgehog v HonestCat. Game is [[User:Piphi/Chess/Game_21|here]]. Discussion is [[User talk:Piphi/Chess/Game_21|here]].<br />
<br />
* Game #22 aops-g5-gethsemanea2 & piphi v CreativeHedgehog & Angrybird029. Game is [[User:Piphi/Chess/Game_22|here]]. Discussion is [[User talk:Piphi/Chess/Game_22|here]].<br />
<br />
* Game #23 zhuqingzhang v Bluedevils. Game is [[User:Piphi/Chess/Game_23|here]]. Discussion is [[User talk:Piphi/Chess/Game_23|here]].<br />
<br />
* Game #24 Cozzmo v Angrybird029. Game is [[User:Piphi/Chess/Game_24|here]]. Discussion is [[User talk:Piphi/Chess/Game_24|here]].<br />
<br />
* Game #25 Pi31 v EDKthepro. Game is [[User:Piphi/Chess/Game_25|here]]. Discussion is [[User talk:Piphi/Chess/Game_25|here]].<br />
<br />
* Game #26 ARay10 v CreativeHedgehog. Game is [[User:Piphi/Chess/Game_26|here]]. Discussion is [[User talk:Piphi/Chess/Game_26|here]].<br />
<br />
* Game #27 CreativeHedgehog v piphi. Game is [[User:Piphi/Chess/Game_27|here]]. Discussion is [[User talk:Piphi/Chess/Game_27|here]].<br />
<br />
[[Category:Chess_Piphi]]</div>Justinsun