https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Nikoislife&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T00:20:50ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2008_AMC_12A_Problems/Problem_23&diff=1110712008 AMC 12A Problems/Problem 232019-11-10T05:38:11Z<p>Nikoislife: /* Solution */</p>
<hr />
<div>==Problem==<br />
The solutions of the equation <math>z^4+4z^3i-6z^2-4zi-i=0</math> are the vertices of a convex polygon in the complex plane. What is the area of the polygon?<br />
<br />
<math>\mathrm{(A)}\ 2^{\frac{5}{8}}\qquad\mathrm{(B)}\ 2^{\frac{3}{4}}\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 2^{\frac{5}{4}}\qquad\mathrm{(E)}\ 2^{\frac{3}{2}}</math><br />
<br />
==Solution==<br />
Looking at the coefficients, we are immediately reminded of the binomial expansion of <math>{\left(x+1\right)}^{4}</math>.<br />
<br />
Modifying this slightly, we can write the given equation as:<br />
<cmath> {\left(z+i\right)}^{4}=1+i=2^{\frac{1}{2}}\cos \frac {\pi}{4} + 2^{\frac{1}{2}}i\sin \frac {\pi}{4} </cmath><br />
We can apply a translation of <math>-i</math> and a rotation of <math>-\frac{\pi}{4}</math> (both operations preserve area) to simplify the problem:<br />
<cmath>z^{4}=2^{\frac{1}{2}}</cmath><br />
<br />
Because the roots of this equation are created by rotating <math>\frac{\pi}{2}</math> radians successively about the origin, the quadrilateral is a square.<br />
<br />
We know that half the diagonal length of the square is <math>{\left(2^{\frac{1}{2}}\right)}^{\frac{1}{4}}=2^{\frac{1}{8}}</math><br />
<br />
Therefore, the area of the square is<br />
<math> \frac{{\left( 2 \cdot 2^{\frac{1}{8}}\right)}^2}{2}=\frac{2^{\frac{9}{4}}}{2}=2^{\frac{5}{4}} \Rightarrow D. </math><br />
<br />
==See Also==<br />
{{AMC12 box|year=2008|ab=A|num-b=22|num-a=24}}<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=110622List of BBCodes2019-10-27T11:55:16Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[cpp]</code></td><td>Formats and highlights the text as C++ code</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[javascript]</code></td><td>Formats and highlights the text as JavaScript code.</td><br />
</tr><br />
<tr><br />
<td><code>[latex]</code></td><td>Formats and highlights the text as LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[tex]</code></td><td>Formats and highlights the text as TeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[tip]</code></td><td>Creates the text that could be hovered over with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table><br />
<br />
For examples on how these tags are used, one can see the usage for most of these BBCode tags [https://artofproblemsolving.com/community/c10h1892169 here.]</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=2018_AMC_8_Problems/Problem_17&diff=1104672018 AMC 8 Problems/Problem 172019-10-20T13:23:58Z<p>Nikoislife: /* Solution 3 */</p>
<hr />
<div>==Problem 17==<br />
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is <math>2</math> miles, which is <math>10,560</math> feet, and Bella covers <math>2 \tfrac{1}{2}</math> feet with each step. How many steps will Bella take by the time she meets Ella?<br />
<br />
<math>\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520</math><br />
<br />
==Solution==<br />
Since Ella rides 5 times as fast as Bella, Ella rides at a rate of <math>\frac{25}{2}</math> or <math>12 \tfrac{1}{2}</math>. Together, they move <math>15</math> feet towards each other every unit. Dividing <math>10560</math> by <math>15</math> to find the number of steps Bella takes results in the answer of <math>\boxed{\textbf{(A) }704}</math><br />
<br />
==Solution 2==<br />
<br />
Since Ella rides 5 times faster than Bella, the ratio of their speeds is 5:1. This means that Bella travels 1/6 of the way, and 1/6 of 10560 feet is 1760 feet. Bella also walks 2.5 feet in a step, and 1760 divided by 2.5 is <math>\boxed{\textbf{(A) }704}</math>.<br />
<br />
==Solution 3==<br />
<br />
We know that Ella's speed is <math>12.5</math> st/s (steps per second) and Bella's speed is <math>2.5</math> st/s. Let the speeds in which they are walking and riding at be the slope of two equations, one for Bella and one for Ella. We make <math>s</math> the amount of steps she takes and <math>d</math> distance from their houses. Since we are finding where they meet, we need to solve for the intersections of the two equations for Ella and Bella. The equations we get are:<br />
<br />
<cmath>d = -12.5s + 10560</cmath><br />
<cmath>d = 2.5s</cmath><br />
<br />
Subtracting both equations from each other gives us:<br />
<br />
<cmath>-15s + 10560 = 0</cmath><br />
<br />
Solving that equation grants us:<br />
<br />
<cmath>s = 704</cmath><br />
<br />
Since we are finding the number of steps they take until they meet each other, we can stop there. The solution is <br />
<br />
<math>\boxed{\textbf{(A) }704}</math><br />
<br />
==See Also==<br />
{{AMC8 box|year=2018|num-b=16|num-a=18}}<br />
<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12A_Problems/Problem_15&diff=1104662005 AMC 12A Problems/Problem 152019-10-20T13:22:36Z<p>Nikoislife: /* Solution 5 */</p>
<hr />
<div>== Problem ==<br />
Let <math>\overline{AB}</math> be a [[diameter]] of a [[circle]] and <math>C</math> be a point on <math>\overline{AB}</math> with <math>2 \cdot AC = BC</math>. Let <math>D</math> and <math>E</math> be [[point]]s on the circle such that <math>\overline{DC} \perp \overline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the [[ratio]] of the area of <math>\triangle DCE</math> to the area of <math>\triangle ABD</math>?<br />
<br />
<asy><br />
unitsize(2.5cm);<br />
defaultpen(fontsize(10pt)+linewidth(.8pt));<br />
dotfactor=3;<br />
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);<br />
pair D=dir(aCos(C.x)), E=(-D.x,-D.y);<br />
draw(A--B--D--cycle);<br />
draw(D--E--C);<br />
draw(unitcircle,white);<br />
drawline(D,C);<br />
dot(O);<br />
clip(unitcircle);<br />
draw(unitcircle);<br />
label("$E$",E,SSE);<br />
label("$B$",B,E);<br />
label("$A$",A,W);<br />
label("$D$",D,NNW);<br />
label("$C$",C,SW);<br />
draw(rightanglemark(D,C,B,2));</asy><br />
<br />
<math>(\text {A}) \ \frac {1}{6} \qquad (\text {B}) \ \frac {1}{4} \qquad (\text {C})\ \frac {1}{3} \qquad (\text {D}) \ \frac {1}{2} \qquad (\text {E})\ \frac {2}{3}</math><br />
<br />
== Solution ==<br />
<br />
===Solution 1===<br />
<br />
Notice that the bases of both triangles are diameters of the circle. Hence the ratio of the areas is just the ratio of the heights of the triangles, or <math>\frac{CF}{CD}</math> (<math>F</math> is the foot of the [[perpendicular]] from <math>C</math> to <math>DE</math>).<br />
<br />
Call the radius <math>r</math>. Then <math>AC = \frac 13(2r) = \frac 23r</math>, <math>CO = \frac 13r</math>. Using the [[Pythagorean Theorem]] in <math>\triangle OCD</math>, we get <math>\frac{1}{9}r^2 + CD^2 = r^2 \Longrightarrow CD = \frac{2\sqrt{2}}3r</math>. <br />
<br />
Now we have to find <math>CF</math>. Notice <math>\triangle OCD \sim \triangle OFC</math>, so we can write the [[proportion]]:<br />
<br />
<div style="text-align:center;"><math>\frac{OF}{OC} = \frac{OC}{OD}</math><br /><math>\frac{OF}{\frac{1}{3}r} = \frac{\frac{1}{3}r}{r}</math><br /><math>OF = \frac 19r</math></div><br />
<br />
By the [[Pythagorean Theorem]] in <math>\triangle OFC</math>, we have <math>\left(\frac{1}{9}r\right)^2 + CF^2 = \left(\frac{1}{3}r\right)^2 \Longrightarrow CF = \sqrt{\frac{8}{81}r^2} = \frac{2\sqrt{2}}{9}r</math>.<br />
<br />
Our answer is <math>\frac{CF}{CD} = \frac{\frac{2\sqrt{2}}{9}r}{\frac{2\sqrt{2}}{3}r} = \frac 13 \Longrightarrow \mathrm{(C)}</math>.<br />
<br />
===Solution 2===<br />
<br />
Let the center of the circle be <math>O</math>.<br />
<br />
Note that <math>2 \cdot AC = BC \Rightarrow 3 \cdot AC = AB</math>.<br />
<br />
<math>O</math> is midpoint of <math>AB \Rightarrow \frac{3}{2}AC = AO \Rightarrow CO = \frac{1}{3}AO \Rightarrow CO = \frac{1}{6} AB</math>.<br />
<br />
<math>O</math> is midpoint of <math>DE \Rightarrow</math> Area of <math>\triangle DCE = 2 \cdot</math> Area of <math>\triangle DCO = 2 \cdot (\frac{1}{6} \cdot</math> Area of <math>\triangle ABD) = \frac{1}{3} \cdot</math> Area of <math>\triangle ABD \Longrightarrow \mathrm{(C)}</math>.<br />
<br />
===Solution 3===<br />
<br />
Let <math>r</math> be the radius of the circle. Note that <math>AC+BC = 2r</math> so <math>AC = \frac{2}{3}r</math>.<br />
<br />
By [[Power of a Point Theorem]], <math>CD^2= AC \cdot BC = 2\cdot AC^2</math>, and thus <math>CD = \sqrt{2} \cdot AC = \frac{2\sqrt{2}}{3}r</math><br />
<br />
Then the area of <math>\triangle ABD</math> is <math>\frac{1}{2} AB \cdot CD = \frac{2\sqrt{2}}{3}r^2</math>. Similarly, the area of <math>\triangle DCE</math> is <math>\frac{1}{2}(r-AC) \cdot 2 \cdot CD = \frac{2\sqrt{2}}{9}r^2</math>, so the desired ratio is <math>\frac{\frac{2\sqrt{2}}{9}r^2}{\frac{2\sqrt{2}}{3}r^2} = \frac{1}{3} \Longrightarrow \mathrm{(C)}</math><br />
<br />
===Solution 4===<br />
<asy><br />
unitsize(2.5cm);<br />
defaultpen(fontsize(10pt)+linewidth(.8pt));<br />
dotfactor=3;<br />
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);<br />
pair D=dir(aCos(C.x)), e=(-D.x,-D.y);<br />
pair H=(e.x,0);<br />
draw(A--B--D--cycle);<br />
draw(D--e--C);<br />
draw(unitcircle,white);<br />
drawline(D,C);<br />
dot(O);<br />
clip(unitcircle);<br />
draw(unitcircle);<br />
label("$E$",e,SSE);<br />
label("$B$",B,NE);<br />
label("$A$",A,W);<br />
label("$D$",D,NNW);<br />
label("$C$",C,SW);<br />
label("$H$",H,SE);<br />
draw(e--H,dashed);<br />
label("O",(0,0),NE);<br />
label("1",(C--O),N);<br />
label("1",(H--O),N);<br />
label("2",(A--C),N);<br />
label("2",(H--B),N);<br />
label("3",(O--D),NE);<br />
label("3",(O--e),NE);<br />
label("$2\sqrt{2}$",(D--C),W);<br />
label("$2\sqrt{2}$",(H--e),E);<br />
draw(rightanglemark(e,(e.x,0),A,2));<br />
draw(rightanglemark(D,C,B,2));</asy><br />
<br />
Let the center of the circle be <math>O</math>.<br />
Without loss of generality, let the radius of the circle be equal to <math>3</math>. Thus, <math>AO=3</math> and <math>OB=3</math>. As a consequence of <math>2(AC)=BC</math>, <math>AC=2</math> and <math>CO=1</math>. Also, we know that <math>DO</math> and <math>OE</math> are both equal to <math>3</math> due to the fact that they are both radii. Thus from the Pythagorean Theorem, we have DC being equal to <math>\sqrt{3^2-1^2}</math> or <math>2\sqrt{2}</math>. Now we know that the area of <math>[ABD]</math> is equal to <math>\frac{(3+2+1)(2\sqrt{2})}{2}</math> or <math>6\sqrt{2}</math>. Know we need to find the area of <math>[DCE]</math>. By simple inspection <math>[COD]</math> <math>\cong</math> <math>[HOE]</math> due to angles being equal and CPCTC. Thus <math>HE=2\sqrt{2}</math> and <math>OH=1</math>. Know we know the area of <math>[CHE]=\frac{(1+1)(2\sqrt{2})}{2}</math> or <math>2\sqrt{2}</math>. We also know that the area of <math>[OHE]=\frac{(1)(2\sqrt{2})}{2}</math> or <math>\sqrt{2}</math>. Thus the area of <math>[COE]=2\sqrt{2}-\sqrt{2}</math> or <math>\sqrt{2}</math>. We also can calculate the area of <math>[DOC]</math> to be <math>\frac{(1)(2\sqrt{2})}{2}</math> or <math>\sqrt{2}</math>. Thus <math>[DCE]</math> is equal to <math>[COE]</math> + <math>[DOC] </math> or <math>\sqrt{2}+\sqrt{2}</math> or <math>2\sqrt{2}</math>. The ratio between <math>[DCE]</math> and <math>[ABD]</math> is equal to <math>\frac{2\sqrt{2}}{6\sqrt{2}}</math> or <math>\frac{1}{3}</math> <math>\Longrightarrow \mathrm{(C)}</math>.<br />
===Solution 5===<br />
We will use the shoelace formula. Our origin is the center of the circle. Denote the ordered pair for <math>D (x,y)</math>, and notice how <math>E</math> is a 180 degree rotation of <math>D</math>, using the rotation matrix formula we get <math>E = (-x,-y)</math>. WLOG say that this circle has radius <math>3</math>. We can now find points <math>A</math>, <math>B</math>, and <math>C</math> which are <math>(-3,0)</math>, <math>(-1,0)</math>, and <math>(3,0)</math> respectively.<br />
By shoelace the area of <math>CED</math> is <math>Y</math>, and the area of <math>ADB</math> is <math>3Y</math>. Using division we get that the answer is <math> \mathrm{(C)}</math>.<br />
<br />
== See also ==<br />
{{AMC12 box|year=2005|num-b=14|num-a=16|ab=A}}<br />
<br />
[[Category:Introductory Geometry Problems]]<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12A_Problems/Problem_15&diff=1104652005 AMC 12A Problems/Problem 152019-10-20T13:22:14Z<p>Nikoislife: /* Solution 5 (Easiest and best using shoelace) */</p>
<hr />
<div>== Problem ==<br />
Let <math>\overline{AB}</math> be a [[diameter]] of a [[circle]] and <math>C</math> be a point on <math>\overline{AB}</math> with <math>2 \cdot AC = BC</math>. Let <math>D</math> and <math>E</math> be [[point]]s on the circle such that <math>\overline{DC} \perp \overline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the [[ratio]] of the area of <math>\triangle DCE</math> to the area of <math>\triangle ABD</math>?<br />
<br />
<asy><br />
unitsize(2.5cm);<br />
defaultpen(fontsize(10pt)+linewidth(.8pt));<br />
dotfactor=3;<br />
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);<br />
pair D=dir(aCos(C.x)), E=(-D.x,-D.y);<br />
draw(A--B--D--cycle);<br />
draw(D--E--C);<br />
draw(unitcircle,white);<br />
drawline(D,C);<br />
dot(O);<br />
clip(unitcircle);<br />
draw(unitcircle);<br />
label("$E$",E,SSE);<br />
label("$B$",B,E);<br />
label("$A$",A,W);<br />
label("$D$",D,NNW);<br />
label("$C$",C,SW);<br />
draw(rightanglemark(D,C,B,2));</asy><br />
<br />
<math>(\text {A}) \ \frac {1}{6} \qquad (\text {B}) \ \frac {1}{4} \qquad (\text {C})\ \frac {1}{3} \qquad (\text {D}) \ \frac {1}{2} \qquad (\text {E})\ \frac {2}{3}</math><br />
<br />
== Solution ==<br />
<br />
===Solution 1===<br />
<br />
Notice that the bases of both triangles are diameters of the circle. Hence the ratio of the areas is just the ratio of the heights of the triangles, or <math>\frac{CF}{CD}</math> (<math>F</math> is the foot of the [[perpendicular]] from <math>C</math> to <math>DE</math>).<br />
<br />
Call the radius <math>r</math>. Then <math>AC = \frac 13(2r) = \frac 23r</math>, <math>CO = \frac 13r</math>. Using the [[Pythagorean Theorem]] in <math>\triangle OCD</math>, we get <math>\frac{1}{9}r^2 + CD^2 = r^2 \Longrightarrow CD = \frac{2\sqrt{2}}3r</math>. <br />
<br />
Now we have to find <math>CF</math>. Notice <math>\triangle OCD \sim \triangle OFC</math>, so we can write the [[proportion]]:<br />
<br />
<div style="text-align:center;"><math>\frac{OF}{OC} = \frac{OC}{OD}</math><br /><math>\frac{OF}{\frac{1}{3}r} = \frac{\frac{1}{3}r}{r}</math><br /><math>OF = \frac 19r</math></div><br />
<br />
By the [[Pythagorean Theorem]] in <math>\triangle OFC</math>, we have <math>\left(\frac{1}{9}r\right)^2 + CF^2 = \left(\frac{1}{3}r\right)^2 \Longrightarrow CF = \sqrt{\frac{8}{81}r^2} = \frac{2\sqrt{2}}{9}r</math>.<br />
<br />
Our answer is <math>\frac{CF}{CD} = \frac{\frac{2\sqrt{2}}{9}r}{\frac{2\sqrt{2}}{3}r} = \frac 13 \Longrightarrow \mathrm{(C)}</math>.<br />
<br />
===Solution 2===<br />
<br />
Let the center of the circle be <math>O</math>.<br />
<br />
Note that <math>2 \cdot AC = BC \Rightarrow 3 \cdot AC = AB</math>.<br />
<br />
<math>O</math> is midpoint of <math>AB \Rightarrow \frac{3}{2}AC = AO \Rightarrow CO = \frac{1}{3}AO \Rightarrow CO = \frac{1}{6} AB</math>.<br />
<br />
<math>O</math> is midpoint of <math>DE \Rightarrow</math> Area of <math>\triangle DCE = 2 \cdot</math> Area of <math>\triangle DCO = 2 \cdot (\frac{1}{6} \cdot</math> Area of <math>\triangle ABD) = \frac{1}{3} \cdot</math> Area of <math>\triangle ABD \Longrightarrow \mathrm{(C)}</math>.<br />
<br />
===Solution 3===<br />
<br />
Let <math>r</math> be the radius of the circle. Note that <math>AC+BC = 2r</math> so <math>AC = \frac{2}{3}r</math>.<br />
<br />
By [[Power of a Point Theorem]], <math>CD^2= AC \cdot BC = 2\cdot AC^2</math>, and thus <math>CD = \sqrt{2} \cdot AC = \frac{2\sqrt{2}}{3}r</math><br />
<br />
Then the area of <math>\triangle ABD</math> is <math>\frac{1}{2} AB \cdot CD = \frac{2\sqrt{2}}{3}r^2</math>. Similarly, the area of <math>\triangle DCE</math> is <math>\frac{1}{2}(r-AC) \cdot 2 \cdot CD = \frac{2\sqrt{2}}{9}r^2</math>, so the desired ratio is <math>\frac{\frac{2\sqrt{2}}{9}r^2}{\frac{2\sqrt{2}}{3}r^2} = \frac{1}{3} \Longrightarrow \mathrm{(C)}</math><br />
<br />
===Solution 4===<br />
<asy><br />
unitsize(2.5cm);<br />
defaultpen(fontsize(10pt)+linewidth(.8pt));<br />
dotfactor=3;<br />
pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);<br />
pair D=dir(aCos(C.x)), e=(-D.x,-D.y);<br />
pair H=(e.x,0);<br />
draw(A--B--D--cycle);<br />
draw(D--e--C);<br />
draw(unitcircle,white);<br />
drawline(D,C);<br />
dot(O);<br />
clip(unitcircle);<br />
draw(unitcircle);<br />
label("$E$",e,SSE);<br />
label("$B$",B,NE);<br />
label("$A$",A,W);<br />
label("$D$",D,NNW);<br />
label("$C$",C,SW);<br />
label("$H$",H,SE);<br />
draw(e--H,dashed);<br />
label("O",(0,0),NE);<br />
label("1",(C--O),N);<br />
label("1",(H--O),N);<br />
label("2",(A--C),N);<br />
label("2",(H--B),N);<br />
label("3",(O--D),NE);<br />
label("3",(O--e),NE);<br />
label("$2\sqrt{2}$",(D--C),W);<br />
label("$2\sqrt{2}$",(H--e),E);<br />
draw(rightanglemark(e,(e.x,0),A,2));<br />
draw(rightanglemark(D,C,B,2));</asy><br />
<br />
Let the center of the circle be <math>O</math>.<br />
Without loss of generality, let the radius of the circle be equal to <math>3</math>. Thus, <math>AO=3</math> and <math>OB=3</math>. As a consequence of <math>2(AC)=BC</math>, <math>AC=2</math> and <math>CO=1</math>. Also, we know that <math>DO</math> and <math>OE</math> are both equal to <math>3</math> due to the fact that they are both radii. Thus from the Pythagorean Theorem, we have DC being equal to <math>\sqrt{3^2-1^2}</math> or <math>2\sqrt{2}</math>. Now we know that the area of <math>[ABD]</math> is equal to <math>\frac{(3+2+1)(2\sqrt{2})}{2}</math> or <math>6\sqrt{2}</math>. Know we need to find the area of <math>[DCE]</math>. By simple inspection <math>[COD]</math> <math>\cong</math> <math>[HOE]</math> due to angles being equal and CPCTC. Thus <math>HE=2\sqrt{2}</math> and <math>OH=1</math>. Know we know the area of <math>[CHE]=\frac{(1+1)(2\sqrt{2})}{2}</math> or <math>2\sqrt{2}</math>. We also know that the area of <math>[OHE]=\frac{(1)(2\sqrt{2})}{2}</math> or <math>\sqrt{2}</math>. Thus the area of <math>[COE]=2\sqrt{2}-\sqrt{2}</math> or <math>\sqrt{2}</math>. We also can calculate the area of <math>[DOC]</math> to be <math>\frac{(1)(2\sqrt{2})}{2}</math> or <math>\sqrt{2}</math>. Thus <math>[DCE]</math> is equal to <math>[COE]</math> + <math>[DOC] </math> or <math>\sqrt{2}+\sqrt{2}</math> or <math>2\sqrt{2}</math>. The ratio between <math>[DCE]</math> and <math>[ABD]</math> is equal to <math>\frac{2\sqrt{2}}{6\sqrt{2}}</math> or <math>\frac{1}{3}</math> <math>\Longrightarrow \mathrm{(C)}</math>.<br />
===Solution 5===<br />
We will use the shoelace formula. Our origin is the center of the circle. Denote the ordered pair for <math>D (x,y)</math>, and notice how <math>E</math> is a 180 degree rotation of <math>D</math>, using the rotation matrix formula we get <math>E = (-x,-y)</math>. WLOG say that this circle has radius <math>3</math>. We can now find points <math>A,B, and C</math> which are <math>(-3,0)</math>, <math>(-1,0)</math>, and <math>(3,0)</math> respectively.<br />
By shoelace the area of <math>CED</math> is <math>Y</math>, and the area of <math>ADB</math> is <math>3Y</math>. Using division we get that the answer is <math> \mathrm{(C)}</math>.<br />
<br />
== See also ==<br />
{{AMC12 box|year=2005|num-b=14|num-a=16|ab=A}}<br />
<br />
[[Category:Introductory Geometry Problems]]<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=110464List of BBCodes2019-10-20T13:15:27Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[cpp]</code></td><td>Formats and highlights the text as C++ code</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[javascript]</code></td><td>Formats and highlights the text as JavaScript code.</td><br />
</tr><br />
<tr><br />
<td><code>[latex]</code></td><td>Formats and highlights the text as LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[tex]</code></td><td>Formats and highlights the text as TeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table><br />
<br />
For examples on how these tags are used, one can see the usage for most of these BBCode tags [https://artofproblemsolving.com/community/c10h1892169 here.]</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=110183List of BBCodes2019-10-08T11:34:14Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[cpp]</code></td><td>Formats and highlights the text as C++ code</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[latex]</code></td><td>Formats and highlights the text as LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[tex]</code></td><td>Formats and highlights the text as TeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Spam&diff=110182AoPS Wiki:Spam2019-10-08T11:32:50Z<p>Nikoislife: </p>
<hr />
<div>'''Spam''' is a message, post, or article that is, for the most part, irrelevant. Context is the best guide for relevance.<br />
<br />
<br />
==Spam on the AoPS message boards==<br />
Spam on the [http://www.artofproblemsolving.com/Forum/index.php?f=155 AoPS message boards] can usually be classified as one of two major types. However, there may be more types which we would appreciate if you refrained from doing on the AoPS message boards. The first type is advertising and the second is to raise one's post count or be annoying or unproductive. The first is grounds for an immediate ban from the message boards and will always be deleted when caught. The second is generally discouraged, but the consequences are not so harsh.<br />
<br />
==Examples of spam==<br />
<br />
Spam can include:<br />
<br />
*Needlessly repeating the same word / letter / phrase over and over<br />
*Useless off-topic posting<br />
*Violating [http://www.artofproblemsolving.com/company/tos ToS] guidelines<br />
*Advertising<br />
*Using post extenders (i.e. /8char)<br />
*Posting deliberately to wreak havoc (Trolling)<br />
<br />
==Spam on AoPSWiki==<br />
There is zero tolerance for spam on the AoPSWiki. Examples of spam include content irrelevant to a topic in that particular article, posting links to irrelevant websites, etc.<br />
<br />
However, the AoPSWiki is not opposed to relevant commercial links, so long as they are appropriately placed and topical.<br />
<br />
The following is always considered spam:<br />
* Links to websites that are collections of links and advertisements. Simply link the most appropriate links where they belong in the [[Main_Page|AoPSWiki]].<br />
<br />
<br />
<br />
[[Category:AoPSWiki|Spam]]</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_27&diff=1101711988 AHSME Problems/Problem 272019-10-07T08:03:37Z<p>Nikoislife: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
In the figure, <math>AB \perp BC, BC \perp CD</math>, and <math>BC</math> is tangent to the circle with center <math>O</math> and diameter <math>AD</math>. <br />
In which one of the following cases is the area of <math>ABCD</math> an integer?<br />
<br />
<asy><br />
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2));<br />
draw(unitcircle);<br />
dot(O);<br />
draw(A--B--C--D--A);<br />
label("$A$",A,dir(A));<br />
label("$B$",B,dir(B));<br />
label("$C$",C,dir(C));<br />
label("$D$",D,dir(D));<br />
label("$O$",O,dir(45));<br />
</asy><br />
<br />
<math>\textbf{(A)}\ AB=3, CD=1\qquad<br />
\textbf{(B)}\ AB=5, CD=2\qquad<br />
\textbf{(C)}\ AB=7, CD=3\qquad\\<br />
\textbf{(D)}\ AB=9, CD=4\qquad<br />
\textbf{(E)}\ AB=11, CD=5 </math><br />
<br />
<br />
<br />
==Solution==<br />
Let <math>E</math> and <math>F</math> be the intersections of lines <math>AB</math> and <math>BC</math> with the circle. One can prove that <math>BCDE</math> is a rectangle, so <math>BE=CD</math>.<br />
<br />
In order for the area of trapezoid <math>ABCD</math> to be an integer, the expression <math>\frac{(AB+CD)BC}2=(AB+CD)BF</math> must be an integer, so <math>BF</math> must be rational.<br />
<br />
By Power of a Point, <math>AB\cdot BE=BF^2\implies AB\cdot CD=BF</math>, so <math>AB\cdot CD</math> must be a perfect square. Among the choices, the only one where <math>AB\cdot CD</math> is a perfect square is <math>\textbf{(D)}\ AB=9, CD=4</math><br />
<br />
== See also ==<br />
{{AHSME box|year=1988|num-b=26|num-a=28}} <br />
<br />
[[Category: Intermediate Geometry Problems]]<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_27&diff=1101701988 AHSME Problems/Problem 272019-10-07T07:53:50Z<p>Nikoislife: </p>
<hr />
<div>==Problem==<br />
<br />
In the figure, <math>AB \perp BC, BC \perp CD</math>, and <math>BC</math> is tangent to the circle with center <math>O</math> and diameter <math>AD</math>. <br />
In which one of the following cases is the area of <math>ABCD</math> an integer?<br />
<br />
<asy><br />
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2));<br />
draw(unitcircle);<br />
dot(O);<br />
draw(A--B--C--D--A);<br />
label("$A$",A,dir(A));<br />
label("$B$",B,dir(B));<br />
label("$C$",C,dir(C));<br />
label("$D$",D,dir(D));<br />
label("$O$",O,dir(45));<br />
</asy><br />
<br />
<math>\textbf{(A)}\ AB=3, CD=1\qquad<br />
\textbf{(B)}\ AB=5, CD=2\qquad<br />
\textbf{(C)}\ AB=7, CD=3\qquad\\<br />
\textbf{(D)}\ AB=9, CD=4\qquad<br />
\textbf{(E)}\ AB=11, CD=5 </math><br />
<br />
<br />
<br />
==Solution==<br />
Using power of a point, we easily notice that the product of AB and CD has to be a perfect square. <math>\textbf{(D)}\ AB=9, CD=4</math><br />
<br />
== See also ==<br />
{{AHSME box|year=1988|num-b=26|num-a=28}} <br />
<br />
[[Category: Intermediate Geometry Problems]]<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_27&diff=1101691988 AHSME Problems/Problem 272019-10-07T07:53:32Z<p>Nikoislife: </p>
<hr />
<div>==Problem==<br />
<br />
In the figure, <math>AB \perp BC, BC \perp CD</math>, and <math>BC</math> is tangent to the circle with center <math>O</math> and diameter <math>AD</math>. <br />
In which one of the following cases is the area of <math>ABCD</math> an integer?<br />
<br />
<asy><br />
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2));<br />
draw(unitcircle);<br />
dot(O);<br />
draw(A--B--C--D--A);<br />
label("$A$",A,dir(A));<br />
label("$B$",B,dir(B));<br />
label("$C$",C,dir(C));<br />
label("$D$",D,dir(D));<br />
label("$O$",O,NE);<br />
</asy><br />
<br />
<math>\textbf{(A)}\ AB=3, CD=1\qquad<br />
\textbf{(B)}\ AB=5, CD=2\qquad<br />
\textbf{(C)}\ AB=7, CD=3\qquad\\<br />
\textbf{(D)}\ AB=9, CD=4\qquad<br />
\textbf{(E)}\ AB=11, CD=5 </math><br />
<br />
<br />
<br />
==Solution==<br />
Using power of a point, we easily notice that the product of AB and CD has to be a perfect square. <math>\textbf{(D)}\ AB=9, CD=4</math><br />
<br />
== See also ==<br />
{{AHSME box|year=1988|num-b=26|num-a=28}} <br />
<br />
[[Category: Intermediate Geometry Problems]]<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_27&diff=1101681988 AHSME Problems/Problem 272019-10-07T07:53:08Z<p>Nikoislife: </p>
<hr />
<div>==Problem==<br />
<br />
In the figure, <math>AB \perp BC, BC \perp CD</math>, and <math>BC</math> is tangent to the circle with center <math>O</math> and diameter <math>AD</math>. <br />
In which one of the following cases is the area of <math>ABCD</math> an integer?<br />
<br />
<asy><br />
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2));<br />
draw(unitcircle);<br />
dot(O);<br />
draw(A--B--C--D--A);<br />
label("$A$",A,dir(A));<br />
label("$B$",B,dir(B));<br />
label("$C$",C,dir(C));<br />
label("$D$",D,dir(D));<br />
label("$O$",O,N);<br />
</asy><br />
<br />
<math>\textbf{(A)}\ AB=3, CD=1\qquad<br />
\textbf{(B)}\ AB=5, CD=2\qquad<br />
\textbf{(C)}\ AB=7, CD=3\qquad\\<br />
\textbf{(D)}\ AB=9, CD=4\qquad<br />
\textbf{(E)}\ AB=11, CD=5 </math><br />
<br />
<br />
<br />
==Solution==<br />
Using power of a point, we easily notice that the product of AB and CD has to be a perfect square. <math>\textbf{(D)}\ AB=9, CD=4</math><br />
<br />
== See also ==<br />
{{AHSME box|year=1988|num-b=26|num-a=28}} <br />
<br />
[[Category: Intermediate Geometry Problems]]<br />
{{MAA Notice}}</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=110166List of BBCodes2019-10-07T03:35:12Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[latex]</code></td><td>Formats and highlights the text as LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[tex]</code></td><td>Formats and highlights the text as TeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=110165List of BBCodes2019-10-07T03:33:22Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[latex]</code></td><td>Formats and highlights the text as LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[tex]</code></td><td>Formats and highlights the text as TeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=Asymptote:_Help&diff=110164Asymptote: Help2019-10-07T03:30:15Z<p>Nikoislife: </p>
<hr />
<div>{{Asymptote}}<br />
<br />
==Troubleshooting==<br />
'''Problem 1:''' You have added the Asymptote tool to TeXnicCenter, but when you hit your hotkey, you get an 'Cannot execute the command' error. <br />
<br />
'''What might be going on:''' TeXnicCenter doesn't see the Asymptote executable.<br />
<br />
'''Possible Solution:''' Go back to the Tools -> Customize window (click Tools, then Customize in TeXnicCenter). Then, click the Tools tab, then select Asymptote. Instead of cutting and pasting the text for the Command line, click the ... button after the Command entry box. Then, navigate to the asy.exe executable. You'll probably have to go up several levels to C:, then click Program Files, then Asymptote, then the file asy with the stylized red A next to it.<br />
<br />
<br />
'''Problem 2:''' I have everything installed and the hotkey doesn't give me an error in TeXnicCenter, but when I hit the hotkey to compile my code, all that happens is that a window flashes briefly then disappears. Nothing else seems to happen.<br />
<br />
'''What might be going on:''' Your file is saved as a .tex file, not a .asy file. (TeXnicCenter will even make it look like it's an .asy file, but it's really saved as an .asy.tex file.)<br />
<br />
'''Solution:''' Click file, then Save As, then type your filename with the .asy extension (such as test.asy), the set the 'Save as type' field to 'All files'. This should make it a .asy file. <br />
<br />
If this does not work then (for Windows XP) go to My Computer, Tools, Folder Options..., File Types. Click on New and type ASY, then OK. Making sure that the file type ASY is highlighted, click the Change button to the right of "Opens with:". From here go to browse and find the Asymptote executable program file click Open then OK. Now, retry the File, Save As extension change method outlined above.<br />
<br />
It would also help to show file extensions so that <tt>file.tex, file.asy, file.pdf</tt> aren't all listed in Explorer and Save boxes as <tt>file</tt>.<br />
<br />
[http://teamtutorials.com/windows-tutorials/show-file-extension-in-windows-xp Show File Extension in Windows XP] gives a simple walkthrough to show how to do this.<br />
<br />
<br />
'''Problem 3:''' You can draw images with Asymptote, but it doesn't compile your file whenever you have a label containing LaTeX code (for example, the line <tt>label("<math>P</math>",(50,50),S);</tt> as in test.asy.)<br />
<br />
'''What might be going on:''' Either Asymptote is not finding your LaTeX compiler, or you have installed your TeX distribution (such as MikTeX) in a folder that Asymptote cannot find.<br />
<br />
'''Solution:''' Let's say you installed miktex in the folder <tt>C:\Program Files\MiKTeX 2.6</tt> (the default). (If you have a different version of MiKTeX, change the 2.6 to the appropriate number. Look in C:\Program Files for the MiKTeX directory to see what this number should be.) Then there should be a subfolder miktex, containing the folder bin, which contains your latex compiler. <br />
When Asymptote is run, it looks for a file called config.asy in the folder <br />
<tt>C:\Documents and Settings\%USERNAME%\.asy\</tt> <br />
where <tt>%USERNAME%</tt> is your Windows user name. (In Windows Vista, this folder is <tt>C:\Users\%USERNAME%\.asy\</tt>, where <tt>%USERNAME%</tt> is your Windows username.) <br />
Double-click on the .asy folder, right click and choose New <math>\rightarrow</math> Text Document. This will create a file called New Text Document.txt. Right-click on this file to rename it as config.asy, then open this file with Notepad or a similar editor. Type or copy-and-paste the following lines into the document:<br />
import settings;<br />
texpath="C:\Program Files\MiKTeX 2.6\miktex\bin";<br />
dvips="C:\Program Files\MiKTeX 2.6\miktex\bin\dvips";<br />
When you have finished, save the file config.asy.<br />
<br />
As mentioned in the solution to Problem 1, you must make sure that the config.asy file is an ASY file, and not a TXT file that is named config.asy. Look in the File Type column in the Details view to see what type of file config.asy is. If your regular text editor insists on making config.asy a TXT document, use TeXnicCenter to create config.asy.<br />
<br />
<br />
'''Problem 4:''' You run Asymptote and get <tt>error: could not load module 'plain'</tt>. <br />
<br />
'''What might be going on:''' Asymptote cannot find the files it needs to run. It may need telling which directory it is installed in.<br />
<br />
'''Solution:''' You need to create or edit the config.asy file. See the solution to Problem 3 for how to find or create it. Assuming that Asymptote is installed at <tt>C:\Program Files\Asymptote</tt> then<br />
<br />
a) if you already have a config.asy file add the following line somewhere after "import settings;": <br />
dir="C:\Program Files\Asymptote";<br />
''or''<br />
<br />
b) if you have created a new config.asy file then add the following lines:<br />
import settings;<br />
dir="C:\Program Files\Asymptote";<br />
<br />
See the note at the end of Problem 3 regarding creating config.asy.<br />
<br />
<br />
'''Problem 4b:''' You run Asymptote and get<br/><br />
<tt>error: could not load module 'plain'<br/><br />
error: could not load module 'file'</tt>. <br />
<br />
'''What might be going on:''' Asymptote does not like file names that contain spaces.<br />
<br />
'''Solution:''' Replace all spaces in the file name with underscores.<br />
<br />
<br />
'''Problem 5:''' Asymptote suddenly starts misbehaving giving strange errors, going into infinite loops, and not recognizing standard package commands.<br />
<br />
'''What might be going on:''' A file with the same name as one of the standard package files (like graph.asy or math.asy) was created either by you or by <math>\LaTeX</math> in your working directory or some other directory which Asymptote searches before going to its own directory, so that file is imported instead of the package file.<br />
<br />
'''Prevention:''' Never name your asy or tex files graph.asy, math.tex and such (these two names seem to be the most common)<br />
<br />
'''Solution:''' Find all extra files with the same name as standard packages and either .asy or .tex extension and rename or delete them.<br />
<br />
==External Sources==<br />
The following are excellent resources on Asymptote for topics not covered in this guide:<br />
<br />
*[http://asymptote.sourceforge.net The Asymptote home page]<br />
* [http://asymptote.sourceforge.net/doc/ Asymptote documentation]<br />
* [http://asymptote.sourceforge.net/FAQ/ Asymptote FAQ]<br />
* [http://sourceforge.net/forum/forum.php?forum_id=409349 Asymptote forum]<br />
<br />
[[Asymptote: Useful functions|Next: Useful functions]]</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=109296List of BBCodes2019-08-23T11:07:23Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[tex]</code></td><td>Formats and highlights the text as TeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=108687List of BBCodes2019-08-10T11:54:47Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=108686List of BBCodes2019-08-10T11:54:39Z<p>Nikoislife: Blanked the page</p>
<hr />
<div></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=108685List of BBCodes2019-08-10T11:52:11Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
Please note that this list might not be complete, as there may be some BBCode tags which aren't listed here. If you discover any new BBCode tag that's not listed here, feel free to edit and add it in the list below.<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=108684List of BBCodes2019-08-10T11:48:04Z<p>Nikoislife: </p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=BBCode:Tutorial&diff=108683BBCode:Tutorial2019-08-10T11:47:08Z<p>Nikoislife: </p>
<hr />
<div>=The Basics=<br />
[[Bbcode|BBCode]] is a way to format text in forum posts. It is in some ways similar to HTML. It uses tags, which are little pieces of code enclosed in brackets, such as [b] to make bold text. For every tag, you need a closing tag too. The closing tag is the same as the opening tag except that it has a / symbol at the front. For example, the tag [/b] is code for "stop making the text bold." There is text between the opening and closing tags, that is the text that is modified by the tags. <br />
<br />
For example, If you want to say "I love pizza" and want the word "love" to be in bold, you would type<br />
<br />
:<code>I [b]love[/b] pizza</code>.<br />
<br />
This would create, "I <b>love</b> pizza."<br />
<br />
Below are some of the most commonly used BBCode tags around AoPS forums. For a comprehensive list of all BBCode tags, see [[List of BBCodes|List of BBCode tags]]<br />
<br />
=Writing Messages in the AoPS Community=<br />
<br />
==The Buttons==<br />
[[File:Post a reply.PNG|800px]]<br />
<br />
This is a picture of what you see when you create a topic or post a reply. On very small screens (such as a smartphone) these buttons will not appear, so you will have to type the BBCode in manually.<br />
===The Bold Button===<br />
The [[file:Bold button.jpeg|25px]] button is for bold tags. When you click on it, your screen should look like this:<br />
<br />
[[File:Clicked on bold button.PNG|800px]]<br />
<br />
These are called bold tags. Then, you can type in any text you want:<br />
[[File:Type in bold text.PNG|800px]]<br />
<br />
When you submit it, it should look like this:<br />
[[File:Submitted Bold Text.PNG]]<br />
<br />
===The Italics Button===<br />
The [[File:Italics button.jpeg|25px]] button is the italics button. When you click on it, your screen should look like this:<br />
<br />
[[File:Clicked on italics button.jpeg|700px]]<br />
<br />
Then you can type in your italicized message:<br />
<br />
[[File:Italics message.jpeg|200px]]<br />
<br />
When you submit, it should look like this:<br />
<br />
[[File:Italics result.jpeg|700px]]<br />
<br />
===The Underline Button===<br />
The [[File:Underline button.jpeg|25px]] button is for underlining text. When you click on it, the symbols [u] and [/u] appear. Then you can type in whatever you want to be underlined:<br />
<br />
[[File:Underline typing.jpeg|300px]]<br />
<br />
When you submit it, it should look like this:<br />
<br />
[[File:Underline result.jpeg|700px]]<br />
<br />
Note: you can also highlight text and then click the underline button.<br />
<br />
===The Quote Button===<br />
The [[File:Quote button.jpeg|25px]] button is for quoting text. You can write an anonymous quote, or quote a specific user or person. When you click on it, you get [quote] and [/quote]. Then type in the quote:<br />
<br />
[[File:Typing generic quote.jpeg|300px]]<br />
<br />
When you hit submit, it looks like this:<br />
<br />
[[File:Published generic quote.jpeg|600px]]<br />
<br />
Or, alternatively, you can say who quoted it:<br />
<br />
[[File:Specific quote typing.jpeg|400px]]<br />
<br />
And you get this:<br />
<br />
[[File:Specific quote published.jpeg|600px]]<br />
<br />
You can also quote a post by clicking this button in the forum.<br />
<br />
[[File:Arrow to quote button.jpeg|650px]]<br />
<br />
You will be taken to a reply page that looks like this:<br />
<br />
[[File:Specific quote typing.jpeg|400px]]<br />
<br />
===The Size Button===<br />
The button with the three T's is for controlling the font size. You can click on either "Too Small", "Small", "Medium", "Large", or "Too Large" depending on what size you want. After you click on one of them, you should get something like [size=75] and [/size].<br />
<br />
===The Hide Button===<br />
The [[File:Hide button.jpeg|25px]] button embeds text in a little interactive clicky. This is useful if you want your text only visible if the reader wants to see it. For example, if you are answering a math problem, you can hide the text so that you don't give away the answer to someone who doesn't want to know it.<br />
<br />
Start by clicking the hide button. You should get [hide] and [/hide].<br />
Then put in your text:<br />
<br />
[[File:Type hide generic.jpeg|600px]]<br />
<br />
When you submit, you get this:<br />
<br />
[[File:Click for hidden text.jpeg|400px]]<br />
<br />
If you click on the words "Hidden Text", you get this:<br />
<br />
[[File:Open hidden text.jpeg|400px]]<br /><br />
<br />
Or you can label your hidden text. First, write in the code of the hidden text something like:<br />
:<code>[hide=answer]Because 6x=y, and y is 6, then x must be 1.[/hide]</code><br />
<br />
<br />
And then submit, you get:<br /><br />
<br />
[[File:Labeled closed hidden text.jpeg|300px]]<br />
<br />
[[File:Published open hidden text.jpeg|400px]]<br />
<br />
===The Code Button===<br />
The [[File:Code_button.JPEG|25px]] button is for posting code that you don't want to be rendered, ie [[LaTeX]] or [[BBCode]]. When you click on it, your screen should display:<br />
<br />
:<code>[code][/code]</code><br />
<br />
Type in your code:<br />
<br />
:<code>[code]$10+9=21$[/code]</code><br />
<br />
And it should display:<br />
<br />
[[File:code submitted.JPEG]]<br />
<br />
Notice that the LaTeX code is not rendered; that is, it does not show up as <math>10+9=21</math>.<br />
<br />
===The Image Button===<br />
The [[File:Image Button.JPEG|25px]] button is for embedding images in posts. If you have this image:<br /><br />
http://artofproblemsolving.com/assets/images/logo-ludicrous.gif<br /><br />
And you want it to show up like this:<br />
<br />
[[File:AoPS loading.GIF]]<br />
<br />
You use [img] tags. Click on the [[File:Image Button.JPEG|25px]] button and your screen should show:<br />
<br />
:<code>[img][/img]</code><br />
<br />
Now insert the link to the image. Make sure it is a [http://www.artofproblemsolving.com/Wiki/index.php/Direct_Image_Link direct link].<br /><br />
:<code>[img]<nowiki>http://artofproblemsolving.com/assets/images/logo-ludicrous.gif[/img]</nowiki></code><br /><br />
When you submit, you should get the image!<br /><br />
<br />
===The URL Button===<br />
The [[File:Link Button.PNG]] button is for parsing and naming URLs. If you click it, your screen should look like this:<br />
<br />
:<code>[url][/url]</code><br />
<br />
Now type in your URL:<br /><br />
:<code><nowiki>[url]http://www.aops.com/[/url]</nowiki></code><br />
<br />
When you submit it, it should be parsed, like this:<br />
<br />
[http://www.aops.com/ http://www.aops.com/]<br />
<br />
Alternatively, if you want a link that goes somewhere but is not actually named the URL, do this:<br />
<br />
<br />
:<code><nowiki>[url=http://www.aops.com/] Click here.[/url]</nowiki></code><br />
<br />
You get<br />
<br />
[http://www.aops.com/ Click here.]<br />
<br />
==Other Stuff==<br />
There are other things you can do with BBCode that do not have buttons in the AoPS communtiy.<br />
<br />
<br />
<br />
===Unordered List===<br />
An unordered list is a simple bullet point list. To use it, you simply use the [list] tag. Before each new list item, type [*]. The [*] tag is for a list item. You don't need a closing tag for the [*] tag. For example, typing<br />
<br />
:<code><br />
:[list]<br />
:[*] pizza<br />
:[*] pasta<br />
:[*] rice<br />
:[*] cereal<br />
:[/list]</code><br />
creates:<br />
<ul><br />
<li>pizza</li><br />
<li>pasta</li><br />
<li>rice</li><br />
<li>cereal</li></ul><br />
<br />
===Ordered List===<br />
The bbcode for the following uses the [list] tag. To tell you want a numbered list, type [list=1]. For example, typing, <br />
:<code><br />
:Top 6 most important people of all time:<br />
:[list=1]<br />
:[*]Gmass<br />
:[*]Richard Rusczyk<br />
:[*]David Patrick<br />
:[*]Isaac Newton<br />
:[*]George Washington<br />
:[*]Thomas Edison<br />
:[/list]</code><br />
<br />
creates<br />
<br />
:Top 6 most important people of all time:<br />
<ol><br />
:<li>Gmass</li><br />
:<li>Richard Rusczyk</li><br />
:<li>David Patrick</li><br />
:<li>Isaac Newton</li><br />
:<li>George Washington</li><br />
:<li>Thomas Edison</li><br />
</ol><br />
<br />
Alternatively, if you want to make a lettered list, start with [list=A] or [list=a]. For a roman numeral list, use [list=i] or [list=I].<br />
<br />
===Python Tags===<br />
If you are taking an AoPS programming with a Python class, you may need to post some Python code in the class forum. To do this you use the [python] tags.<br />
These tags are like the [code] tags except that they show the python code in colors.<br />
For example, typing<br />
<br />
:<code>[python]<br />
:print("hello"+" world")<br />
:print("Hello, world!")<br />
:[/python]<br />
</code><br />
<br />
Yields<br />
<br />
[[File: Python tags.PNG]]<br />
<br />
If you want to have a PYTHON renderer in a post in a forum, you can use [pywindow] tags. <br />
For example, typing<br />
<br />
:<code>[pywindow]<br />
:print("Hello World!")<br />
:[/pywindow]</code><br />
<br />
yields a code box that can be run and edited.<br />
<br />
===Aops Font===<br />
<i>Main article: [[AoPS font]]</i><br />
<br />
Say you want to type the <span class="aops-font">Y</span> symbol on a forum post. You can use the [aops][/aops] tags to create symbols like this. For example, typing <br />
<br />
:<code>[aops]Y[/aops]</code><br />
<br />
makes the <span class="aops-font">Y</span> symbol. For a guide to how to type other symbols, see the [[AoPS font]] page.<br />
<br />
===Strike Through===<br />
The Strike Through tag looks like this:<br />
<ol><br />
Tags:<br />
<li>[[File:Strike_Through.jpg]]</li><br />
Content:<br />
<li>[[File:Strike_Through2.jpg]]</li><br />
Result:<br />
<li>[[File:Strike_Through3.jpg]]</li></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=108682List of BBCodes2019-08-10T11:46:33Z<p>Nikoislife: /* Known BBCodes */</p>
<hr />
<div>This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=User:Nikoislife&diff=108659User:Nikoislife2019-08-10T01:01:23Z<p>Nikoislife: </p>
<hr />
<div>owo<br />
<br />
<font style="border-style:solid;border-width:10px 20px 30px 40px;border-color:orange yellow violet pink">Yayayayayay</font><br />
<br />
Don't mind me.</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=108658List of BBCodes2019-08-10T00:54:32Z<p>Nikoislife: /* Known BBCodes */</p>
<hr />
<div>=Known BBCodes=<br />
<br />
This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Inserts a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>$</code></td><td>Inserts a dollar sign</td><br />
</tr><br />
<tr><br />
<td><code>\<b></b>\</code></td><td>Inserts a line break.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=List_of_BBCodes&diff=108657List of BBCodes2019-08-10T00:50:53Z<p>Nikoislife: Created page with "=Known BBCodes= This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, se..."</p>
<hr />
<div>=Known BBCodes=<br />
<br />
This article shows a list of BBCodes that could be used in AoPS forums. For a much more detailed explanation on how to use some of the common tags on AoPS, see [[BBCode:Tutorial|BBCode Tutorial]].<br />
<br />
A summary of the known BBCode tags on AoPS are the following:<br />
<br />
<table><br />
<tr><br />
<td><code>[*]</code></td><td>List item. This tag is used with <code>[list]</code>.</td><br />
</tr><br />
<tr><br />
<td><code>[aops]</code></td><td>Changes the text's font face to AoPS font.</td><br />
</tr><br />
<tr><br />
<td><code>[aopshtml]</code></td><td>Renders HTML code. This feature is exclusive only for site admins.</td><br />
</tr><br />
<tr><br />
<td><code>[aopsnowrap]</code></td><td>Prevents text from breaking into a new line.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[asy]</code></td><td>Renders Asymptote code.</td><br />
</tr><br />
<tr><br />
<td><code>[b]</code></td><td>Formats the text in bold style.</td><br />
</tr><br />
<tr><br />
<td><code>[br]</code></td><td>Adds a line break.</td><br />
</tr><br />
<tr><br />
<td><code>[c]</code></td><td>Formats and highlights the text as C code.</td><br />
</tr><br />
<tr><br />
<td><code>[center]</code></td><td>Aligns text to the center.</td><br />
</tr><br />
<tr><br />
<td><code>[code]</code></td><td>Formats the text in monospace font.</td><br />
</tr><br />
<tr><br />
<td><code>[css]</code></td><td>Formats and highlights the text as CSS code.</td><br />
</tr><br />
<tr><br />
<td><code>[email]</code></td><td>Turns the text into a hyperlink with the specified E-mail address.</td><br />
</tr><br />
<tr><br />
<td><code>[font=(font face)]</code></td><td>Changes the text's font to the specified font style.</td><br />
</tr><br />
<tr><br />
<td><code>[haskell]</code></td><td>Formats and highlights the text as Haskell code.</td><br />
</tr><br />
<tr><br />
<td><code>[hide]</code></td><td>Hides the text using a hide tag with the specified title.</td><br />
</tr><br />
<tr><br />
<td><code>[html]</code></td><td>Formats and highlights the text as HTML code.</td><br />
</tr><br />
<tr><br />
<td><code>[i]</code></td><td>Formats the text in italics style.</td><br />
</tr><br />
<tr><br />
<td><code>[img width=(width%) height=(height%) alt=(alt)]</code></td><td>Inserts images with specified percent width and percent height, and the alternate name.</td><br />
</tr><br />
<tr><br />
<td><code>[indent]</code></td><td>Inserts an indent before the text.</td><br />
</tr><br />
<tr><br />
<td><code>[java]</code></td><td>Formats and highlights the text as Java code.</td><br />
</tr><br />
<tr><br />
<td><code>[left]</code></td><td>Aligns text to the left.</td><br />
</tr><br />
<tr><br />
<td><code>[list]</code></td><td>Creates a bulleted list, or a numbered list with the specified type. <math>\text{[*]}</math> is used to indicate a list item.</td><br />
</tr><br />
<tr><br />
<td><code>[math]</code></td><td>Renders LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>[php]</code></td><td>Formats and highlights the text as PHP code.</td><br />
</tr><br />
<tr><br />
<td><code>[python]</code></td><td>Formats and highlights the text as Python code.</td><br />
</tr><br />
<tr><br />
<td><code>[pywindow]</code></td><td>Creates an interactive Python window.</td><br />
</tr><br />
<tr><br />
<td><code>[quote=(name)]</code></td><td>Inserts text as a quote with the specified name.</td><br />
</tr><br />
<tr><br />
<td><code>[ruby]</code></td><td>Formats and highlights the text as Ruby code.</td><br />
</tr><br />
<tr><br />
<td><code>[rule]</code></td><td>Creates a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>[right]</code></td><td>Aligns text to the right.</td><br />
</tr><br />
<tr><br />
<td><code>[u]</code></td><td>Adds underline to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[url=(url)]</code></td><td>Turns text into hyperlink with the specified URL.</td><br />
</tr><br />
<tr><br />
<td><code>[s]</code></td><td>Adds strikethrough to the text.</td><br />
</tr><br />
<tr><br />
<td><code>[size=(size%)]</code></td><td>Changes the size of the text in the specified percent size.</td><br />
</tr><br />
<tr><br />
<td><code>[verbatim]</code></td><td>Writes plaintext without any BBCode formatting.</td><br />
</tr><br />
<tr><br />
<td><code>[youtube]</code></td><td>Embeds a YouTube video.</td><br />
</tr><br />
<tr><br />
<td><code>-----</code></td><td>Creates a horizontal line.</td><br />
</tr><br />
<tr><br />
<td><code>$</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>$$</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\(</code></td><td>Renders inline LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\[</code></td><td>Renders display mode LaTeX code.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align}</code></td><td>Renders LaTeX code using the align environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{align*}</code></td><td>Renders LaTeX code using the align* environment.</td><br />
</tr><br />
<tr><br />
<td><code>\begin{tabular}</code></td><td>Renders LaTeX code using the tabular environment.</td><br />
</tr><br />
<tr><br />
<td><code>\LaTeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\TeX</code></td><td>Inserts the LaTeX logo.</td><br />
</tr><br />
<tr><br />
<td><code>\today</code></td><td>Inserts the date today.</td><br />
</tr><br />
</table></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=BBCode:Tutorial&diff=108656BBCode:Tutorial2019-08-10T00:27:50Z<p>Nikoislife: </p>
<hr />
<div>=The Basics=<br />
[[Bbcode|BBCode]] is a way to format text in forum posts. It is in some ways similar to HTML. It uses tags, which are little pieces of code enclosed in brackets, such as [b] to make bold text. For every tag, you need a closing tag too. The closing tag is the same as the opening tag except that it has a / symbol at the front. For example, the tag [/b] is code for "stop making the text bold." There is text between the opening and closing tags, that is the text that is modified by the tags. <br />
<br />
For example, If you want to say "I love pizza" and want the word "love" to be in bold, you would type<br />
<br />
:<code>I [b]love[/b] pizza</code>.<br />
<br />
This would create, "I <b>love</b> pizza."<br />
<br />
=Writing Messages in the AoPS Community=<br />
<br />
==The Buttons==<br />
[[File:Post a reply.PNG|800px]]<br />
<br />
This is a picture of what you see when you create a topic or post a reply. On very small screens (such as a smartphone) these buttons will not appear, so you will have to type the BBCode in manually.<br />
===The Bold Button===<br />
The [[file:Bold button.jpeg|25px]] button is for bold tags. When you click on it, your screen should look like this:<br />
<br />
[[File:Clicked on bold button.PNG|800px]]<br />
<br />
These are called bold tags. Then, you can type in any text you want:<br />
[[File:Type in bold text.PNG|800px]]<br />
<br />
When you submit it, it should look like this:<br />
[[File:Submitted Bold Text.PNG]]<br />
<br />
===The Italics Button===<br />
The [[File:Italics button.jpeg|25px]] button is the italics button. When you click on it, your screen should look like this:<br />
<br />
[[File:Clicked on italics button.jpeg|700px]]<br />
<br />
Then you can type in your italicized message:<br />
<br />
[[File:Italics message.jpeg|200px]]<br />
<br />
When you submit, it should look like this:<br />
<br />
[[File:Italics result.jpeg|700px]]<br />
<br />
===The Underline Button===<br />
The [[File:Underline button.jpeg|25px]] button is for underlining text. When you click on it, the symbols [u] and [/u] appear. Then you can type in whatever you want to be underlined:<br />
<br />
[[File:Underline typing.jpeg|300px]]<br />
<br />
When you submit it, it should look like this:<br />
<br />
[[File:Underline result.jpeg|700px]]<br />
<br />
Note: you can also highlight text and then click the underline button.<br />
<br />
===The Quote Button===<br />
The [[File:Quote button.jpeg|25px]] button is for quoting text. You can write an anonymous quote, or quote a specific user or person. When you click on it, you get [quote] and [/quote]. Then type in the quote:<br />
<br />
[[File:Typing generic quote.jpeg|300px]]<br />
<br />
When you hit submit, it looks like this:<br />
<br />
[[File:Published generic quote.jpeg|600px]]<br />
<br />
Or, alternatively, you can say who quoted it:<br />
<br />
[[File:Specific quote typing.jpeg|400px]]<br />
<br />
And you get this:<br />
<br />
[[File:Specific quote published.jpeg|600px]]<br />
<br />
You can also quote a post by clicking this button in the forum.<br />
<br />
[[File:Arrow to quote button.jpeg|650px]]<br />
<br />
You will be taken to a reply page that looks like this:<br />
<br />
[[File:Specific quote typing.jpeg|400px]]<br />
<br />
===The Size Button===<br />
The button with the three T's is for controlling the font size. You can click on either "Too Small", "Small", "Medium", "Large", or "Too Large" depending on what size you want. After you click on one of them, you should get something like [size=75] and [/size].<br />
<br />
===The Hide Button===<br />
The [[File:Hide button.jpeg|25px]] button embeds text in a little interactive clicky. This is useful if you want your text only visible if the reader wants to see it. For example, if you are answering a math problem, you can hide the text so that you don't give away the answer to someone who doesn't want to know it.<br />
<br />
Start by clicking the hide button. You should get [hide] and [/hide].<br />
Then put in your text:<br />
<br />
[[File:Type hide generic.jpeg|600px]]<br />
<br />
When you submit, you get this:<br />
<br />
[[File:Click for hidden text.jpeg|400px]]<br />
<br />
If you click on the words "Hidden Text", you get this:<br />
<br />
[[File:Open hidden text.jpeg|400px]]<br /><br />
<br />
Or you can label your hidden text. First, write in the code of the hidden text something like:<br />
:<code>[hide=answer]Because 6x=y, and y is 6, then x must be 1.[/hide]</code><br />
<br />
<br />
And then submit, you get:<br /><br />
<br />
[[File:Labeled closed hidden text.jpeg|300px]]<br />
<br />
[[File:Published open hidden text.jpeg|400px]]<br />
<br />
===The Code Button===<br />
The [[File:Code_button.JPEG|25px]] button is for posting code that you don't want to be rendered, ie [[LaTeX]] or [[BBCode]]. When you click on it, your screen should display:<br />
<br />
:<code>[code][/code]</code><br />
<br />
Type in your code:<br />
<br />
:<code>[code]$10+9=21$[/code]</code><br />
<br />
And it should display:<br />
<br />
[[File:code submitted.JPEG]]<br />
<br />
Notice that the LaTeX code is not rendered; that is, it does not show up as <math>10+9=21</math>.<br />
<br />
===The Image Button===<br />
The [[File:Image Button.JPEG|25px]] button is for embedding images in posts. If you have this image:<br /><br />
http://artofproblemsolving.com/assets/images/logo-ludicrous.gif<br /><br />
And you want it to show up like this:<br />
<br />
[[File:AoPS loading.GIF]]<br />
<br />
You use [img] tags. Click on the [[File:Image Button.JPEG|25px]] button and your screen should show:<br />
<br />
:<code>[img][/img]</code><br />
<br />
Now insert the link to the image. Make sure it is a [http://www.artofproblemsolving.com/Wiki/index.php/Direct_Image_Link direct link].<br /><br />
:<code>[img]<nowiki>http://artofproblemsolving.com/assets/images/logo-ludicrous.gif[/img]</nowiki></code><br /><br />
When you submit, you should get the image!<br /><br />
<br />
===The URL Button===<br />
The [[File:Link Button.PNG]] button is for parsing and naming URLs. If you click it, your screen should look like this:<br />
<br />
:<code>[url][/url]</code><br />
<br />
Now type in your URL:<br /><br />
:<code><nowiki>[url]http://www.aops.com/[/url]</nowiki></code><br />
<br />
When you submit it, it should be parsed, like this:<br />
<br />
[http://www.aops.com/ http://www.aops.com/]<br />
<br />
Alternatively, if you want a link that goes somewhere but is not actually named the URL, do this:<br />
<br />
<br />
:<code><nowiki>[url=http://www.aops.com/] Click here.[/url]</nowiki></code><br />
<br />
You get<br />
<br />
[http://www.aops.com/ Click here.]<br />
<br />
==Other Stuff==<br />
There are other things you can do with BBCode that do not have buttons in the AoPS communtiy.<br />
<br />
<br />
<br />
===Unordered List===<br />
An unordered list is a simple bullet point list. To use it, you simply use the [list] tag. Before each new list item, type [*]. The [*] tag is for a list item. You don't need a closing tag for the [*] tag. For example, typing<br />
<br />
:<code><br />
:[list]<br />
:[*] pizza<br />
:[*] pasta<br />
:[*] rice<br />
:[*] cereal<br />
:[/list]</code><br />
creates:<br />
<ul><br />
<li>pizza</li><br />
<li>pasta</li><br />
<li>rice</li><br />
<li>cereal</li></ul><br />
<br />
===Ordered List===<br />
The bbcode for the following uses the [list] tag. To tell you want a numbered list, type [list=1]. For example, typing, <br />
:<code><br />
:Top 6 most important people of all time:<br />
:[list=1]<br />
:[*]Gmass<br />
:[*]Richard Rusczyk<br />
:[*]David Patrick<br />
:[*]Isaac Newton<br />
:[*]George Washington<br />
:[*]Thomas Edison<br />
:[/list]</code><br />
<br />
creates<br />
<br />
:Top 6 most important people of all time:<br />
<ol><br />
:<li>Gmass</li><br />
:<li>Richard Rusczyk</li><br />
:<li>David Patrick</li><br />
:<li>Isaac Newton</li><br />
:<li>George Washington</li><br />
:<li>Thomas Edison</li><br />
</ol><br />
<br />
Alternatively, if you want to make a lettered list, start with [list=A] or [list=a]. For a roman numeral list, use [list=i] or [list=I].<br />
<br />
===Python Tags===<br />
If you are taking an AoPS programming with a Python class, you may need to post some Python code in the class forum. To do this you use the [python] tags.<br />
These tags are like the [code] tags except that they show the python code in colors.<br />
For example, typing<br />
<br />
:<code>[python]<br />
:print("hello"+" world")<br />
:print("Hello, world!")<br />
:[/python]<br />
</code><br />
<br />
Yields<br />
<br />
[[File: Python tags.PNG]]<br />
<br />
If you want to have a PYTHON renderer in a post in a forum, you can use [pywindow] tags. <br />
For example, typing<br />
<br />
:<code>[pywindow]<br />
:print("Hello World!")<br />
:[/pywindow]</code><br />
<br />
yields a code box that can be run and edited.<br />
<br />
===Aops Font===<br />
<i>Main article: [[AoPS font]]</i><br />
<br />
Say you want to type the <span class="aops-font">Y</span> symbol on a forum post. You can use the [aops][/aops] tags to create symbols like this. For example, typing <br />
<br />
:<code>[aops]Y[/aops]</code><br />
<br />
makes the <span class="aops-font">Y</span> symbol. For a guide to how to type other symbols, see the [[AoPS font]] page.<br />
<br />
===Strike Through===<br />
The Strike Through tag looks like this:<br />
<ol><br />
Tags:<br />
<li>[[File:Strike_Through.jpg]]</li><br />
Content:<br />
<li>[[File:Strike_Through2.jpg]]</li><br />
Result:<br />
<li>[[File:Strike_Through3.jpg]]</li></div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality&diff=107973Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality2019-07-28T12:35:54Z<p>Nikoislife: </p>
<hr />
<div>The '''Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality''' (RMS-AM-GM-HM), is an [[inequality]] of the [[root-mean square]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says:<br />
<br />
<cmath>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</cmath><br />
<br />
with equality if and only if <math>x_1=x_2=\cdots=x_n</math>. This inequality can be expanded to the [[power mean inequality]].<br />
<br />
As a consequence we can have the following inequality:<br />
If <math>x_1,x_2,\cdots,x_n</math> are positive reals, then <br />
<cmath>(x_1+x_2+\cdots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots \frac{1}{x_n}\right) \geq n^2</cmath><br />
with equality if and only if <math>x_1=x_2=\cdots=x_n</math>; which follows directly by cross multiplication from the AM-HM inequality. This is extremely useful in problem solving.<br />
<br />
The Root Mean Square is also known as the [[quadratic mean]], and the inequality is therefore sometimes known as the QM-AM-GM-HM Inequality. <br />
<br />
== Proof ==<br />
<br />
The inequality <math>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}</math> is a direct consequence of the [[Cauchy-Schwarz Inequality]];<br />
<cmath>(x_1^2+x_2^2+\cdots +x_n^2)(1+1+\cdots +1)\geq (x_1+x_2+\cdots +x_n)^2</cmath><br />
<cmath>\frac{x_1^2+x_2^2+\cdots +x_n^2}{n}\geq \left(\frac{x_1+x_2+\cdots +x_n}{n}\right)^2</cmath><br />
<cmath>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}</cmath><br />
Alternatively, the RMS-AM can be proved using Jensen's inequality:<br />
Suppose we let <math>F(x)=x^2</math> (We know that <math>F(x)</math> is convex because <math>F'(x)=2x</math> and therefore <math>F''(x)=2>0</math>). <br />
We have:<br />
<cmath>F\left(\frac{x_1}{n}+\cdots+\frac{x_n}{n}\right)\le \frac{F(x_1)}{n}+\cdots+\frac{F(x_n)}{n}</cmath><br />
Factoring out the <math>\frac{1}{n}</math> yields:<br />
<cmath>F\left(\frac{x_1+\cdots+x_n}{n}\right)\le \frac {F(x_1)+\cdots+F(x_n)}{n}</cmath><br />
<cmath>\left(\frac{x_1+\cdots+x_n}{n}\right)^2 \le \frac{x_1^2+\cdots+x_n^2}{n}</cmath><br />
Taking the square root to both sides (remember that both are positive):<br />
<cmath>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n} \blacksquare.</cmath><br />
<br />
<br />
<br />
The inequality <math>\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}</math> is called the AM-GM inequality, and proofs can be found [[Proofs of AM-GM|here]].<br />
<br />
<br />
The inequality <math>\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</math> is a direct consequence of AM-GM; <math>\frac{\sum_{i=1}^{n}\sqrt[n]{\frac{x_1x_2\cdots x_n}{x_i^n}}}{n}\geq 1</math>, so <math>\sqrt[n]{x_1x_2\cdots x_n}\frac{\sum_{i=1}^{n}\frac{1}{x_i}}{n}\geq 1</math>, so <math>\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</math>.<br />
<br />
Therefore the original inequality is true.<br />
<br />
===Geometric Proof===<br />
<br />
<asy>size(250);<br />
pair O=(0,0),A=(-1,0),B=(0,1),C=(1,0),P=(1/2,0),Q=(1/2,sqrt(3)/2),R=foot(P,Q,O);<br />
draw(B--O--C--arc(O,C,A)--O--R--P); rightanglemark(O,P,R);<br />
draw(O--B,red);<br />
draw(P--Q,blue);<br />
draw(B--P,green);<br />
draw(R--Q,magenta);<br />
draw((A-(0,0.05))--(P-(0,0.05)),Arrows);<br />
draw((P-(0,0.05))--(C-(0,0.05)),Arrows);<br />
label("AM",(O+B)/2,W,red);<br />
label("GM",(P+Q)/2,E,blue);<br />
label("HM",(R+Q)/2,unit(P-R),magenta);<br />
label("RMS",(3B+P)/4,unit(foot(O,B,P)),green);<br />
label("$a$",(A+P)/2,3*S);<br />
label("$b$",(P+C)/2,3*S);</asy><br />
<br />
The inequality is clearly shown in this diagram for <math>n=2</math><br />
<br />
{{stub}}<br />
[[Category:Inequality]]<br />
[[Category:Theorems]]</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=User:Nikoislife&diff=107918User:Nikoislife2019-07-25T08:57:54Z<p>Nikoislife: Created page with "owo"</p>
<hr />
<div>owo</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=Legendre%27s_Formula&diff=107005Legendre's Formula2019-06-27T01:59:27Z<p>Nikoislife: </p>
<hr />
<div>'''Legendre's Formula''' states that<br />
<br />
<cmath>e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor =\frac{n-S_{p}(n)}{p-1}</cmath><br />
<br />
where <math>p</math> is a prime and <math>e_p(n)</math> is the [[exponent]] of <math>p</math> in the [[prime factorization]] of <math>n!</math> and <math>S_p(n)</math> is the [[sum]] of the [[digit]]s of <math>n</math> when written in [[base]] <math>p</math>.<br />
<br />
==Examples==<br />
Find the largest integer <math>k</math> for which <math>2^k</math> divides <math>27!</math><br />
<br />
===Solution 1===<br />
Using the first form of Legendre's Formula, substituting <math>n=27</math> and <math>p=2</math> gives<br />
<cmath>\begin{align*}e_2(27!)=&\left\lfloor\frac{27}{2}\right\rfloor+\left\lfloor\frac{27}{2^2}\right\rfloor+\left\lfloor\frac {27}{2^3}\right\rfloor+\left\lfloor\frac{27}{2^4}\right\rfloor\\<br />
=&13+6+3+1\\<br />
=&23\end{align*}</cmath><br />
which means that the largest integer <math>k</math> for which <math>2^k</math> divides <math>27!</math> is <math>\boxed{23}</math>.<br />
<br />
===Solution 2===<br />
Using the second form of Legendre's Formula, substituting <math>n=27</math> and <math>p=2</math> gives<br />
<cmath>e_2(27!)=\frac{27-S_2(27)}{2-1}=27-S_2(27)</cmath><br />
The number <math>27</math> when expressed in Base-2 is <math>11011</math>. This gives us <math>S_2(27)=1+1+0+1+1=4</math>. Therefore,<br />
<cmath>e_2(27!)=27-S_2(27)=27-4=23</cmath><br />
which means that the largest integer <math>k</math> for which <math>2^k</math> divides <math>27!</math> is <math>\boxed{23}</math>.<br />
<br />
==Proofs==<br />
=== Part 1 ===<br />
We use a counting argument.<br />
<br />
We could say that <math>e_p(n!)</math> is equal to the number of multiples of <math>p</math> less than <math>n</math>, or <math>\left\lfloor \frac{n}{p}\right\rfloor</math>. But the multiples of <math>p^2</math> are only counted once, when they should be counted twice. So we need to add <math>\left\lfloor \frac{n}{p^2}\right\rfloor</math> on. But this only counts the multiples of <math>p^3</math> twice, when we need to count them thrice. Therefore we must add a <math>\left\lfloor \frac{n}{p^3}\right\rfloor</math> on. We continue like this to get <math>e_p(n!)=\sum_{i=1}^{\infty} \left\lfloor \dfrac{n}{p^i}\right\rfloor</math>. This makes sense, because the terms of this series tend to 0.<br />
<br />
=== Part 2 ===<br />
Let the base <math>p</math> representation of <math>n</math> be <cmath>e_xe_{x-1}e_{x-2}\dots e_0</cmath> where the <math>e_i</math> are digits in base <math>p.</math> Then, the base <math>p</math> representation of <math>\left\lfloor \frac{n}{p^i}\right\rfloor</math> is <cmath>e_xe_{x-1}\dots e_{x-i}.</cmath> Note that the infinite sum of these numbers (which is <math>e_p(n!)</math>) is<br />
<br />
<cmath>\begin{align*} \sum_{j=1}^{x} e_j\cdot(p^{j-1}+p^{j-2}+\cdots +1) &= \sum_{j=1}^{x} e_j \left( \frac{p^j-1}{p-1} \right) \\<br />
&=\frac{\sum_{j=1}^{x} e_jp^j -\sum_{j=1}^{x} e_j}{p-1} \\<br />
&=\frac{(n-e_0)-(S_p(n)-e_0)}{p-1} \\<br />
&=\frac{n-S_p(n)}{p-1}.<br />
\end{align*}</cmath><br />
<br />
===Problems===<br />
==== Introductory====<br />
====Olympiad====<br />
* Let <math>b_m</math> be numbers of factors <math>2</math> of the number <math>m!</math> (that is, <math>2^{b_m}|m!</math> and <math>2^{b_m+1}\nmid m!</math>). Find the least <math>m</math> such that <math>m-b_m = 1990</math>. (Turkey TST 1990)<br />
{{stub}}<br />
[[Category:Number theory]]<br />
[[Category:Theorems]]<br />
[[Category:Definition]]</div>Nikoislifehttps://artofproblemsolving.com/wiki/index.php?title=CSS:_Basics&diff=105085CSS: Basics2019-04-02T05:49:23Z<p>Nikoislife: /* Elements Usually Edited With CSS In AoPS User Blogs */</p>
<hr />
<div>{{CSS}}<br />
==Syntax==<br />
As mentioned beforehand, the CSS syntax is<br />
<br />
<div style="background-color: #DFF; padding: 5px;"><br />
'''Example:'''<br />
<pre><nowiki>selector { <br />
property: value;<br />
}</nowiki></pre><br />
</div><br />
<br />
<br />
<div style="font: 13px/1.75 Verdana,Geneva,sans-serif;"> After every <pre style="display: inline;padding:0;line-height:inherit;border:none;margin:0 1px;">property:value</pre> statement, there must be a semicolon, a ";". The last statement in an element does not require a semicolon, but it is a good practice to do so anyway.</div><br />
<br />
CSS ignores whatever whitespace, so you can write CSS many ways:<br />
<br />
<div style="background-color: #DFF; padding: 5px;"><br />
'''Example:'''<br />
<pre><nowiki>/* We can do */<br />
<br />
selector{property: value;}<br />
<br />
/* or we can use */<br />
<br />
selector<br />
{<br />
property: value;<br />
}</nowiki></pre><br />
*Note that /* */ mean comments and does not affect the css.<br />
</div><br />
<br />
However, it is considered good practice to do it the first way instead of the second and last way(s); it's more readable, and it's easy to organize and edit. <br />
<br />
Also, if you are only applying one or two attributes to an element, you can write it this way:<br />
<br />
<code type="css">selector { property: value; }</code><br />
<br />
<br />
==Elements Usually Edited With CSS In AoPS User Blogs==<br />
===The body element===<br />
In the AoPS User Blogs, the body element is most commonly modified for the:<br />
<ul><br />
<li>Background Image/Color</li><br />
<li>Font used generally in the entries and sidebar</li><br />
</ul><br />
===Styling the Div Element===<br />
When styling a div element, you have to specify which element you are styling. If you are working with a div, you can optionally put div in front (if you are working with a span, then put a span, etc.). To specify an element by id, put # then the id, and to specify an element by class, put a period (a .) then the class. For example<br />
<code type="css">#header{background:green;width:700px;height:80px;}</code><br />
will make the header look something like this<br />
<div style="background:green;width:700px;height:80px;">Header Text</div><br />
The tag that was modified would be<br />
<code type="html"><div id="header">Header Text</div></code><br />
====Modifying the Background====<br />
The two most common things done to the background of an AoPS User Blog are setting a background image or color.<br />
Some of the most commonly used properties for a background:<br />
<ul><br />
<li>background-color</li><br />
<li>background-image</li><br />
<li>background-repeat</li><br />
<li>background-attachment</li><br />
<li>background-position</li><br />
<li>background-size</li><br />
</ul><br />
To use them all together, use the shorthand syntax:<br />
<code>background: color position/size repeat attachment image;</code><br />
<br />
<br />
<div style="background-color: #FFAAAA; padding: 5px;font-size:13px;line-height:1.75em;"><br />
'''Warning:'''<br />
<br />
You '''cannot''' omit the slash (/) between the position and size values in the background shorthand syntax, and the size value should always come after the property value.</div><br />
<br />
<br />
Don't worry about "breaking the rules"; this doesn't affect the way CSS thinks about things, and it saves you much code. <br />
<br />
====Text formatting====<br />
<!-- Should we include this? Too much like a stub - information given is totally useless. --><br />
There are some properties used for formatting text.<br />
The most commonly used properties are:<br />
*text-align<br />
*text-transform<br />
Text-align is used for aligning text horizontally. Text-transform is usually used for capitalizing the text.</div>Nikoislife