Difference between revisions of "User:Johnxyz1"
(→Miscellaneous) |
(→Representing Actions as Permutations) |
||
| Line 99: | Line 99: | ||
==Representing Actions as Permutations== | ==Representing Actions as Permutations== | ||
| − | ''The idea is that if you must do a fixed number of operations of multiple types, you can make those operations letters, and permutate them.'' For example, if you have a grid of | + | ''The idea is that if you must do a fixed number of operations of multiple types, you can make those operations letters, and permutate them.'' For example, if you have a grid of \(4\times 6\) and you want to walk from one corner to the opposite one, WLOG you need to go up \(4\) times and right \(6\) times. You can do that in any order, so basically you are arranging |
UUUURRRRRR | UUUURRRRRR | ||
which simplifies the problem. | which simplifies the problem. | ||
Example: 2024 AMC 8 Problems/Problem 13. In this problem you can treat going up as \(U\) and going down as \(D\). Since you have to end up on the ground in \(6\) steps you have \(3\) U's and \(3\) D's; same as above. | Example: 2024 AMC 8 Problems/Problem 13. In this problem you can treat going up as \(U\) and going down as \(D\). Since you have to end up on the ground in \(6\) steps you have \(3\) U's and \(3\) D's; same as above. | ||
Revision as of 15:51, 6 September 2024
Favorite topic: ![\[\text{Counting \& Probability}\]](http://latex.artofproblemsolving.com/b/c/c/bcc448ef67345b970329c6e4921492fb5e4261cc.png)
for which I am reading AOPS intermediate book on
Favorite color: ![\[\text{\textcolor{green}{Green}}\]](http://latex.artofproblemsolving.com/6/c/8/6c8a353f688e76bb1ed7de3f8a0640984e6070c1.png)
Favorite software: ![\[MS\ \text{Excel}\]](http://latex.artofproblemsolving.com/e/2/2/e22328b2e74d53ca51923df707336ccc00545712.png)
Favorite Typesetting Software: ![\[\text{\LaTeX}\]](http://latex.artofproblemsolving.com/6/6/c/66cd718efc440a635cf4cf51cb4f64ecfa2af622.png)
Favorite Operating System: Linux (although I am rarely on one)
Below are some stuff I am doing to practice 
. That does not mean I know all of it (actually the only ones I do not know yet is the cubic one and the 
one)
![\[\text{If }ax^2+bx+c=0\text{, then }x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]](http://latex.artofproblemsolving.com/7/c/b/7cbf48dbdce18ee42c347668a9314410de651207.png)
![\[e^{i\pi}+1=0\]](http://latex.artofproblemsolving.com/2/6/c/26cc6b1c790a7db2144e4f5948158479292c55e6.png)
![\[\sum_{x=1}^{\infty} \frac{1}{x}=2\]](http://latex.artofproblemsolving.com/e/e/5/ee5784ec0e67d8ddb8d721443e25e5a727e1cf76.png)
![\begin{align*} x &= \sqrt[3]{\left(\frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}\right) + \sqrt{\left(\frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}\right)^2 + \left(\frac{c}{3a} - \frac{b^2}{9a^2}\right)^3}} \\ & + \sqrt[3]{\left(\frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}\right) - \sqrt{\left(\frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a}\right)^2 + \left(\frac{c}{3a} - \frac{b^2}{9a^2}\right)^3}} - \frac{b}{3a} \\ &\text{(I copied it from another website but I typeset it myself;}\\ &\text{I am pretty sure those are not copyrightable. I still need \textit{years} to even understand this.)}\\ &\text{This is the cubic formula, although it is \textit{rarely} actually used and memorized a lot. The equation is}\\ &ax^3+bx^2+cx+d=0 \end{align*}](http://latex.artofproblemsolving.com/f/0/8/f08f932bc16a96f0869fb3aafcfc578fde10bda5.png)
Source code for equations:
https://1drv.ms/t/c/c49430eefdbfaa19/EQw12iwklslElg9_nCMh0f0BVthxSSl-BOJAwsXtGbbhPg?e=1LfZJm
Asymptote test (with autoGraph):
![[asy]/* AUTO-GRAPH V-4 beta by PythonNut*/ /* Customizations: feel free to edit */ import math; import graph; /* x maximum and minimum */ int X_max = 10; int X_min =-10; /* y maximum and minimum */ int Y_max = 10; int Y_min = -10; /* linewidth */ real line_width = 0.75; /* graph color */ pen graph_color = magenta; /* special */ bool mark_lattice = false; bool show_grid = true; real X_tick_density = 1; real Y_tick_density = 1; real ratio = 1; real resolution = 0.0001; int size = 300; /* graph function */ real f(real x) { return sin(x)*sin(x); /* type function to be graphed here */ } /* The Code. Do not disturb unless you know what you are doing */ bool ib(real t){ return (Y_min <= f(t) && f(t) <= Y_max); } size(size);unitsize(size*ratio,size);Label l;l.p=fontsize(6); xaxis("$x$",X_min,X_max,Ticks(l,X_tick_density,(X_tick_density/2),NoZero),Arrows); yaxis("$y$",Y_min,Y_max,Ticks(l,Y_tick_density,(Y_tick_density/2),NoZero),Arrows);// if (show_grid){add(shift(X_min,Y_min)*grid(X_max-X_min,Y_max-Y_min));} real t, T1, T2; for (T1 = X_min ; T1 <= X_max ; T1 += resolution){ while (! ib(T1) && T1 <= X_max){T1 += resolution;} if(T1 > X_max){break;} T2 = T1; while ( ib(T1) && T1 <= X_max){T1 += resolution;} T1 -= resolution; draw(graph(f,T2,T1,n=2400),graph_color+linewidth(line_width),Arrows); } if (mark_lattice){ for (t = X_min; t <= X_max; ++t){ if (f(t)%1==0 && ib(t)){ dot((t,f(t)),graph_color+linewidth(line_width*4)); } } } dot((0,0));[/asy]](http://latex.artofproblemsolving.com/d/2/a/d2ac85a7a4174cadc3d62edf70c10b86c43ea503.png)
If you want to typeset your own LaTeX equation in a STANDALONE PDF like AOPS does (although they do images), use the standalone documentclass.
Miscellaneous
On an AMC 8, you can use a ruler to measure things out. Sometimes that works!
Representing Actions as Permutations
The idea is that if you must do a fixed number of operations of multiple types, you can make those operations letters, and permutate them. For example, if you have a grid of \(4\times 6\) and you want to walk from one corner to the opposite one, WLOG you need to go up \(4\) times and right \(6\) times. You can do that in any order, so basically you are arranging
UUUURRRRRR
which simplifies the problem.
Example: 2024 AMC 8 Problems/Problem 13. In this problem you can treat going up as \(U\) and going down as \(D\). Since you have to end up on the ground in \(6\) steps you have \(3\) U's and \(3\) D's; same as above.
