User:SamuelRiv/TeXbox

Math problems
Theorem to prove (induction seems to be problematic)


 * $$3^N ~=~ \sum_{i=0}^{N} 2^i \binom{N}{i}$$

where $$\tbinom{N}{i} = \tfrac{N!}{i!(N-i)!}$$.

Induction gives


 * $$3\cdot 3^N ~\sim~ 2^{N+1} + \sum_{i=0}^{N} \frac{N+1}{N+1-i} \cdot 2^i \binom{N}{i}$$

Suggestions?

Symbolificating

 * $$\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A$$
 * $$\int\!\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\bigcirc\,\,\mathbf D\;\cdot\mathrm{d}\mathbf A$$

T.M.I. (s15e04)
The T.M.I. System: Randy Marsh tells the students that they can get their T.M.I. rating with the following formula: "Length times diameter plus weight over girth divided by angle of the tip squared."
 * $$ {L \times d + {W \over G} \over \angle {\alpha}_t^2 }\,\!$$

However according to the U.S. Surgeon General the correct formula for T.M.I. is "Length times girth over angle of the shaft divided by mass over width."
 * $$ {{L \times G \over \angle {\alpha}_s} \div {M \over W}}$$

Cartman's psychiatrist claims "I have found a direct correlation between anger and T.M.I." and presents the equation "'Anger equals length times width over mass-squared minus angle of the shaft plus yaw'"
 * $$\mathrm{Anger} = \frac{L\times W}{M^2 - \angle {\alpha}_s} + \mathrm{yaw}\,\!$$

However, the Surgeon General claims their official formula is "true solid science" and "100% accurate". To resolve the anger crisis, the Surgeon General goes on television to say the national average has lowered from 6 inches to 1.5 inches, thus anyone with a T.M.I. larger than 1.5 inches is considered "above average".

Note that T.M.I. is given in inches, a unit of length. Of the formulas given, only the Surgeon General's has consistent dimensions, but these are of length3 / mass. However, by redefining the measurement of some variables (such as defining "width" as a linear mass density), length alone can be produced. The scale of T.M.I. is similar to that of penis size.

Cash For Gold (s16e02)
Cartman gives the "Chemical equation for gold," as referenced in alchemy:


 * $$gcfs_2 - puc_2 j_2 + \frac{cb(sn)}{d_1 \mathcal{O}_p} = GOLD$$

where $gcfs$ is "guys with cash-for-gold signs", $pucj$ is "people's unwanted crappy jewelry", $cbsn$ is "cable-based shopping network", and $d O_{p}$ is "demented old people".