Activity for TheCodidacter, or rather ACodidacterâ€
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Comment | Post #291255 |
> Since floor is monotonic, we also have $x^2 + 1 \geq rx$ implies $\lfloor x^2 + 1 \rfloor \geq \lfloor rx \rfloor$.
Oh goodness, I never thought of that. 2 hours in 2 minutes. Thank you very much! (more) |
— | about 1 month ago |
| Edit | Post #291254 | Initial revision | — | about 1 month ago |
| Question | — |
Prove that $\forall x\in\Bbb R:\lfloor x^2\rfloor-\lfloor rx\rfloor\ge-1\iff|r|\le2$. > Prove that the inequality $\lfloor x^2\rfloor-\lfloor rx\rfloor\ge-1$ holds true for all real numbers $x$ if and only if $|r|\le2$. My (proposed) solution, which took me around 2 hours: > First, we reframe the inequality to $\lfloor x^2\rfloor-\lfloor rx\rfloor+1\ge0$.Suppose $r=a$ satisfies th... (more) |
— | about 1 month ago |
| Comment | Post #291104 |
Update: looked for a while, didn't find anything (more) |
— | about 2 months ago |
| Comment | Post #291104 |
Indeed, the only properties I've used are relating to the angles. I suspected nothing will come out from most other properties.
Interesting observation though, there might be something lurking from that $CD\perp MD$. I'll check. (more) |
— | about 2 months ago |
| Edit | Post #291104 |
Post edited: Geo*****G******ebra. |
— | about 2 months ago |
| Edit | Post #291104 |
Post edited: Geo**G**ebra. The more you know. |
— | about 2 months ago |
| Edit | Post #291104 |
Post edited: Length |
— | about 2 months ago |
| Edit | Post #291104 | Initial revision | — | about 2 months ago |
| Question | — |
Find length $MP$ in trapezoid Trapezoid $ABCD$ is given such that $AB\parallel CD$ and $AD=BD$. Let $M$ be the midpoint of $AB$ and $P$ be the intersection of diagonal $AC$ with the circumcircle of $\triangle BCD$. Given that $BC=27$, $CD=25$, and $AP=10$, find $MP$. Here's some visualization that I made in GeoGebra: image ... (more) |
— | about 2 months ago |
| Comment | Post #289826 |
My code runs on an online IDE, which makes it _much_ slower 🙂
I have tested just 3 candidates on 10,000 simulations, and admittedly to my surprise, you were **absolutely right!**
So that's what I've been confusing this whole time–randomly choosing the number of votes per candidates makes it mor... (more) |
— | 8 months ago |
| Comment | Post #289826 |
Continued:
```
Sample probability [average population 2999153595322]: 0.008285
Sample probability [average population 29998082716178]: 0.008359
Sample probability [average population 300075885268942]: 0.008453
Sample probability [average population 2999350982695126]: 0.008428
Sample probability... (more) |
— | 8 months ago |
| Comment | Post #289826 |
Initially my code only relied on `random.random()` for population size. I then tweaked it to make the population size vary (as you said) by using `random.randint()`, and it does converge around $\frac1{(n-1)!}$. The code starts at a small population size then increases it by powers of 10. Interesting... (more) |
— | 8 months ago |
| Comment | Post #289826 |
If I understand your reasoning correctly: the standard deviation of a binomial distribution is proportional to the square root of the population, which in turn makes it more likely for the votes cast between participants to be roughly equal as the population grows.
I'd say I agree with this... The... (more) |
— | 8 months ago |
| Edit | Post #289823 |
Post edited: Clearer wording |
— | 8 months ago |
| Edit | Post #289823 | Initial revision | — | 8 months ago |
| Question | — |
Strange behavior in elections and pie charts So, a friend asked me the probability for a candidate to get at least 50% of the total votes in an election consisting of 5 candidates (let's pretend everyone picks at random 🙂). I thought for a bit and then presented an answer: > Let's first generalize so that the election has $n$ candidates. > ... (more) |
— | 8 months ago |
| Edit | Post #288411 |
Post edited: Slight notational mistake |
— | 11 months ago |
| Suggested Edit | Post #288411 |
Suggested edit: Slight notational mistake (more) |
helpful | 11 months ago |
| Comment | Post #288411 |
Very clear explanation, thank you 🙂
I'll rephrase your explanation about using $r^2$ on the integral (to make sure I understand correctly): dividing $C_1$ into concentric rings (like [this](https://upload.wikimedia.org/wikipedia/commons/1/17/WA_80_cm_archery_target.svg)), we can visualize that the... (more) |
— | 11 months ago |
| Edit | Post #288409 | Initial revision | — | 11 months ago |
| Question | — |
Average distance from circle's center to a point > What is the average distance from a point inside a circle to the circle's center? I came across this problem, and I've heard the solution is $\frac23R$ for the radius $R$. So, I tried to tackle this myself: Let there be two concentric circles $C1$ and $C2$ with radii $R$ and $r$ ($R\gt r$).... (more) |
— | 11 months ago |