Competify Hub provides high quality problems monthly for the reddit server, we will provide the solution in the next month's post.

October POTM Solution: 65/8. Let H and O be the orthocenter and circumcenter of ∆ABC, respectively. Since H is one of the foci, O must be the other focus because H and O are isogonal conjugates. Now, let H’ be the reflection of H over BC. It is well-known that H’ lies on the circumcircle of ∆ABC, so the length of the major axis is OH’ = (13)(14)(15)/(4[ABC]). The semiperimeter of ∆ABC is (13 + 14 + 15)/2 = 42/2 = 21, so by Heron’s Formula, we get [ABC] = √(21 * (21 - 13) * (21 - 14) * (21 - 15)) = √(21 * 8 * 7 * 6) = 84. Thus, the length of the major axis is (13)(14)(15)/(4 * 84) = 65/8.

November POTM If A is a point on the graph of y = x^2 and B is a point on the graph of y = 2x - 5, find the minimum possible distance from A to B. Express your answer as a common fraction in simplest radical form.

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Hi Everyone
I need tester for my Entusia Math Exercise Application
All helps appreciated please test my app each day and if you can also follow our social media links that would be awesome.
Let me know if anything needed

Join Google Group First
https://groups.google.com/g/entusia-tester/

Then Download From Play Store
https://play.google.com/store/apps/details?id=com.entusia.entusia

Web Link
https://play.google.com/apps/testing/com.entusia.entusia

Reddit Community
https://www.reddit.com/r/Entusia/

Instagram
https://www.instagram.com/entusia_app

Discord
https://discord.gg/BW4Xg7nDdU

This one popped into my head very randomly. So I asked an AI and am still waiting on an answer. Perhaps Reddit will be faster. I'll write a program, in bash first then C++ if that takes to long, when I have the time.

So this question will assume that 0 AD equals 0000/01/01 00:00:00, time zone agnostic, in the format YYYY/MM/DD HH:MM:SS, a year is 365 days, and a day is 24 hours. All to make it as simple as possible. I don't care that the Gregorian calendar has changed, or that the first year was 0001, or for sidereal days for that matter. I will include leap years just because but not leap days.

So the question is based on the number of characters in the format YYYYMMDDHHMMSS, 14, vs the number of characters in the seconds since 0 AD. As of 2024/10/26 21:27:30 that would be 61,236,059,250 seconds or 11 characters. When will the number of characters in the seconds since 0 AD be longer than the number of characters in the full date and time?

In thinking on it I first did a manual calculation for the date 10000/01/01 24:00:00 to see where the number of seconds was. I was disappointed and found it to be 315,360,086,400 or only 12 characters long to the formats now 14 characters. Well I'll let the AI make its attempt, it's only been a few hours so far, while I put together a program myself to run on my PC or maybe on my offices HPC if the AI takes way too long. Though I might need to try a different prompt first before I "waste" precious CPU time.

I would want to know that date and how long it is till then in years, days, hours, minutes and seconds. The time needed to calculate this would also be interesting if it really takes a long time. I don't think it would though.

Anyways how would you try this? I'm not great at math in any way beyond writing a program with basic functions. I'll drop my own program algorithm, in the "blob of text that reads like a flowchart" format, in the comments tomorrow. Till then I would love to hear of your own attempts or algorithms. I'll also drop the AIs time when I hear back or if I try a different prompt. I'll probably share those as well just because.

Hi, first of all, I'm no mathematician so maybe this is a dumb observation, but here I go.

I was watching a YouTube video about Pokémon games where you have a 1 in 8192 chance of encountering a "shiny" Pokémon (that is a Pokémon with different color than usual), and I started wondering what were the odds of encountering one if I were to trigger 8192 encounters, thinking it should be pretty high.

So I started calculating more simple cases, like the odds of rolling 6 dice and getting a six, or flipping two coins and getting heads at least once.

The case with the coins is really easy, there are 4 possible combinations and three of them have at least one heads, so 75% chance of success.

For the dice, I calculated the odds of not getting a 6 on any of them: (5/6)x(5/6)x(5/6)... so (5/6)^6, and then substracted it from 1, and got 66.51%, almost 2/3, thats way lower than I anticipated, but the interesting part is that now I had a general formula:

1 - ((x-1)/x)^x

If I roll 100 100-sided dice, the odds of getting at least one '100' are ~63.4%.

If I trigger 8192 encounters in Pokémon, the chances of one of them being a 'shiny' are ~63.21%.

Then I used an onine calculator to know the limit of that formula if x tends to infinity and I simply got:

1 - (1/e)

So if you roll an infinite number of infinite sided dice, your chances of rolling 'infinite' (or any other specific number) at least once are 1 - (1/e), or ~63.21%.

Also, if you want to know how many tries you need to get your desired result, you can just do:

1 - ((x-1)/x)^y (x being the possible outcomes and y being the number of tries)

So to know the chances of rolling a 6 with different numbers of dice you'd get:

1 die: 16.66%

6 dice: 66.51%

10 dice: 83.85%

15 dice: 93.51%

20 dice: 97.39%

Is there any "theorem" which was accepted, say in XV Century, and later when proofs became more rigourous was disproven?

I am collecting different ways of simplifying a value or expression so far this is all what I got

• combining like terms
• reducing fractions into their lowest term
• removing operation that doesn't change the value (ex: divided by one, multiplied by one, +0)

Do you guys have other activities or processes that simplifies an expression or value, that I can add into this list?

What can I do to determine if a number is a constant (n) or a variable (x)?

We all know that the term exponential is used mostly wrongly these days. I can usually live with it, but... sometimes it's too much, and I can't stop myself from informing people.

Paraphrasing a recent example, "war drones can be produced exponentially now." No, dude, that's like letting drones mate... (Edit: to be more precise, like letting clone themselves.)That was a funny thought, but sadly I'm the funniest person at parties...

I've a 20.00 bill with 2 sets of double numbers, fancy serial numbers website said it was 92% cool but not what it's worth....FOR what it's worth the check out at Walmart wouldn't accept 2 of my old 20s someone had to change them. I've actually 4 1977 20s but this was the only one that was cool

I found one that is perfect for the mod I'm making in a game but the game doesn't allow exponentiation

f(x) = - 10^(-.002(x-1500)) +1000

any tapering function that has a section that is similar should work but I am limited to add/subtract/multiply/division/modulus/squareroot/round

anyone know a tapering function limited to those operations? It can't be piecemeal either.

Hello, Competify Hub provides high quality problems monthly for this reddit server, we will provide the solution in the next month's post.

September POTM Solution: (√6)/2. Let f be the transformation that stretches the plane by a factor of OB/OA in the direction of OA, and let Q be the projection of P onto OA.

Also, let A’ = f(A), P’ = f(P), and Q’ = f(Q).

Note that under f, the ellipse becomes a circle with center O and radius OB, so 10∠P’OA’ = (360°)(1/6) = 60° because of the area condition.

Therefore, OA/OB = OA/OA’ = OQ/OQ’ = (cos 60°)/(cos 45°) = (√3)/(√2) = (√6)/2.

October POTM

Problem: In ∆ABC with AB = 13, BC = 14, and CA = 15, there is an ellipse inscribed in ∆ABC such that one focus is the orthocenter of ∆ABC. Find the length of the major axis of this ellipse as a common fraction.