Had a test earlier today and got stumped by the question 6^3x = 64. For reference, this was a no calculator test, and no logarithms (technically haven't been taught them yet). I asked a bunch of friends afterwards and none of them could figure it out. I searched it up, Gemini wasn't helpful even when asked to try without logarithms, ending at an answer of 'between 1 and 0'.

Is it realistic to reach the answer of ~0.7737 without a calculator or logarithms within the time constraints of 10-15 minutes?

Edit: probably a typo or I read it wrong ;-;

So in Catching Fire, the 75th Hunger Games reaps one male and one female tribute from each of 12 districts for a total of 24. Based on what we know from a Ballad of Songbirds and Snakes, no tribute is around from before the 10th Hunger Games, so there's 64 total games to pull from.

So what are the odds that in 64 games, each district has at least one male and at least one female victor to reap?

Basically, I have 24 objects and I pull them randomly out of a bag. With replacement, what are the odds that after 64 pulls, I've gotten each object at least once?

I thought I have to subtract my inverse answer from 1 to get the original value but apparently I am incorrect. My teacher refuses to post answer keys to ungraded practice sheets so I'm relying on AI to check my answers to see if I'm doing everything correctly unfortunately.

https://www.desmos.com/calculator/6bidgwpvku

I have tried accomplishing this graphically in desmos but am interested in a better solution than a symmetrical case: https://www.desmos.com/calculator/d5ecywv34w
as it is kind of boring and doesn't look very 3D

For the spring I may be taking real analysis and topology together, as well as differential equations, a functional programming course, and physics. Is this too much or fine?

I have some proof experience and am currently taking Calc 3, linear algebra, an intro programming class, and a psych gen ed, and I have a 4.0 currently. Ive self studied to chapter 2 of baby rudin.

Hi yall, my teacher is really nice but super ass at answering emails. I want to send this here to make sure I’ve got all the info I need— I know how to do all the steps, but my issue stems from actually remembering what those steps are and often mix up methods and shit. Unfortunately I won’t have another lecture and won’t be able to attend office hours until the exam, so I can’t ask him in person.

Also, how am I supposed to prove that ak is positive? He left a note on an old test that said I need to do more beyond saying that a1(or whatever the starting value is)>0 but I’m not sure exactly what he wants and I can’t find any info on my notes about it.

So I’m writing this math paper for school. I solved it graphically but I really wanted to do it by hand so that the work could be shown, cuz it’s like extra swag yk? Anyways, I reached out to a teacher about it and she was like “I don’t think it’s possible to solve it algebraically” and I was like “I don’t think that’s possible” anyways it goes like this 145.04714x(0.991479)^(x-1)>-1 
Then solving it for x.
If anyone wants to read the paper and give feedback that would be very helpful. (Please if you are like transphobic or something, please just look at the math, this girl in my class is transphobic and it was kinda just a waste of time because her feedback was so obviously biased)
I already know that I don’t have the axis labeled btw.

Suppose 91 distinct positive integers greater than 1 are given such that there are at least 456 pairs among them which are relatively prime. Show that one can find four integers a, b, c, d among them such that gcd(a, b) = gcd(b, c) = gcd(c, d) = gcd(d,

a) = 1. (RMO 2019)

The original question was this, which, due to the largeness of 456 was easy to prove but then i wondered how would one find the answer to the title question

I was imagining a large number descending to 0 in smaller steps.

Say S is the starting integer of a sequence.

If S= 0, then the sequence is 0.

If S= 1, then the sequence is 1 to 0

If S= 2, then the sequence is 2 to 1 to 0

If S= 3, then the sequence is 3 to 2 to 1 to 0

If S= 81,000, then the sequence 81,000 to 80,919 to 80,903 to 80,902 to 80,901 to 80,900 to 80,899 to 80,898 to 80,897 to 80,896 and so on to... 0.

If S= 256,000, then the sequence is 256,000 to 255,744 to 255,663 to 255,647 to 255,646 to 255,645 to 255,644 to 255,643 to 255,642 to 255,641 and so on to... 0.

If S= 625,000, then the sequence is 625,000 to 624,375 to 624,119 to 624,038 to 624,022 to 624,021 to 624,020 to 624,019 to 624,018 to 624,017 and so on to... 0.

If S= 10,000,000, then the sequence is 10,000,000 to 9,990,000 to 9,983,439 to 9,979,343 to 9,976,942 to 9,975,646 to 9,975,021 to 9,974,765 to 9,974,738 to 9,974,722 to 9,974,722 to 9,974,721 to 9,974,720 to 9,974,719 to 9,974,718 and so on to... 0.

So if we take N to be an integer and S= (1000)*(N^4), then S is 1,000 times power of 4. If S= 1000(N^4), then the first step to descend is N^4, then (N-1)^4, then (N-2)^4, (N-3)^4 until the sequence is descending by 1 each step and heads towards 0. Sorry if the ttile is confusing, I'll repost with a better title if someone can think of a better one.

Would there ever be a number S such that 0 is skipped? So for example an S where at the end of the sequence it goes something like 7,796 to 3,700 (descending 8^4) to 12,99 (descending 7^4) to 3 (descending 6^4) and now the next step would be descending by 625, but that skips 0.

I suspect there will never be an S such that 0 is skipped. The harmonic series diverges but the infinite sum 1/N^4 converges to π^4/90 and my hunch is that this is related to this descending sequence I made up.

Edit someone suggested "let N be the largest integer (if any) such that S >= 1000*(N^4), or 1 if there is no such integer; then subtract N^4, (N-1)^4, (N-2)^4, etc. until you reach 1^4, then keep subtracting 1^4" which is a nice way to put this statement, thank you u/johnpeters42

I’m in university advanced functions in grade 12. My avg is like 87-88 because on every test I make small mistakes, but more importantly, I get every thinking question wrong. I asked my teacher what I should do and she said look up questions online, do the given reviews, and try to think about how to connect and extend what you’ve learned in the question. I always do all the review, I haven’t searched up questions, but I’ll start doing it, and I try to think about how I can extend my knowledge. None of it ever helps, I don’t understand how people can make these connections.

Tldr: For people actually good and thinking questions, how do you answer “true” thinking questions that you’ve never seen before

The proof:

Questions:

1. What is the proof idea?
2. Is this example 1 (below) able to illustrate the proof? If not, why?
3. Are my calculations for L_{i,j} ^k (below) given the example 1 correct? Here is a more verbose version.
4. Given my example (when n=4 and j=4) and my calculations, L(A) ⊆ L_{1,4} ^4 is also true (the proof claims L_{1,4} ^4 ⊆ L(A)). Is this true?
5. What purpose does k have? Can this be the example 2 (below, a variation of example 1) that illustrates what k does? {image - example then traveling through intermediate state}
6. Given that n=4 and j=4, how is it that L(A) is the union of all the sets L_{1,4} ^4, where s_4 is an accepting state since there is only one L_{1,4} ^4 set? Or this is a mistake: the statement should be ... all the sets L_{1,4} ^k ...?

The crux of my misunderstanding is k in L_{i,j} ^k. What am I missing?

---
Example 1:

Calculations for L_{i,j} ^k :

L_{1,1} ^k = {x ∈ I* | x is defined by regex (0|1)*}
L_{1,2} ^k = {x ∈ I* | x is defined by regex (0|1)*0}
L_{1,3} ^k = {x ∈ I* | x is defined by regex (0|1)*01}
L_{1,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011}
L_{2,2} ^k = {x ∈ I* | x is defined by regex (0|1)*0}
L_{2,3} ^k = {x ∈ I* | x is defined by regex (0|1)*01}
L_{2,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011}
L_{3,3} ^k = {x ∈ I* | x is defined by regex (0|1)*01}
L_{3,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011}
L_{4,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011}

Example 2:

Let i = 1, j = 2.

L_{1,2} ^1 = {x ∈ I* | x is defined by regex (0|1)*(0|10)} because there is an intermediate state s_3 for which 3 > 1.
L_{1,2} ^2 = {x ∈ I* | x is defined by regex (0|1)*(0|10)} because there is an intermediate state s_3 for which 3 > 2.
L_{1,2} ^3 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state s_h for which h > 3.
L_{1,2} ^4 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state s_h for which h > 4.

Probably something related to matrix or algebra . Ok ireally cannot make it more specific why is the first rule stopping me?

From Bartles and Sheberts book on Real Analysis. For proof of a implies b, why x* belongs to V is needed? we know x* is infimum of set V, so x*+ e will belong to V and x*- e will not. Is it a misprint ?

I just saw this somewhere, btw. (Edit: sorry, not drawn to scale)

The problem is to find its perimeter. Assumption is that all sides meet at right angles. How do you solve it intuitively?

I only made it until: P = 96 - 2x, where x is that little horizontal segment at the middle. I was thinking I could get it from the areas of hypothetical rectangles,, or making up equations from the smaller perimeters, but I can't... (well it takes too long)

(I already have a degree in engineering but it annoys me that I can't solve this kind of problem anymore..)

EDIT: Thanks for all your answers! It's great and at the same time refreshing to see how everyone explained it differently, but still the same point. Whatever x is, it was just taken from the top side.

Anyway, I also understood where I went wrong. I thought:

Top: D = 27-x Sides: 2(21) Bottom: silly of me to assume as D as well

So I got P = 2(21) + 2D, hence the -2x I mentioned. I could have just added each segment: +13 +14 +x to D, and I would have gotten it correctly.

Overthinking is bad. Real bad.

Hey folks. Saw this question in our university’s integration bee, though it wasn’t actually used. Thought it looked fairly easy so I figured I’d try it later but I’m not making any headway.

After trying for a while I thought I’d see what integral-calculator gives, but they use the Gauss error function, which doesn’t make a whole lot of sense since the problems are supposed to be designed to be solvable with the basic integration techniques taught in 1st year calc (IBP, u-sub, trig sub, PFD etc.).

If you guys have any ideas I’d appreciate them.

Thanks!

Apparently, the answer is eight??

This equation was in standard form so I found the AOS, which would be the X point for the maximum, but my teacher doesn’t do that, he does the same thing that you do an absolute inequality solutions or something like that? I don’t really remember

, but why the heck does he sub in zero for t, no he doesn’t even sub in zero, he makes it “0=..”

What did i do wrong ahhh

First slide is key second slide is mine

Hey, I really need some help using Newton raphson (I believe) techniques to solve transcendental equations. I think my derivations are correct, but I seem to end up with an absurdly high or low number. The answer for Y should be between 0 and 1, however, I ended up with 1066 using my calcutor. Any ideas?

I'm trying to divide a right triangle in peices in a way that each peice is of equal area. I tried this myself bu just eyeballing if it's right or wrong please tell me!

In these pictures I am playing the game Clues By Sam (https://cluesbysam.com/) which is one of those one-a-day puzzles with the following rules:

• Your goal is to figure out who is a criminal and who is innocent.
• Based on what you know, tap on a suspect to choose if they are innocent or criminal. They might reveal a new clue.
• You cannot guess! Just like in real life, you can't convict someone based on a 50/50 hunch. There is always a logical next choice, even when you think there isn't!
• If you're prevented from making something you believe is a logical choice, you missed something. The puzzles are verified by foolproof algorithms, the puzzles are always 100% correct.

The puzzles are easier at the start of the week, and get harder as the week goes on. These printscreens come from the game on 17 October, so shouldn’t be a spoiler for anybody!

At the start of the game, Q’s clue reveals S and I solve the game as Q -> S -> P -> I -> K -> M -> W -> U -> …

During the game, I’m making notes so

• Isaac tells us that F+H = 1C
• Isaac tells us that R+T = 1C
• Kumar tells us that V+Z = 1C
• Sam tells us that D+T = 1C
• Umar tells us that A+E+J = 1C
• Umar tells us that B+C+H = 2C
• Wanda tells us that each row has upto 2C
• Given Isaac + Wanda‘s clues, looking in row 2, Esha is innocent
• David tells us that C+H = 1C
• Given David + Umar’s clues, Bonnie is a criminal
• Bonnie tells us that D+H+F = 1C
• Given Bonnie + Isaac’s clues, David is innocent
• Given David is innocent, Tom is a criminal
• Given Tom is a criminal, Ruby is innocent
• Ruby tells us that A+C+J+Z = 1C
• Given Esha is innocent, Umar tells us A+J = 1C so Chris is innocent and Zach is also innocent
• …

I have two questions.

1. What sort of maths is this?
2. What sort of notation should I be using?

Currently, for hard and tricky puzzles I end up writing Sam -> DT/C and then cross out letters and criminals that I can simplify, but is there an easier way?

Didn't really know how to flair this as it is such a useful theorem. I was wondering this as it was recently a pythagorean triple day. My intuition says it should go to 100% of numbers. Is this correct? And if so does anyone have a proof?

edit: guys i meant "with all natural numbers" sorry

Please help me settle this dispute with a friend.

The question is about averages and determining a combined average from 2 groups where the values are weighted unevenly between the groups.

This is pertaining to a rating system where the rating options can be 1-6. After the groups gave their ratings

Group A ( average rating from 10 members)Has a weighted value of 60% of total rating

Group B (average rating from 20 members) Has a weighted value of 40% of total rating

Group A average rating is 1
Group B average rating is 1

What would the combined average rating be?

My friend says that it would be less than 1 and I say that it would still average as 1 despite group A having a higher weighting in the total.

Who is right?

My understanding is that you take (A x .6) + (B x .4)= weighted average

My friend says you do it this way
{(A x .6) + (B x .6)}/2 = weighted average.