help-27

[north glen]

our second dice roll isnt influenced by the first one

[uncut crow]

yea

[tender wharf]

independent events

[uncut crow]

do you understand all that?

[north glen]

I do understand that if we have a bag with numbers 1,2,3,4,5,6,1,2,3,4,5,6, then the second choice is dependent from the first one, but this doesn't happen with dice rolls.

[uncut crow]

and do you understand that with your 72 choose 12 thing, you are introducing a similar dependence?

[north glen]

yeah

makes sense

Now... the probability of having at least one 6 after rolling 12 dice:

[woven radish]

Chuti | Argentina

[uncut crow]

there must be a typo there

[north glen]

why?

yeagyh

yeah so the 12 covers 5 and 6

$(\frac{5}{6})^{12}$

[woven radish]

Chuti | Argentina

[north glen]

like that

[tender wharf]

yeah this works

[north glen]

then the probability of having one 6 is $1 - (\frac{5}{6})^{12}$

[woven radish]

Chuti | Argentina

[north glen]

But the book says:

what's the logic behind that?

[tender wharf]

atleast 1

[tender wharf]

you choose 1 die out of 12 which will show 6

apart from others that die will have probability 1/6

and 11 others(5/6)^11

[north glen]

but thats the same as suming both probabilities?

[tender wharf]

here you are essentially counting all the cases of one 6 and multiple 6

[tender wharf]

both?

[tender wharf]

you mean no 6 and one 6?

[north glen]

yeah

[tender wharf]

yeah so for atleast 2 we need 1-( no6 + exactly one 6)

[north glen]

Okay so now I'm trying to calculate C:

I will calculate the probability of having at most, two 6's. Then I want to calculate: \
Probability of no 6's \
$\frac{5^{18}}{6^{18}}$ \ \

Probabilty of one 6. \
$1 - \frac{5^{18}}{6^{18}}$ \ \

Probabilty of two 6. \
idk

[woven radish]

Chuti | Argentina

[north glen]

ok now I understood

thanks to everyone

.close

[devout snow]

Channel closed

Closed by @north glen

Use .reopen if this was a mistake.

[tender wharf]

again, atmost two 6 means 0 + 1 + 2

[tender wharf]

that is probability of atleast one

and not exactly one

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[gray oriole]

Are they talking about a<b relation?

[ionic harness]

I think so

[faint gorge]

Yes

[gray oriole]

then why is there a <= in part c?

(s x s, <=)

[gray oriole]

how do i determine the lesser pairs

1,2 1,3 1,4 1,5

[faint gorge]

that def isnt the usual less or equal sign, but i am not familiar with what it is either

[gray oriole]

are these the lesser ones

[ionic harness]

Lexicographic order, so consider first < first, then second < second

Lexicographic = alphabetical

[gray oriole]

but these are numbers

[ionic harness]

Alpha < alpho because a < o, checking first letter then second letter, etc.

[gray oriole]

what does alphabetical have to do anything wiht this

[ionic harness]

Right, that's why I gave it in terms of numbers

122 < 123 because 2 < 3, you check first number against first then second against second

[faint gorge]

You first check the first entries, well 1<1 is never valid, so then you check the second entries

The first entry is always prioritized, or rather the previous one

[gray oriole]

1,2 1,3 1,4 2,3 2,4 3,4

[faint gorge]

given it works out

[gray oriole]

are you saying this is the order with which i write th eordered pairs

[ionic harness]

Yes

[gray oriole]

so smaller than 2,3 would be 1,2 1,3 1,4

right

[ionic harness]

And 2,1 2,2

[robust dust]

given that it says it's a poset, and is using the preccurlyeq symbol, it probably means the reflexive closure of that

[gray oriole]

how does 2,1 satisfy the less than relation?

[ionic harness]

2,1 < 2,3 because 1 < 3

[gray oriole]

but 2,1 ordered pair is not in the relation

why are we considering it?

[ionic harness]

OH I'm stupid

Yeah

I thought we were ordering S^2

Not ordering S

[gray oriole]

alright

now time for hasse diagram

4
|
3
|
2
|
1

is this it

[ionic harness]

S x S looks like you're supposed to be ordering S^2

[gray oriole]

isnt it gonna be the same

its the <= relation on S x S

[ionic harness]

(2,1) in S x S

[ionic harness]

The diagram you've drawn is <= on S

[gray oriole]

i dont think so man

i drew a hasse diagram

for a partial order relation

a relation is a cartesian product

u sure

cuz im confused

[ionic harness]

Right, in this case the product (SxS)x(SxS)

[gray oriole]

really

4-tuples?

[ionic harness]

According to a) you're comparing the ordered pairs

If we write (a,b) in R as a < b, then ((1,1), (2,3)) in R for example

[gray oriole]

why is it gonna be (SxS)x(SxS)?

[ionic harness]

Consider this

$R$ here is a relation on $\RR$

[woven radish]

Coolempire93

[ionic harness]

And $R \subseteq \RR ^2$

[gray oriole]

R squared

[ionic harness]

Right

[gray oriole]

isnt it

[woven radish]

Coolempire93

[ionic harness]

Jesus I can't type anything correctly today

[ionic harness]

Right

Therefore the relation $R$ we have here on $S \cross S$ must be $R \subseteq (S \cross S)^2$

[woven radish]

Coolempire93

[ionic harness]

Does that make sense

[gray oriole]

We have R on S x S?

I thought it was on S

[ionic harness]

Find all pairs in $S \cross S$ less than $(2,3)$

[woven radish]

Coolempire93

[ionic harness]

Since we are comparing a pair and a pair

The relation is on pairs

[gray oriole]

the relation itself has pairs, i thought

[ionic harness]

Yes, in this case pairs of pairs

[gray oriole]

does the relation have pairs of pairs?

[ionic harness]

Yes

[gray oriole]

what

So you are saying the 'less than' relation discussed here consists of pairs of pairs?

[ionic harness]

Yes, based on the notation of a, b and c

[gray oriole]

dang

alr im skipping this one then

too hard

ty

[ionic harness]

Why?

[gray oriole]

I dont want to operate in 4 dimensions

[ionic harness]

It's not particularly difficult (and still results in a total order), same as your previous questions

[ionic harness]

If you imagine the ordered pairs (a,b) as a.b you actually get the same relation

a.b being decimal

Like (1,1) -> 1.1

(1,1) < (1,2) then because 1.1 < 1.2

So even though it's "4-dimensional" we work with this relation every day

The lexicographic relation is how we compare numbers, alphabetical order, etc.

But if you really don't want to do it then .close

[gray oriole]

thanks dude

.close

[devout snow]

Channel closed

Closed by @gray oriole

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[livid bough]

How do i prove this?
I first examined the hint, pointing out that multiplying the term together |g| times will cause the a^-1 in the end of the expression cancel out the a at the start of the expression, so we will be left with a g^|g| a^-1 = a * e * a^-1 = a * a^-1 = e, ergo the |aga^-1| = |g|
Im not sure how or where to apply this, though.

[wind mason]

to apply it, let h=ga^{-1}

magic, boom

|g| is the order of element g right?

[livid bough]

yuh

[wind mason]

i should do this as well

[ionic harness]

AH so that's what it was

[livid bough]

then we get |gga^-1| = |ga^-1g| don't we

[ionic harness]

I was trying to figure it out 😂

[livid bough]

i get the idea though

[wind mason]

yeah but there's a more direct way

solve for g, blah blah

[livid bough]

uh
im confused on hwhat to do

[wind mason]

if h=ga^{-1}, solve for g, then use |aga^{-1}|=|g| to subsitute

[livid bough]

solve for g in h=ga^{-1}?

g = ah then isn't it

or
ha

[wind mason]

ha, right inverse

HA

yes

[livid bough]

so
we substitute it in and get |ah| = |ha|
wowzers!

[wind mason]

waow

want to try something

[livid bough]

try what

[wind mason]

something I'm trying to prove right now

for group theory

[livid bough]

uh oyu can dm it to me
i'll close the channel

.close

[devout snow]

Channel closed

Closed by @livid bough

Use .reopen if this was a mistake.

[jagged socket]

Hello

I kinda need help

it is for trigono,etry

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[stray orbit]

sorry if this isnt an allowed question for "help" i can close if so

[acoustic leaf]

we don't allow questions about piracy on this server due to discord TOS.

.close

[devout snow]

Channel closed

Closed by @acoustic leaf

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[stray orbit]

word my bad

[stray orbit]

.close

[devout snow]

Channel closed

Closed by @stray orbit

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rough nova]

How do I rotate a point by a line onto a plane?

[steel sage]

What do you mean? A point P, a plane L. What do you want to do?

[ionic harness]

In what system

[rough nova]

Like if I have a point A outside of plane (P) and a line (d) on that plane

[ionic harness]

(like is this analytic and we have coordinates)

[rough nova]

I want to rotate point A by line d onto plane P

[rough nova]

Oxyz coordinates

[steel sage]

You mean projection?

Projection of A on P?

Or the reflection

[ionic harness]

I assumed they meant rotation as in the line is an axis

[steel sage]

Sorry, got it

[rough nova]

Like I'd get 2 points A' and A'' on P so that d(A,d)=d(A',d)=d(A'',d)

[ionic harness]

Yep exactly

Since you have coordinates then all you need to do is project the point down onto the plane to get the direction and then put points at the two distances

[steel sage]

Yeah, B=projection of A on P, C is reflection of B by d on P. B,C are the two points

[ionic harness]

Well the projection may not actually be a rotation point

[steel sage]

It is though

[ionic harness]

But both rotations will be at that same angle from the projection of the point onto L

[rough nova]

The projection wouldn't be the rotation unless point A is on the plane, which it isn't

[ionic harness]

Right

[steel sage]

Shit

[limber elm]

Hi

[steel sage]

I didn’t have energy drink. So i say dumb things

[ionic harness]

All that gives you is a direction vector and then you scale that

to d and -d

[rough nova]

Think of it like an open book, I take 1 point on a page that's being flipped and I want to know where it will end up if i flip it to the left or the right

[ionic harness]

Was my procedure not enough

[rough nova]

Oh this?

[ionic harness]

Yes that's how I would do it

[rough nova]

Idk what you meant by "get the direction"

[ionic harness]

Ah

To be clearer

[steel sage]

B=proj(A on P), O=proj(B on d)
On line OB, find two points Q1 and Q2 such that |OQ1|=|OQ2|=|OA|

[ionic harness]

Let Q be the closest point to P on the line (so QP is perpedicular to L)

[rough nova]

You draw a perpendicular line from the projection to d?

Yep

[ionic harness]

Then if R is the projection of P onto A

The end of QRhat*d and the end of QRhat*-d are the points you seek

[rough nova]

What's that in coordinates though?

Seems like quite a bit of work

Eh...

I guess i should try with a specific case before figuring out the general "formula"?

[ionic harness]

Yeah

But you should just be able to apply the list of formulas

[steel sage]

I used projection only because projection is very easy to calculate

[ionic harness]

You could try with specific cases but I think you could do it generally just like that

[rough nova]

Well I'll tell you what I'm doing this for

I'm trying to take a geometric extrema problem and turn it into an inequality, but I need help with the coordinates stuff

Here's my idea, given point A and line d on plane P

And a point B outside of the plane

If i rotate point B by line d onto the side of plane P without point A (call that point C)

And AC intersects d at X'

Then X' is the point on line d that minimizes AX+XB

Now I want to convert that to the algebraic form, but I want to figure out the "formula" first so I can adjust the points and line such that the equality case is beautiful

[steel sage]

A is just an arbitrary point? Why AC intersects d?

A is on P?

[rough nova]

A on plane P

[steel sage]

Oh

[rough nova]

I hope you understand my idea here

[steel sage]

What are given, fixed and what are we minimizing again? And what is X? (I only saw definition of X’)

[rough nova]

X is a moving point on d, X' is the position of X that minimizes AX+XB

Everything else is fixed

[steel sage]

Understood

[rough nova]

For example, A(0,1,2), P(x+y+z=3), line d on plane P with (x+2y+3z=10), point B(1,2,3) then the inequality i get would be:

x+y+z=3, x+2y+3z=10, minimize: \sqrt{(x^2+(y-1)^2+(z-2)^2}+\sqrt{(x-1)^2+(y-2)^2+(z-3)^2}

But that's kinda ass

[steel sage]

Your conclusion is exactly right, X’ indeed should be this intersection point (when A is on the other side of d from C, if not replace A with its reflection A’ by d)

[rough nova]

Maybe i pick the point X' first, then take a point C on AX' so that when i rotate C by d it could get beautiful coordinates

[steel sage]

Or you always assume C is the one point that is on the other side of d than A

Then your statement is correct

[rough nova]

There's also another idea, which is with 2 points B and C outside of the place, find point X that minimizes XA+XB+XC

Which i'd assume is a known result

[steel sage]

Another question?

[rough nova]

Well I'm trying to create an inequality here, i'm just throwing out my ideas

Maybe one of them will be easy to calculate the coordinates

But maybe more points is just more difficult

Back to the few points, any ideas?

[steel sage]

Let me first finish your original question
B=(0,1,h), X=(x,0,0), A=(a,-b,0), you are minimizing sqrt(x^2+(1+h^2))+sqrt((x-a)^2+b^), let r=sqrt(1+h^2), this is minimizing |(x,0) (0,r)|+|(x,0) (a,-b)| thus (x,0) should be the intersection of (0,r) (a,-b) and x-axis and (0,r) is exactly the rotation of B on P on the different side of d than A so one point B case you are correct

[rough nova]

So that the coordinates of A,B,X, the equation of the plane P and line d all have "beautiful numbers"

My rotating argument is just the triangle inequality

Or the minkowski inequality in the algebraic form

AX+XB=AX+XC≥AC=AX'+X'C

[steel sage]

Oh I see

Makes sense

[ionic harness]

Okay back let me read

[steel sage]

And you are considering now n point cases B1, B2,…,Bn minimizing |XA|+|XB1|+…+|XBn|?

[ionic harness]

I'll still do this just for demonstration

I've been out of calc 3 for a while though so I'll have to look them up XD

[rough nova]

Nah scrap that

That isn't solvable by hand

[rough nova]

My goal here is to get a nice inequality

Like this one, but with some set of numbers that yields a nice result

Like the coordinates of X' being natural numbers or something like that

Or maybe extend it to 2 points outside the plane so that after rotating onto the plane, the Fermat's point of the triangle lies on line d

[rough nova]

This sounds so much better omg

[steel sage]

But it could happen that

The Fermat point of C1, C2, A doesn’t lie on d for any those four choices of (C1, C2)

[rough nova]

Which is why i'm picking the points

[steel sage]

(Ci is one of the two rotation from Bi)

[rough nova]

I'm not solving the general problem, i'm finding a specific algebraically beautiful case to turn into an inequality

[steel sage]

Then I guess two points case doesn’t let you obtain some nice inequality you want to obtain

[rough nova]

I'll recap my two ideas:
Idea 1: On a plane P exists point A and line d, point B outside the plane. The rotation of point B by d onto the half of P that doesn't contain A is B', AB' cuts d at X'. X' is the position of point X on line d that minimizes XA+XB

Idea 2:
On a plane P exists point A and line d, points B, C outside the plane. The rotations of point B, C by d onto the half of P that doesn't contain A are B', C'. If the Fermat point X' of ∆AB'C' lies on d then X' is the position of point X on line d that minimizes XA+XB+XC

Idea 2 seems cooler

[steel sage]

I am very confused, idea of what?

[rough nova]

Idea of creating an inequality, which is my goal here

[steel sage]

First one already let you obtained an inequality

|AX|+|XB|>=|AB’|

[rough nova]

I still haven't found specific points, plane, and line that gives a nice inequality (in algebraic form)

[ionic harness]

You know this is actually more complicated than I thought XD

Given a line $l$ defined by $l = \vec{l_0} + t\vec{u}$, define a vector from the line to the point $P$ as $\vec{v} = \vec{P} - \vec{l_0}$. We can project this vector onto the line as $\vec{w} = \frac{\vec{v} \cdot \vec{u}}{\vec{u} \cdot \vec{u}}\vec{u}$, so the vector from line to the point is $\vec{d} = \vec{v} - \vec{w}$. We project that point down onto the plane $A$ defined by normal vector $\hat{n}$ to get the projection of $\vec{d}$ onto the plane $\vec{p} = \vec{d} - \frac{\vec{d} \cdot \vec{n}}{\vec{n} \cdot \vec{n}}\vec{n}$. So the rotated points of $P$ in the plane are $d\hat{p}$ and $-d\hat{p}$.

[woven radish]

Coolempire93

[rough nova]

Like in this example i gave, the result is probably ugly though i haven't checked

[rough nova]

Oh you use vectors

[steel sage]

I see no line here

Intersection of those two planes?

[rough nova]

The line is the intersection of x+y+z=3 and x+2y+3z=10

Yeah

Because that form is more beautiful i guess

Actually the plane doesn't have to be given

You only need the points and the line

The plane is just part of the solution

[rough nova]

I didn't know the formula of projection on a plane

That seems quite complicated

Does there exist a triangle so that its vertices and Fermat point all of integer coordinates?

Probably not

[rough nova]

What does the hat p mean?

[steel sage]

a subspace W (like a plane), W has a basis a_1, …, a_r as column vectors, M=(a_1,…,a_r)
Projection of column vector v on W is Pv where P=M (M^t M)^-1 M^t

[ionic harness]

Unit vector of p

[rough nova]

Is there a general formula for the coordinates of the Fermat point of a triangle with no angles ≥120°?

[steel sage]

If you gram-Schmidt M first P=M M^t

[rough nova]

I think there is

[rough nova]

Sorry i haven't reached that level of math yet

[steel sage]

Yes but point being what? Fermat point likely not on d

[ionic harness]

This is all linear algebra

[rough nova]

I'm in highschool

[swift spindle]

??

[rough nova]

I'm creating an inequality problem

I just pick the points and line

Then convert the problem into algebraic form

[ionic harness]

Gram-schmidt is LA, I'm surprised you could understand my work haha

[rough nova]

But i need to pick specific points and line so that the numbers aren't too ugly (as they normally are)

[steel sage]

Anyway you seem don’t have a specific question, more like exploring ideas. So i am just lost in the process. Hope you can obtain some inequalities in the end

[ionic harness]

We want to consider a point $A$ in the plane, let's define it with vector $\vec{a}$, so that $\vec{x} = d\hat{p} - \vec{a}$ intersects $l$ at a point $X$, so $X$ minimizes $||\vec{AX}|| + ||\vec{XP}|| = ||\vec{a} - \vec{x}|| + ||\vec{x} - \vec{P}|| = ||-d\vec{p}|| + ||\vec{x} - \vec{P}||$ (this can be simplified a little but man...) ... but at least proving it in this manner will be quite complicated, cogwheel's method might be easier

[rough nova]

I did a few papers on graph theory which required me to study a bit of linear algebra on my own, but that's just me in my free time

[woven radish]

Coolempire93

[ionic harness]

Ah wait -dp(hat) is just d

Yeah this problem would be better with some drawings haha

[rough nova]

Well the question is to find the general formula for coordinates of point X

From which I can adjust the constants to make it "beautiful"

[rough nova]

I'm graphing stuff in desmos but it's not easy to see

[steel sage]

So your original question?
M=proj(B on P), O=proj(M on d)
Q on line OM such that |OQ|=|OB| and OQ•OA<0, X is intersection of AQ and d

[rough nova]

Would the calculations be much easier if we pick A as the center O(0,0,0)?

[steel sage]

Yeah

This way P and d both become shbspaces , projections become very easy

[steel sage]

Projection is here

[rough nova]

Consider this small problem: Points A,B and a line d. Find X on d to minimize XA+XB.

The answer X' is the intersection of AB' with d (B' being the rotation of B onto the plane that contains both A and d)

[steel sage]

your B’ is my Q

[rough nova]

I'm trying to find a case where points A,B,X' and the line all have beautiful coordinates

[rough nova]

Got it

[steel sage]

So you are not having a specific question. You have obtained a result and want to explore what you can do with it. Kind of vague

[rough nova]

The result is in the form of geometry

I'm looking for a case with beautiful (preferably integers) case of the algebraic form

Of course if point B is directly on top of line d wrt plane P then it works

But other than that

And the way i'm trying to find that beautiful case is by figuring out the general formula for the coordinates of point X

I hope you understand what i'm trying to do @steel sage

[steel sage]

I can’t… i suppose if you really expand everything algebraically you will end up with a trivial triangle inequality |XP|+|XQ|>=|PQ|.

[rough nova]

My other idea with the Fermat point is that because it's a known result, maybe there's a formula for its coordinates given the coordinates of the vertices

algebraically it'd be a Minkowski i guess

But the solver would still need the "rotating onto a plane" part, no?

[rough nova]

But if that's still too easy then the fermat point will work

Idk the formula for Fermat point (if there's one) but i have tried a simple case

With the points (-2,-2), (-1,1), (1,-1) (i just picked some simple points to test) then the fermat point is at (-1/√3, -1/√3)

Which isn't enough to see any pattern ofc

[steel sage]

To put it this way, I don’t understand what you are doing because one point case, {inequalities you can obtain} is a subset of {inequalities of the form |XP|+|XQ|>=|PQ|}, yours are special cases of the latter, it’s just the way to obtain these P, Q are complicated, rotation of some point by some line landing on some plane…
Two points case {the inequalities you can obtain} is still a subset of {inequalities of the form |XP|+|XQ|+|XR|>=some constant related to the Fermat point of PQR}
So why do I complicate the way of defining PQR I just choose them properly in the beginning

[rough nova]

Yes exactly, I can just choose the points and the line

But what points A,B',C' do i choose so that the fermat point, the line, and the coordinates of points B,C (not in the plane of A and d) have nice coordinates?

[steel sage]

Maybe that guy can understand. I really can’t

[rough nova]

Sorry, i'm probably not conveying it well

[rough nova]

Well for example, if i just pick random points A, B', C'

Take the Fermat point of that, its coordinates are probably really ugly

Then i pick a point through that

Rotate points B', C' up by some degree by line d to get points B,C for the "puzzle"

Those points are going to have very ugly coordinates too

[rough nova]

I don't even know how to calculate the coordinates of that fermat point

I'll have lunch and ask again later after i make new discoveries

.close

[devout snow]

Channel closed

Closed by @rough nova

Use .reopen if this was a mistake.

[ionic harness]

Okay after drawing it out I agree this is likely true

[rough nova]

It's true by triangle Inequality

AX+XB=AX+XB'≥AB'=AX'+X'B

[ionic harness]

Yeah

[rough nova]

Algebraically it's Minkowski

My goal is to create an inequality problem

I can just pick random points based on this idea and get one, but the results will be very ugly

I need to cherry pick the points

@ionic harness i guess my other idea with Fermat point is also true?

Just to to sure

[ionic harness]

That I can't say because I'm not familiar with fermat points 😅

[rough nova]

Are you familiar with the definition?

https://en.wikipedia.org/wiki/Fermat_point

I guess this will be helpful

[ionic harness]

Yeah I'll take a look at that

👍

[rough nova]

The construction part and the proof by vectors part seem useful

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[slate peak]

how would one factor 100a+10b+c=abc+ab+bc+ac+a+b+c

[slate peak]

it doesn't seem like the identity (a+1)(b+1)(c+1) will be useful here

1≤a,b,c≤9 btw

oh

seems like a system of equations works

[steel sage]

It’s impossible I think

Left is not symmetric

Right hand side is

Or you want find solutions for this

[tender wharf]

Try (a+1)(b+1)(c+1) = 100a+10b+c + 1

[slate peak]

this

wanted to factor it for solutions

[steel sage]

Oppenheimer rewrote it nicely. Though I can’t think of another method than brutal force

[ionic harness]

Brutal force 😂

[slate peak]

6th sense tells me to factor it into the form
(a-x)(b-y)(c-z)

should be possible

[tender wharf]

btw do we need integer sols

[slate peak]

strictly necessary

[tender wharf]

Den brutal force

[slate peak]

im lazy

🤧

it requires me to find all values (a,b,c) which satisfies the equation

[steel sage]

Oh I think I can find a way to solve it. You first express a in terms of b and c.

Do it

[slate peak]

sure

euh brain crash

[steel sage]

a(b+1)(c+1)+(b+1)(c+1)=100a+10b+c+1

Move terms with a to one side

a=?

[steel sage]

a=?

(Don’t need to expand)

[slate peak]

... I forgot how to isolate a for this case

[steel sage]

You don’t know ua+v=u’a+v’ then
a=(v’-v)/(u-u’)?

[slate peak]

cant recall

[steel sage]

……

Troll?

[slate peak]

im for real rn brain crash

literally cannot perform any math in head rn

[steel sage]

2a+7=a+11
2a-a=11-7
a=4
You never did any operation like this kind?

Then you should close it

[slate peak]

oh

.close

[devout snow]

Channel closed

Closed by @slate peak

Use .reopen if this was a mistake.

[steel sage]

When you are recovered
a=||b(c-9)/(100-(b+1)(c+1))||
if not b, c ||both 9||, then the denominator is well defined and positive, numerator ||b(c-9)<=0||, contradiction. Thus ||b=c=9||

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[fallow copper]

I need help with my homework: An engineer starts working and receives a salary of 10 million per month. Every 3 years, their salary increases by 15%. Each month, they save 20% of their salary in a bank account with an interest rate of 0.5%/month compounded. Knowing that they receive the money at the beginning of the month and the savings are immediately transferred to the bank, how much money will they have saved in 10 years from the start of their job? (Round the digits accordingly)

[severe prairie]

Lol

!status

[devout snow]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[fallow copper]

what

[severe prairie]

Nothin

u dont know

how to start?

[fallow copper]

yes

I’m new

this is my first time here actually

[severe prairie]

np

[fallow copper]

then what should i do?

explain what I have tried?

[severe prairie]

yeah

show me what uve tried

I can js walk u thru

process asw

its quite quick

[fallow copper]

ok

[severe prairie]

alr look

first u want to find what goes into the bank each month

whats the starting salary?

[fallow copper]

10 mil

[severe prairie]

and they save 20% each month

so whats that

[fallow copper]

ye so 2 mil goes into the bank right?

[severe prairie]

yes

and the salary goes up

15%

every year

[fallow copper]

wait

[severe prairie]

so u want what goes into the bank

[fallow copper]

it said every 3 years

[severe prairie]

oops

sorry

3 years i meant

[fallow copper]

ok

[severe prairie]

so now

u wana find

what goes into the bank

years 1-3 4-6 7-9 and year 10

[fallow copper]

yes

[severe prairie]

1-3 is 2 mil

what would 4-6 be

[fallow copper]

2,3

[severe prairie]

yes

7-9

[fallow copper]

2,645

[severe prairie]

10

[fallow copper]

3,04175

[severe prairie]

yep

althought lets js round that to 3.042

now

the total time is 10 years

which is 120 months

[fallow copper]

yes

[severe prairie]

and the interest is .5% a month

so now you wanna add

all the savings up

WITH interest included

so the numbers u js gave (in terms of millions dont forget that)

[fallow copper]

yes?

[severe prairie]

what u get

watd u get

as ur final result

[fallow copper]

like this?

[severe prairie]

Uh

idk what u did there

look

ill show u how to do

the first block

and u do the rest

years 1-3

[fallow copper]

ok

[severe prairie]

month is 1-36

Monthly saving is 2mil

n of months is 36

the formula for

future value

would be

2000000[{(1.005)^36 -1}\0.005]*1.005

idk how to do the writing thing onhere lol

hopefully u understand that

[fallow copper]

wow

[severe prairie]

😭😭

[fallow copper]

where does that formula come from?

[severe prairie]

well

Its kinda from

compound intereet

but on the monthly saving u dont deposit js once u deposit it monthly

and i js basically used the geometric series sum

wait

what grade r u in

[fallow copper]

11 grade

[severe prairie]

have u learnt

abt the geometric series

[fallow copper]

sigma and stuff?

[severe prairie]

i used this formula

[fallow copper]

oh i knew this

[severe prairie]

yeah

well its just that

[fallow copper]

ohhh

I used that for the first 3 years

t1 t2 t3 is month 1 2 3

well my final result is about 374.57

is that correct?

[devout snow]

@fallow copper Has your question been resolved?

[fallow copper]

<@&286206848099549185>

[daring crown]

Yeah?

[fallow copper]

I need help with this

[fallow copper]

oh I just recalculated it and got 384 mil

[tender wharf]

Do you know this formula

[stiff stone]

this looks like the future value of an annuity due

[fallow copper]

yes

[tender wharf]

K

So substitute values for each time period of 3 years

[fallow copper]

thanks

I got it

[tender wharf]

Np

Btw what is your final answer?

[fallow copper]

around 384 mil

[tender wharf]

For 10 years?

[fallow copper]

yes

[tender wharf]

Hmm..

I got 725 mil for first 3 years

Principal is 2 mil

Period is 36 months

[fallow copper]

what?

[tender wharf]

Ye

How'd you get 384 mil

Ah fol

Mb

[fallow copper]

this

[tender wharf]

384 is right

[fallow copper]

ok

[tender wharf]

Yeah you are right

[fallow copper]

aight thank you

.close

[devout snow]

Channel closed

Closed by @fallow copper

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[desert saffron]

Hi

[desert saffron]

could you help me figure out this pattern

[sand quarry]

Not a maths question, but whatever

What do you observe of the central object (the hollowed-out, upside-down arc)'s colour

[desert saffron]

Sorryyyy

[desert saffron]

it changes colour

i'm fine with that one

[sand quarry]

Okay

[desert saffron]

Its just that sweet looking shape

Like idk what the pattern is

[sand quarry]

So for the object on the sides, you notice how they appear first as yellow, then the next time-step they become purple

[desert saffron]

No, sorry, if you mean the right most sweet on the 2nd row

I observe it stays purple

[sand quarry]

It does not, though

Compare second time-step with the third time-step

What happened to the leftmost object

[desert saffron]

Yes now that one became purple

Ok so that one should be yellow? for the 4th image

dont mind the ugly drawing 😭

Am i correct to assume this?

[sand quarry]

no you cannot

The rightmost item never turned to yellow

Its only that yellow -> purple not that purple -> yellow

[desert saffron]

Idk if this picture is inverted

but the rightmost shape has been purple consistently

[sand quarry]

Thats...what I said, yes

[desert saffron]

sorry 😭 but i think your lefts and rights are wrong

so you mean leftmost?

[sand quarry]

The rightmost item never turned to yellow

This means that the leftmost, purple item in the third time-step should not turn to yellow like you had drawn in your fourth time-step

[desert saffron]

but whats the correlation between the left and right ying yang looking shape

so ur say this?

[desert saffron]

but the right yingyang shape did turn yellow on the 2nd image

Sorry if im being confusing, this is just a practice test

To develop my logical thinking

Could you just show me where you'd place it i think im overcomplicating it

[sand quarry]

a yellow appeared a vacent spot
a purple appeared where a yellow was

do you agree with this

[desert saffron]

Oohhh okay

Yes i do

[sand quarry]

ok

[desert saffron]

so the purple should override the yellow one?

at the top?

[sand quarry]

so in the third image, you have 1 yellow spot and 1 vacent spot

what does that mean?

[sand quarry]

yes

[desert saffron]

Ohhhhh

Ur so smart

Who would think of this 😭

so like this?

[sand quarry]

no

where did the left one go

[desert saffron]

ohhh

how did you go about finding this out?

especially once i actually do it in timed conditions

60 seconds to find the pattern, would you please share any tips

[sand quarry]

this is not correct yet

[desert saffron]

Oh 💀

[sand quarry]

according to this what happens to a vacent spot

[desert saffron]

a yellow appears

[sand quarry]

right

[desert saffron]

hmmm

wait

[sand quarry]

i genuinely would not know how to answer this, sorry! Those types of questions are very gimmicky. Only tip I could give you is to do a lot of them

[desert saffron]

so is the yellow moving clockwise

so would it be that

[sand quarry]

no

[desert saffron]

fair enough

thanks anyways

[desert saffron]

omds

[sand quarry]

a yellow appears in a vacent spot

[desert saffron]

yeah so any vacant spot?

[sand quarry]

yes

[desert saffron]

Finally

And was this just any vacant spot?

My thought process was to do the opposite of the yellow yingyang

[desert saffron]

Thank you so much for your help, much appreciated!

[devout snow]

@desert saffron Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[sand quarry]

\lm So, I understand that in 1D Fourier Analysis, the partial sums $S_N(f)$ converge to $f$ in the $L^p$ norm for any $1<p<\infty$ thanks to the boundedness of the Hilbert transform

\medskip
But I am pretty confused by what happens in higher dimensions. Like, why does the geometry of the cutoff matter so much? If you sum frequecies over a growing square, convergence still holds. If you sum them over a growing disk (which \emph{feels} more natural and rotationally symmetric) convergence fails \textbf{unless} $p =2$

[sand quarry]

like, whats happening?

[vagrant skiff]

damn it's lex

[frozen aurora]

one explanation I've heard is that the L^p norm is rotationally invariant only for p=2 (unit ball is a circle) but i think that's more of a consequence of some other behaviour rather than something that implies your behaviour :/

it's a good question

[vagrant skiff]

The characteristic function of the ball is not an L^p multiplier for any p ≠ 2, p in (1,infty) hen d>= 2 So the disk partial sums S_ball,N f cannot converge unless p = 2.

hmm

So in 1D the partial Fourier sums
[
S_Nf(x) = bsum_{abs{x} <= N} hat f(n) e^{inx}
]
corresponds to a multiplier
[
m_N(xi) = 1_{[-N,N]}(xi).
]

[woven radish]

クーリー

[vagrant skiff]

the discontinuity of of m_N is just two points and the associated convolution kernel is essentially the Dirichlet kernel, whose bad behavior is controlled by cancellation

and you know that this control is encoded analytically in the boundedness of the Hilbert transform, which is a 1D Calderón-Zygmund operator

hm this is quite delicate

[sand quarry]

i mean i guess i somewhat get all of that

but it doesnt really explain why the geometry distinguishes the square frmo the disk

[vagrant skiff]

tsk tsk

you have no patience

[sand quarry]

sorry, sire

[vagrant skiff]

you are forgiven

hm

okay

diamond$K_E(x) = int_E e^{ix cd xi} dd xi$

[woven radish]

クーリー

[vagrant skiff]

fdathe convolution kernel associated to a cutoff region E

hm

hmm

The boundary partialE dictates where E is larg because the integral is dominated by stationary phase points

for a cube partialE is made of flat faces

each face has a constant normal

meaning

the worst oscillatory contributions come from only finitely many directional configurations

so the operator is an iteration/tensoring of 1D cutoffs

I think

see for disk and ball

you get a continuum of directions contributin comparably

[sand quarry]

oh like

[vagrant skiff]

and in L^2 Plancherel makes multiplier boundedness depend only on norm{m}_infty, so 1_E s alwatys fine

I think

[sand quarry]

since hte normal points in every direction in the disk case

you cant tensor product your way out of it

[vagrant skiff]

yes

I think

there is no coordinate system in which the multiplier factors, so you cannot reduce the operator to iterated 1D problems

I think

[gilded valley]

(btw, i dont know enough fourier analysis to answer this q, but #advanced-analysis would probably be a better place to ask this after youre done the discussion here)

[vagrant skiff]

worry not---I am the most advanced analyst around here

[sand quarry]

nah dw about it this guy wants me to post here so he gets helpful points

[vagrant skiff]

[sand quarry]

anyways ty pure i got the ideanow

kinda

[vagrant skiff]

Hm

[sand quarry]

i didnt understand half the jargon you threw at me there

but i got the idea for the most part

[gilded valley]

oh fair

LMFAO

[vagrant skiff]

One day you will get them

[sand quarry]

:ke

aight

[vagrant skiff]

[sand quarry]

.close

[devout snow]

Channel closed

Closed by @sand quarry

Use .reopen if this was a mistake.

[vagrant skiff]

blud closed on me

[sand quarry]

what

[vagrant skiff]

anh

it's alright

[sand quarry]

[vagrant skiff]

yeah

1_{|xi| < N} has no factorisation whatsoever

for fisk

disk

in any coordinate system.

[sand quarry]

ok well fair enough

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[spice jetty]

I am currently enrolled in a course "subjects in statistics".
I am working with Markov Random Fields.
I have not had Bayesian statistics, so i dont know the definition of proportional.
I have made my own definition.

My question is:
Is this definition correct without prior knowledge of Bayesian statistics

[ionic harness]

This looks like the same definition I saw in calculus and number theory, not necessarily just statistics

[spice jetty]

Alright, so i guess it is a sufficient definition

Thanks for the help!

[ionic harness]

No problem 🙂

[devout snow]

@spice jetty Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[pastel rose]

Hi can I have a small hint pls

[pastel rose]

I tried doing subtracting them thought partial fractions might be the case

Idt it works

[topaz beacon]

whats missing?

[pastel rose]

Wdym

[topaz beacon]

in the sum on the right, what terms are missing from the sum on the left?

[pastel rose]

Ohhhh 😭 wait I think I can try it from there, lemme just work out thr list of terms first them spot pattern

Oh when n is of the form 2k

like

alright I’ll have a go

Yep got it

lol why did I not think of that 😭 shoud’ve just listed out terms then seen what was missing

Alr Ty

.close

[devout snow]

Channel closed

Closed by @pastel rose

Use .reopen if this was a mistake.

[pastel rose]

.reopen

[devout snow]

✅ Original question: #help-27 message

[pastel rose]

Sorry is this fine btw?

Like it’s for analysis 1, I just wanna make sure I’m not missing anything

The working that someone else got is different to mine so was just wondering

this is what they have

[mortal bear]

If you want you can add a line that says like the sum of squares is even + odd then make it go to the other side

but your work is good

they do the same as you basically

[pastel rose]

Alright thank you

.close

[devout snow]

Channel closed

Closed by @pastel rose

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[jolly raven]

Topic: Monte Carlo Integration

Hello, I asked this question in another channel, but I deleted my latex source and the bot closed it. Here is the question:
https://cdn.discordapp.com/attachments/423244559682764800/1453248157025828990/376242170065321986.png?ex=694cc250&is=694b70d0&hm=acae32d2327c69cffd82ce8cac6b5bf3f38ae54f763b78ef864d9f6282bff4b3& https://cdn.discordapp.com/attachments/423244559682764800/1453248401453092884/376242170065321986.png?ex=694cc28a&is=694b710a&hm=8976bdaa00600d2cb7ffd4aabea156fa5df2b79e08d78706f8c9953472fddaf7&

Here is a link to the channel where the bot deleted my question:
#help-7｜zen1thxyz message

[jolly raven]

The screenshot referenced in the question:

[devout snow]

@jolly raven Has your question been resolved?

[jolly raven]

looks like the question might be a bit specialised, so I'll move to one of the advanced channels

.close

[devout snow]

Channel closed

Closed by @jolly raven

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[potent compass]

Can I ask if this is how this is done

[soft umbra]

which part are you having problem with?

could you be a bit more specific?

[pseudo basin]

second to last line, sqrt(x^4) = x^2 not x^4

[potent compass]

The whole thing

[pseudo basin]

also there's some really frustratingly ambiguous notation in the beginning

[potent compass]

Idk how to solve it

[pseudo basin]

let me highlight

[potent compass]

thank you

[pseudo basin]

is this ^3 meant to go on the x or is it meant to go outside the root?

(not the same thing)

[potent compass]

X^3

sorry

[pseudo basin]

ok then you should make it go clearly inside the root

btw x^-2 is not equal to x

so that should have stayed as x^-2 xor made into 1/x^2

that's your other mistake

[potent compass]

tysm

I got 4

[sand quarry]

,, \4{\3{16x^4}}{x^2} \0b\ne 4x

[potent compass]

Ohh

Could u explain pls

[crystal dawn]

this bottom one looks correct already

[timid igloo]

Guys I need help with the following series

[potent compass]

Oh ok

[crystal dawn]

!occupied, unfortunately

[devout snow]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[crystal dawn]

please use an available channel like #help-49

[timid igloo]

limit as N approaches infinity Sum(F_n) , where the sum varies from n=1 to N

Where F_n denotes the nth Fibonacci Number

[timid igloo]

Wdym?

[crystal dawn]

this help channel is in use by another helpee.

please pick another help channel to send your question in.

[potent compass]

They mean someone else is alr using this channel

[sand quarry]

Ah

Yeah @potent compass its fine

[potent compass]

tysm

.close

[devout snow]

Channel closed

Closed by @potent compass

Use .reopen if this was a mistake.

[potent compass]

.reopem

.reopen

[devout snow]

✅ Original question: #help-27 message

[potent compass]

Can I ask if the cube root of 2 cancel out

[crystal dawn]

what was the original question?

[potent compass]

Question 15)

[crystal dawn]

aight

no, it does not

you'll need another copy of the cube root to do that

[potent compass]

Then idk wht to do

[deep abyss]

you can factor out $\sqrt[3]{2}$ from the numerator

[crystal dawn]

$\sqrt[3]{2}$

[woven radish]

Hanako(x, y); ∂(fox)/∂x

ch3rry

[potent compass]

Wow

This is complicated

[deep abyss]

no cbrt command?

[crystal dawn]

you're gonna have to define one in your preamble if you want that

[crystal dawn]

but yes, that's also a valid strategy, OP

[deep abyss]

awh thats sed

[hollow zealot]

16= 8 × 2
54= 27 × 2

[potent compass]

But when do ik when to factor out

[crystal dawn]

hold up

[deep abyss]

thts ryt

[crystal dawn]

oh ok

mb

[potent compass]

.

Aldo tysm to you all

[crystal dawn]

factor out something that can help you simplify stuff

[deep abyss]

HHere, we could see that it'd help with the denominator

[potent compass]

Ohh

Could I cancel out the cube root two’s

[deep abyss]

yes

[potent compass]

Tysm cherry

.close

[devout snow]

Channel closed

Closed by @potent compass

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[trim pewter]

i dont understand the applications of wlog

[trim pewter]

how.. do i use this

[violet wind]

any time you have symmetric casework

[celest frigate]

WLOG is used to treat many similar cases as 1. Like a<b and b<a have the exact same writing in a typical proof so we can use WLOG, though this is case specific

[violet wind]

So like if I said a b and c are three different numbers that add up to 10, and none of them are 1, prove that their product must be 30

[celest frigate]

It is best learnt by practice

[trim pewter]

could someone

drop me a question

with sample solutions

so i can understand what's happening

[violet wind]

How would you solve this

[trim pewter]

icl no clue

[violet wind]

(positive integers btw)

[trim pewter]

oh

integers

[violet wind]

yeah i should have specified

[trim pewter]

a+b+c=10
prove abc = 30

2 <= a,b,c <= 6

idk

[pseudo basin]

"wlog" is just a word / phrase

it means "without loss of generality"

[trim pewter]

idk how to apply this idea tho

like when is it useful

ykwim

[pseudo basin]

the idea is that if you have like 2 or 3 or however many things whose roles in the problem are symmetric but you have to tell them apart somehow

but you need to pick out the biggest or the smallest or the most red of them all or sth like that

[violet wind]

Well, we're already down to 125 cases

[trim pewter]

wlog, let 2 <= a <= b <= c <= 6 or smth like that

[violet wind]

We could check them all one by one

[pseudo basin]

you can assign one of the names to it "without losing generality", on the pretext that if your assumption turns out to be false as written, it can be made right with a renaming of things

[trim pewter]

im lost

[pseudo basin]

ok look

you got 3 numbers a, b, c

[trim pewter]

yuh

[pseudo basin]

in terms of what you're told about them AND the goal

swapping the NAMES around doesnt do shit

[steel sage]

Or special cases imply the general result
like proving continuity ε-δ language. Any ε>0 you can say, WLOG any 0<ε<1

[trim pewter]

yep

[violet wind]

This is a correct application btw

[trim pewter]

vro what

ur dragging it

[pseudo basin]

so you can say ok lets call whoever is the smallest of them "a"

the next one "b"

and the last one "c"

for the sake of concreteness

[trim pewter]

dunno what the next step is

[violet wind]

since I mentioned that a, b, and c are distinct we can say something a little bit stronger

[pseudo basin]

a+b+c=10
prove abc = 30

2 <= a,b,c <= 6

this is incoherent btw

[trim pewter]

the fact that i didnt say a, b, c are integers?

[pseudo basin]

did you mean:
let a, b, c be INTEGERS between 2 and 6 incl. satisfying a+b+c=10. prove abc = 30

[trim pewter]

ok yeah but i was just laying it out for myself

[pseudo basin]

goal and data are mixed together

[trim pewter]

oh

1 < a < b < c < 7

[violet wind]

yeah

2≤a<b<c≤6 is fine

[pseudo basin]

how do we know none of a, b, c are equal

[violet wind]

Given

[rain summit]

does 2, 2, 6 not satisfying this?

[violet wind]

Distinct a,b,c

[rain summit]

ahhhh

[trim pewter]

wait ur trolling right

like

we're not doing casework

right

[rain summit]

so should be $2 \leq a < b < c \leq 6$

[violet wind]

It is casework

[woven radish]

1 divided by 0 equals Infinity

[violet wind]

But

[trim pewter]

🥀

[rain summit]

that's better

[trim pewter]

yep

[steel sage]

Or like prove ab is even when a, b one odd one even, you can say, WLOG assume a is odd, b is even

[violet wind]

This is like 10 cases

[rain summit]

cool

[trim pewter]

oo

[violet wind]

Then we just think about a

What are the options for a?

[trim pewter]

would i do smth like

wlog assume a is odd, b is even
hence, a can be expressed as a=2k+1 and b=2k, where k is an integer

then do ab = 2k(2k+1)
hence, ab is divisible by 2
thus, ab is even

[trim pewter]

😔

case work 😔

a = 2, 3, 4

[brittle coral]

right idea, but be careful,

a = 2k+1
b = 2m

[trim pewter]

oh u right

i didnt realise that

[steel sage]

Special cases imply the general result I think we can always say WLOG, special cases being symmetric is one example

[trim pewter]

i actually wouldve made that mistake in a test if u didnt point that out rn icl

[rain summit]

goodness

just case work already

[trim pewter]

huh

[rain summit]

but i can reduce more cases

a, b, c are distinct so max c = 6, max b = 5 so max a = 4

so you only need to consider a = 2, 3, 4

[trim pewter]

isn't that what i said

[rain summit]

now for a = 2, try to consider b

[trim pewter]

yeah i can do the casework from here
i got it

wait

i was randomly thinking about it

but is there like

a fast way to prove some am-gm inequalities using wlog

like proving 3 var amgm for example

[rain summit]

idk

[trim pewter]

using wlog

[steel sage]

Too many examples. Another one: Find perfect square m, such that … you can say let m=a^2, WLOG a>=0

[trim pewter]

lwk im lost

like idk what that question is asking me

[brittle coral]

cog is trying to say, we dont need to consider negative a because (-a)^2 = a^2

[trim pewter]

oh wait

sorry im a tweaker

i thought "you can say let m=a^2, WLOG a>=0 " was part of the q

🥀

.close

[devout snow]

Channel closed

Closed by @trim pewter

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[supple narwhal]

I want to show question 11

[weak cove]

Okay what have you tried

[supple narwhal]

I show the first direction idk if its true

[supple narwhal]

.rotate

[weak cove]

, rotate

[supple narwhal]

,rotate