help-39

[vestal tapir]

cool

[smoky gull]

its 2🙏

[vestal tapir]

oh right, it's 1

[supple hill]

For me it's correct. We know that all days=14 and day with people 13 if we use the formula.

[supple hill]

Yeah.

Use that: $$\text{At least one} = S_1 - S_2 + S_3 - S_4 + S_5 - S_6$$

[jolly parrot]

Ga³¹Br³⁵I⁵³9000✞

[supple hill]

Here is correct to.

[pearl pond]

@smoky gull Has your question been resolved?

[supple hill]

Do you have more questions about that?

[pearl pond]

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[rigid mist]

.brother that's what i thought when i first saw him

[shadow elm]

i tried making two cases of n being even and odd

the only thing i can comprehend is that , at $n=1 , a_n = odd$ which might make a change in things , other than that, all other numbers in the sequence must be even

[jolly parrot]

parthisjoking

[graceful marsh]

i have not studied these things formally

but i can tell you from intuition

that the first thing which needs to be proven

can be proved if u assume n is even

i proved it but obv i cant give u the solution

[shadow elm]

yea i got that part aswell

jt just forms a recursion

[graceful marsh]

it didn't form a recursion for me

[shadow elm]

by using the formula of linear homogenous for 2 degree . i got the recursive relation

[shadow elm]

how did you do it then

[graceful marsh]

those are some big words which i dont know

[graceful marsh]

see

if n is even, let n=2k

then tell me, what will $a_{2k}$ be equal to?

[jolly parrot]

Anshuman

[shadow elm]

2^2k + 2^k

[graceful marsh]

yes

right

now what will $a_{2k-1}$ be equal to

[jolly parrot]

Anshuman

[shadow elm]

2^2k-1 + 2^k

[graceful marsh]

no

$2^{2k-1} + 2^{k-1}$

[shadow elm]

no

[jolly parrot]

Anshuman

[shadow elm]

there is a floor function there as well

[graceful marsh]

yeah

floor of (2k-1)/ 2

[shadow elm]

that will be k

[graceful marsh]

will be equal to floor of { 2k/2 - 1/2}

which is floor of (k - 1/2)

which is equal to k-1

[shadow elm]

yeah

mb mb

[graceful marsh]

so now we have: $$a_{2k-1} = 2^{2k-1} + 2^{k-1}$$

[jolly parrot]

Anshuman

[graceful marsh]

no worries

now go ahead and multiply $a_{2k-1}$ by $2$ and see what you get

[jolly parrot]

Anshuman

[graceful marsh]

and compare it with other things

[shadow elm]

so a_2k = 2(a_2k-1)

[graceful marsh]

yes exactly

so for every $k>=1$ it happens

[jolly parrot]

Anshuman

[graceful marsh]

so there are infinitely many

[shadow elm]

but the question says that we have to show that the terms can be expressed as a SUM of DISTINCT numbers in the sequence

[graceful marsh]

im sure there are other individuals who could tell you properly as im just in class 10th and all this is intuition based

oh distinct

i didn't read that sorry

[shadow elm]

sure

no worries

imma study physics , prolly the chat will close on itself

@me if you wanna help

[graceful marsh]

i dont think i can help you with that

you should probably tag helpers or smth

[shadow elm]

i was referring to anyone who wanted to help

but sure ig

<@&286206848099549185>

[blazing marlin]

What can I help with

[blazing marlin]

I actually haven’t done this type of problem before

[shadow elm]

oh

its okay

[pearl pond]

@shadow elm Has your question been resolved?

[pearl pond]

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[pearl pond]

Please don't occupy multiple help channels.

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[still dirge]

How does -4xnm become +4mxn ?

[cosmic sinew]

,, \vec n \cross \vec m = -\vec m \cross \vec n

[jolly parrot]

ghost

[still dirge]

.close

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[shadow elm]

i found out that f(f(n)) = f(n) , can be proved using induction or smm
but idk how to prove the main ques

[dark hound]

how are you not in the mods server

[shadow elm]

huh

whats the mod server

[prisma kernel]

Mathematics olympiad discord server iirc

[shadow elm]

idk about that one

[pearl pond]

@shadow elm Has your question been resolved?

[pearl pond]

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[zenith trench]

and

I'm working

on some textbook questions

struggling on graphs alot

[sterile python]

what have you done broski

[zenith trench]

let me

ss

what i typed out so far

k i cant screenshot my tablett

for some reason so ill just type out what i got so far

for q2 we need at least n=6 verticies where the sum of degrees is 2(n-1) => 2(6-1) = 10 (Handshake Lemma)

[sterile python]

maybe this could help

[zenith trench]

i tried one note

is being

a tad strange

[zenith trench]

but its basically

just this

i didnt get much done

beyond that

for q1 i have no clue where to begin

i watched some videos on isomorphic trees

and found some notes but not much else

from the notes i can understand is that two isomorphic graphs are just the same graph drawn differently

im also reading up on degree sequences which is where i got that number from^

additionally, I found this in the textbook, which is what im looking at for reference

[pearl pond]

@zenith trench Has your question been resolved?

[pearl pond]

@zenith trench Has your question been resolved?

[woven matrix]

https://cdn.discordapp.com/attachments/908078029778083872/1484703534296600628/image.png?ex=69bf316b&is=69bddfeb&hm=5c29b89bf85c0dada36547a9262c902a1735aa1926a39edbe9896f78139bdf72&

when I see a problem like this, the first thing I notice are G and its "opposite" can be seen the other way, like G is also the opposite of "opposite G", they form a partition of all edges, if an edge is not in G its in opposite G and the reverse is also true

the other point is that n >= 5 is necessary

[woven matrix]

from this point, I know that a logical way to go is to assume G has no cycle and it should imply "opposite G" has one (or the reverse)
so graphs without cycles with n >= 5 have a certain property that we have to think of

first intuition, if a graph has too many edges, at some point it will have a cycle (so graph without cycle must have an upper bound of their number of edges)

so the fact that G and opposite G are a partition of edges will imply some problem with that

if one has very few edges, the other has a lot

[woven matrix]

and the n >= 5 must probably be necessary because you need at least that much vertices in order to have enough edges

[woven matrix]

I would say the problem is with the total number of edges, which is n choose 2 since it's the total of possible edges in a graph with n vertices

maybe in your book you've already done something about graph without cycles ? otherwise, try to figure out the upper bound

for your 2nd problem about trees, I would try to make trees big enough with an even number of vertices for each degree

so I can swap them somehow if needed and make a 2nd tree not isomorphic to the first

[pearl pond]

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[zenith trench]

@woven matrix took some time but i completed those two questions just now with ur advice :)

[zenith trench]

here are the solutions for that sake

ignore my crappy paint lol

[frank urchin]

all option seems correct to me

[zenith trench]

my new question is this

[pearl pond]

@zenith trench Has your question been resolved?

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[eager jewel]

..

<@&268886789983436800>

[pearl pond]

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[small barn]

Image 1 integrates to image 2.... however, I'm getting a slightly different answer,
instead of -1/4a, I'm getting -1/2a.... I want to know where I mada the mistake....

[honest spear]

!occupied

[pearl pond]

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).

[honest spear]

show your workings

[small barn]

can I just tell you what I did?

My handwriting is in hindi?

Not english

[honest spear]

I mean numbers are numbers fr

[small barn]

ax + b = u

du = a dx

dx = du/a

integral (sinu cos u du/a)

1/a taken out of integrand

I use identity sin2x = 2 sin x cos x

and get 1/2a outside intgerand and 2sin u cos u du inside

= 1/2a integral(sin 2u du)

I integrate

[honest spear]

do a k sub

[small barn]

and get 1/2a -cos(2ax + 2b)

[honest spear]

this is what went wrong fr

let k=2u

[small barn]

Okay

[small barn]

Oh okay

Got it

[honest spear]

thats the extra 1/2 factor

[small barn]

Okay yeah I see

I'm kinda new, forgive me for the silly mistake

[honest spear]

dw

!done

[pearl pond]

If you are done with this channel, please mark your problem as solved by typing .close

[small barn]

.close

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[violet prism]

can someone teach me how to solve a second order homogenous DE?

[rough forge]

Sure

Matter of fact, what you are essentially solving is a quadratic in disguise

[honest spear]

[rough forge]

I will let u

[honest spear]

no im in game

[violet prism]

😭

[violet prism]

this makes sense

[rough forge]

<@&268886789983436800>

[fathom moat]

yo why r u calling the mods

jackass

[fathom moat]

u r a snitch

[light helm]

Take a day off

[violet prism]

bro tryna promote his soundcloud 😭 🙏

[violet prism]

ode?

[rough forge]

ordinary differential equation

[violet prism]

is that the same as first order?

[rough forge]

no

it is a differential equation with one independent variable

[violet prism]

what does independent variable mean 😭

[rough forge]

like you know a function y=f(x)

x is independent it can be anything we want it to be, but y depends on the value of x

[violet prism]

ohhhh okay

so f(x) will always be dependent on x for example

[rough forge]

hmm?

[violet prism]

if x is an independent variable

then a function of x is always dependenton x?

[rough forge]

that doesnt make sense, x cant be both

[violet prism]

???

wait okay

so

if x is independent

then f(x) is not dependent?

[rough forge]

f(x) is the same as y; what value you will get for it will depend on x

[violet prism]

so... f(x) is dependent on x?

😭

[rough forge]

yes

[violet prism]

okay okay

now what

[rough forge]

when you solve the linear homogeneous diff eq of order 2 here, you can have 3 cases

2 real solutions, 1 real solution or 2 complex solutions

after all you are solving a quadratic equation

[violet prism]

that makes sense

so then what are the solutions to the DE?

[violet prism]

yes no maybe so?

[rough forge]

yes

Now depending on p and q you get real or complex solutions

You can try to figure that out

[violet prism]

using discriminant

?

so.. $p^2 \ge 4q$

[jolly parrot]

keqae

[violet prism]

for real solutions

[pastel umbra]

Yes; I believe you can case-by-case it depending on that inequality

[rough forge]

Notice that in the = case λ1=λ2, so one vector falls out.

Instead we need to do the following

We need to account for the multiplicity

[violet prism]

okay wait

sorry

i understand that

for the second order homogenous DE

your two general solutions are $e^{\lambda_1}x, e^{\lambda_2}x$

[jolly parrot]

keqae

[violet prism]

uh

formatting mistake

assume the x is in the exponent

but how does this lead to

[rough forge]

yes

[violet prism]

ae^... + be^...

[rough forge]

i could try to explain it in terms of algebra, suppose we have solutions y1 and y2 then one could show that any linear combinations is a solution as well

in fact in terms of linear algebra we have a subspace

Easy example: y'=y

[violet prism]

so basically for my preuniversity ahh i should just remember the cases 😭

[rough forge]

so e^x but also Ae^x satisfies the ode

you can verify that algebraically

[rough forge]

i mean yeah if you havent taken linear algebra its hard to understand why the theory works, but it's worth it because it is well thought out concept

[violet prism]

so for example

if y1 and y2 are solutions

then 2y1 + 5y2 is also a solution?

[rough forge]

Yes and you can prove that

[violet prism]

thats interesting

[rough forge]

in fact i have the proof myself

you use the fact that the differential operator is linear

[violet prism]

whats the differential operator?

[rough forge]

d/dx

[violet prism]

oh

right

why is it linear

[rough forge]

in simple terms you use the fact you can pull out scalar or distribute differentiation

[violet prism]

hmm okay

[violet prism]

i understand that

[rough forge]

this is why any combination is also a solution

[violet prism]

and are the distribution properties an exclusive feature of linear operators?

[rough forge]

yes

[violet prism]

whats an example of a nonlinear operator?

[rough forge]

maybe for example taking the square root

[violet prism]

oh yeahhh

therefore squaring also is nonlinear?

or just any type of exponiation

[rough forge]

i wouldnt say therefore but yes squaring is also not linear

[violet prism]

and any logarithmic function

[rough forge]

logs have some similar properties but they are not linear

[violet prism]

okay cool

[rough forge]

this all

[violet prism]

how does the different numbers of solutions

[rough forge]

you learn that in linear algebra

[violet prism]

affect your

solution

or does it not change

say for example the roots to $\lambda \in \mathbb{C}$

[jolly parrot]

keqae

[rough forge]

uhm not sure how to answer that but have you heard of vectors before?

[violet prism]

.. yes 😭

[rough forge]

ok peep this

[rough forge]

thats the same as y=x to understand that

[violet prism]

yeah

[rough forge]

now imagine we found the solution (1,1)

that point is on the line

but so is (2,2) or (π,π) etc

this is what we call a space of solutions

[violet prism]

or just any (x, x)

[rough forge]

yes

Or

[violet prism]

okay

[rough forge]

x(1,1) that is also a linear combination

[rough forge]

you can now think the same of this, you have two vectors that form a plane of solutions so thats why we have the A + B thingy

[violet prism]

i... dont see that

[rough forge]

yeah its abstract for pre uni

id just accept it and move on till linear algebra

[violet prism]

hmm okay

so does complex solutions for $\lambda$ change the form of the general solution?

[jolly parrot]

keqae

[rough forge]

Its the same concept but u take the real and imaginary parts

[violet prism]

oh so it doesnt matter?

[rough forge]

ye

[violet prism]

alright

thanks

for your help

.close

[pearl pond]

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[green aurora]

why x multiplies the first column

[sterile python]

If mult you have two rows ax+by and cx+dy

Ax and cx here can be expressed as x mult by vector a c

Similarly to y and vector b d

[green aurora]

but shouldn't this step be done first?

[sterile python]

Which one

[worn musk]

arent they matrixes?

[worn musk]

if you know matrix multiplication, the number of column should be equal to number of rows

[green aurora]

why did you tell me the final solution right away?

[sterile python]

They are the same lol

[green aurora]

.

[worn musk]

these are matrix multiplications

[sterile python]

Thats the way you multiply, not the answer.

Like you ask me what is 2x2 i tell you 4 then you ask why tell answer

[green aurora]

but I didn't mean the next step after that

[verbal whale]

They are equivalent
You can choose, depending on the context, which way to write the result

[green aurora]

im saying why this is true

[sterile python]

They are equivalent, albert said that

[verbal whale]

This sheds light on the fact that the application of a matrix to a vector can be expressed as the linear combination of the two columns, which might be useful for some cases (such as linear systems)

I hope this clears your doubt

If that was your doubt 😅

[worn musk]

the number one matrix is 2x2 matrix, 2nd one is 1x2 matrix hence, the matrix multiplications says," when matrix A and matrix B has equal columns and rows respectively then they are multiplied by these steps:

[worldly jewel]

in some sense this should be taken to be the definition of matrix multiplication

[worn musk]

fr

[verbal whale]

Don't you agree that $\begin{bmatrix} ax + by \ cx + dy \end{bmatrix} = x\begin{bmatrix} a \ c \end{bmatrix} + y \begin{bmatrix} b \ d \end{bmatrix}$ ??

[jolly parrot]

Alberto Z.

[green aurora]

but why was x multiplied by a and c?

https://klipy.com/gifs/pengu-pudgy-1178

[dark hound]

work it out in terms of algebra and youll see why

[sterile python]

.

[dark hound]

do you know bow to multiply matrices

[sterile python]

I already explained broski

[worn musk]

he doesnt know thats why he is confused

[dark hound]

ah i see

[worn musk]

he must have seen "matrix multiplication" concept on youtube

[green aurora]

but why is it true?

[worn musk]

cmon man ;-;

[sterile python]

Imagine having a system of linear equations

The same concept

[green aurora]

????

can you show the proof pls

[minor cloud]

I think it would be better at this point to accept it as a definition for the time being

[green aurora]

what do I have to do now?

[minor cloud]

you could refer to this vid I suppose for intuition, but I'm not sure how much that would help you.

[green aurora]

What is the basis for a and c?

https://klipy.com/gifs/pengu-pudgy-250

[verbal whale]

Why do you bring up this now? 😅

Do one thing at a time

Especially if you haven't solved the previous doubt!

[green aurora]

I think this is the same thing as above

[verbal whale]

I don't think so at all

[verbal whale]

@green aurora

[rough stream]

I mean it's true because that's what we mean when we write these symbols down

Kind of like asking "why is a dog called a dog"

[verbal whale]

Can you answer this, for the third time? @green aurora
Do you agree with that step?
Yes or no?

[green aurora]

but then why does everyone say different things?

[verbal whale]

No, we are not

[green aurora]

yes

[verbal whale]

Then where is your doubt?

If you agree with that, and you know how to multiply two matrices, your doubt should be now solved

[green aurora]

but I didn't ask for that thing

[verbal whale]

Mmhh

[green aurora]

this 🙁

[verbal whale]

Do you know HOW to multiply two matrices? In particular, a matrix with a vector?
YES
or
NO
????

[green aurora]

for example why don't I also multiply y by a and c?

[rough stream]

Just because that's not what we mean when we multiply two matricies

[green aurora]

What is the basis for doing this?

[rough stream]

None. Common vote.

[minor cloud]

.

[rough stream]

We all have to use the same matrix multiplication, so this is the one we use.

There's no reason why a "dog" is called a "dog" but it would be very confusing if I just started calling them cats.

[green aurora]

I’m not asking how to compute the product once the definition is given. I’m asking why matrix multiplication is defined that way in the first place, instead of some other rule

You’re explaining how to apply the definition. My question is more basic: why is matrix multiplication defined in that specific way at all?

[rough stream]

Like, let's say historically they did decide on another rule. Call it "Rule B"
Then you'd be asking "why do we use rule B and not rule A?"

You might be wondering "why have a matrix multiplication at all?" Which has some interesting answers, but at the moment you're asking "why does matrix multiplication look like that?" Which was decided by popular vote at the time.

[green aurora]

That’s not quite the same. I’m not asking about a bare convention; I’m asking which definition is mathematically justified by the properties we want matrices to have

[rough stream]

You are asking about a bare convention.

[green aurora]

the reason this one is called matrix multiplication is that it is the unique natural rule that makes matrices represent linear maps and makes composition correspond to multiplication. In that sense it is not a bare convention; it is mathematically motivated

[rough stream]

Sure, but there's a ton of ways this could have been represented

You're asking "why this particular representation"?

The answer: "No particular reason"

If we did have y multiply by [a c] that would not have changed anything

[green aurora]

Yes, there are many representations. But for the standard coordinate-matrix representation, row-by-column multiplication is not a free choice; it is the rule induced by composition of linear maps

[rough stream]

I'm sorry you're so steadfast in your mistaken impression.

[green aurora]

I think we’re talking past each other: I’m not denying that other conventions are possible; I’m saying that once the standard coordinate convention is fixed, the multiplication rule is forced.

[rough stream]

And I'm telling you that's not correct

Unless you mean the convention itself is the multiplication rule we're telling you is a convention

[green aurora]

No once the basis and coordinate convention are fixed, the rule is not arbitrary. The columns of the matrix are, by definition, the images of the basis vectors e1,e2,... and linearity then forces the action on any vector. So if you change the multiplication rule, you are no longer using the same representation of the same linear map

[rough stream]

You'd just need to permute the columns to fix it

[green aurora]

The Georgia Tech linear algebra text explicitly says that the definition of matrix multiplication is justified because it corresponds to composition of linear transformations, and even calls that the “one and only reason” for defining it that way. MIT 18.06 also explains that once a basis is fixed, a linear map is determined by the images of the basis vectors, and the columns of the matrix are exactly those images in coordinates. So with the standard coordinate convention fixed, the multiplication rule is not arbitrary; changing it changes the represented map or the convention.

[rough stream]

Where "fix" means get from one arbitrary convention to another arbitary convention

The columns of the matrix are definitely not defined as the image of a basis vector, btw. That's a result

[green aurora]

okay!

.close

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[violet kraken]

x^2 + y^2 =2y
how can i say this a circle

[autumn fossil]

try subtracting the 2y from both sides and then complete the square

the general procedure is grouping all the x things together, then the y things together and then completing the squares

so in this case, you need to move that 2y term to the left side, so that its together with the y^2 term

[violet kraken]

ohh

im dumb af thanks @autumn fossil

.close

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[blissful brook]

in part a

[blissful brook]

when doing partial fractions

why can you sub in x = -2/5 and x = 1/2

when they are restricted

[somber night]

cz u would get a fraction where 0 is the denominator

[green aurora]

what do you mean

[graceful marsh]

du u mean can't?

[somber night]

i think its a typo

atleast i assumed it

[graceful marsh]

yeah

[blissful brook]

no liek

[graceful marsh]

but he said "in part a" so even im confused

[blissful brook]

you can do 50x^2 + 38x + 9 = A(1-2x)(5x+2) + B(1-2x) + C(5x+2)^2

and then sub in x = 1/2 and x = -2/5 and solve

and you get the correct A, B, C values

[somber night]

ohhh

its just for the first part i belive

[toxic remnant]

in the original equation you cant because thats a division by 0, but when we cross multiply it gives a polynomial true for all x

[blissful brook]

okay

makes sense

[somber night]

nice

[blissful brook]

.close

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[elfin stirrup]

In what sense can I take the abstract Mellin Transform of a function where the integral diverges?

[pearl pond]

@elfin stirrup Has your question been resolved?

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[sharp smelt]

A few questions

Here isn't (x)=Q[x,y]

Well,,,

it doesn't need to have identity I suppose

[pseudo oxide]

no then 1 would have to be inside (x)

[sharp smelt]

Got it

[pseudo oxide]

and stuff like y^2 + 1 either, bc that would imply x times f(x, y) = y^2 + 1 -> therefore stuff like that's invertible, and it isn't

[sharp smelt]

so just sums of kx^ij^j

[pseudo oxide]

you mean kx^i y^j?

yes

[sharp smelt]

Cool

[pseudo oxide]

the prime ideal thing is easy enough to prove :3

[sharp smelt]

yea, from defn even

[pseudo oxide]

ye

[sharp smelt]

no need to show Q[x,y](x) is a ID

hmm

how does (x,y) differ

[pseudo oxide]

wdym

i mean, same argument goes for why it's prime

[sharp smelt]

(x)≠(x,y) though right

[pseudo oxide]

no

stuff like y^2 is in (x,y)

not in (x)

[sharp smelt]

ah right

[pseudo oxide]

yep

[sharp smelt]

🤦

[pseudo oxide]

ahahaha

is okay

don't beat urself up

buuuut the prime ideal idea is the same

can you see how to show it's maximal

[sharp smelt]

probably wanna show Q(x)/(x,y) is a field?

[pseudo oxide]

Q[x, y]

correct

but like

that's trivial

[sharp smelt]

well \ but discord doesn't display that

[pseudo oxide]

no it's /

i think \ is like

[sharp smelt]

ooh right, I keep getting confused with set diff.

[pseudo oxide]

the exclusion

yeah

[sharp smelt]

cool

thanks so mcuh!

[pseudo oxide]

yeah

[sharp smelt]

*much

[pseudo oxide]

nah nw

anyway you know what to do

[sharp smelt]

.coose?

ye, i'll close it

.close

[pearl pond]

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[pseudo oxide]

god that reference is peak

[pearl pond]

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[kind lagoon]

I need to find the largest number and provide proof

[cosmic sinew]

what have you tried

[kind lagoon]

I tried squaring all of those and putting it into a^2-2ab+b^2 formula

[light helm]

conisder mutliplying by conjugate/conjguate

[cosmic sinew]

yes multiply and divide by conjugate

[torn flicker]

i hope he knows what conjugate means

[kind lagoon]

imma try

is A the answer then

[cosmic sinew]

seems so

[kind lagoon]

.close

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[rose phoenix]

In last 2nd line why it is 180 -60
Bcuz 180 - theta equals theta
So it should be like 180-120 = 120

Bcuz see the 2nd page attached here since 180 - 60 =sin 60

[light helm]

dont ditch the sin

sin(180 - theta) = sin(theta)

[rose phoenix]

Ya ya i know

[light helm]

sin(120) = sin(180 - 120) = sin(60)

[rose phoenix]

Yes

See the first pht attached

Last 2nd line

For sec

[light helm]

Can you post that again so I can rotate

[light helm]

they're applying supplementary identity for sec (and cos)

cos(theta) = -cos(180 - theta)

sec(theta) = -cos(180-theta)

[rose phoenix]

See
Sec -60
Sec 180 - 120
Sec 120
Right

[light helm]

no

[rose phoenix]

Ohk I got it

[light helm]

sec(180 - theta) isn't sec(theta)

-sec(theta) isn't sec(-theta)

[rose phoenix]

.close

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[shadow elm]

someone gave a solution in the previous channel

but can we ignore the fact that it explicitly says SUM …

@proper nova i think it was u (sorry for pinging)

[proper nova]

hm

do you mean this?

[shadow elm]

yeah 🤯

is it really that deep?

[proper nova]

damn

<@&268886789983436800>

[shadow elm]

idk how to get this in my head

this strays away from the question at hand tbh

[pearl pond]

@shadow elm Has your question been resolved?

[pearl pond]

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[royal galleon]

for c how can I say anything about the relationship between f(n), f(n-2), and f(n+2). I found this stack exchange but there explanation is not clear. They just say tan can't take a value greater than 1 on (0,pi/4) and that implies two inequalites. https://math.stackexchange.com/questions/2368340/how-to-show-the-inequality-for-int-0-pi-4-tannx-dx

[flat onyx]

what step you don't understand ?

[royal galleon]

Is he trying to point to the fact that the height of any part of the tan curve is going to be some decimal value and by multiplying a decimial more times you make it smaller thus the height of the curve that is multiplied by itself more times is smaller and thus the area is smaller and because of that we can say the integral is less

[final hinge]

Remember for $x\in(0,1)$ and $n < m$, we have $x^m < x^n$

[jolly parrot]

π=√g

[royal galleon]

ye so the curve would have higher height

[final hinge]

Yes i guess that is what you said, so yeah use that

[royal galleon]

ok. Thanks

.solved

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[ember rampart]

hello everybody, why is a+b not a^2 + b^2 but 2ab

[verbal whale]

Huh???

[hybrid meteor]

W question

[verbal whale]

!xyp

[pearl pond]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[ember rampart]

a+b not a^2+b^2

[autumn fossil]

You probably meant to ask why is (a+b)^2 = a^2 + b^2 + 2ab, instead of just a^2 + b^2

[warped violet]

One has ^2, the other doesn't

[autumn fossil]

is that right?

[ember rampart]

oh that makes sense

thank you

[supple hill]

Do you have more questions?

[ember rampart]

yes

why a^2+b^2 not a+b

[verbal whale]

It's like asking:
why is 3+4 not 3² + 4²

😅

[ember rampart]

its 7

[verbal whale]

Lol

[supple hill]

Because one have the ^2 and the other does not.

[verbal whale]

I guess you're trolling then!

[ember rampart]

trollin'?

[supple hill]

It’s like 2+3 it’s not 2^2+3^2=13

[hybrid meteor]

a^2 = a*a =/ a

so it doesnt make sense

[ember rampart]

a*a = a \therefore a=1

[hybrid meteor]

a*a = a^2 ;-;

[ember rampart]

a^2 = a => (a^2-a)=0 => a(a-1)=0 => a=0 or a=1

so why a+b \neq a^2+b^2

[hybrid meteor]

is ts ragebaiting?

[autumn fossil]

it can be equal to it for specific values of a and b, but it's not always equal, so it's not an identity

just like a^2 = a isnt always equal, but can be equal for specific values of a (namely 0 and 1)

[smoky gull]

get this man a true

[ember rampart]

thank you so much!!

[ember rampart]

partypooper

[supple hill]

Do you have more questions?

[ember rampart]

nope :love:

[supple hill]

!done

[pearl pond]

If you are done with this channel, please mark your problem as solved by typing .close

[ember rampart]

@hybrid meteor воняет, прям фу

.close

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[subtle phoenix]

This makes absolutely no sense omg

What is this bs

I m about to cry

[rough forge]

Which part does not make sense?

[subtle phoenix]

like how can u take y = root x along x axis

shouldn’t it go along z

as z is missing

from our equation

ykwim?

[rough forge]

no

[subtle phoenix]

then

explain pls

[rough forge]

you are doing a rotation akin to the right

[subtle phoenix]

Where did y = x come from

?

[rough forge]

so in 3-dimensional space (with z-axis forward) you get a solid

[subtle phoenix]

ummm u integrated?

[rough forge]

it's a random example so you understand what it means to rotate a func about x-axis

[subtle phoenix]

Wow

What

This is like what the answer key of my book says

I m confused what do they mean by y >= 0

like can’t y be negative or positive

but we square it in the end to make it positive

[rough forge]

Oh my blud I misread the q totally

[subtle phoenix]

wow

[rough forge]

y cant be negative because you start with y=sqrt(x)

sqrt(x) is non-negative

[subtle phoenix]

trace in the xz plane is confusing me

[rough forge]

Well when you rotate it or rather look at the picture, can you notice a symmetry?

[subtle phoenix]

no but I won’t have an image in the exam

[rough forge]

you can derive that yourself in 5s

The point is your rotation has a constant radius sqrt(x), in 3d you will notice in the yz plane that you have traces in the form of circles, which gives you (sqrt(x))^2=x=y^2+z^2

[pearl pond]

@subtle phoenix Has your question been resolved?

[subtle phoenix]

I don’t get it omg

<@&286206848099549185>

Help

[tulip ore]

you here?

[subtle phoenix]

Yeah I m here

but I m still stuck

[tulip ore]

you wont have a fancy image on the exam, but you can still draw it following a few steps

first, do you know what y = sqrt(x) looks like

[subtle phoenix]

yeah it’s like

like arc

that goes up and and kinda flattens ish

[tulip ore]

yea

[subtle phoenix]

next what

[tulip ore]

then take the shape and spin it around the x-axis

[subtle phoenix]

How

Umm

like spin it how?

[tulip ore]

you see the x-axis, right?

[subtle phoenix]

Yeah I see it

now

[tulip ore]

now what does it means to spin something

[subtle phoenix]

but why is it going down too

[tulip ore]

around something else

the x-axis is not currently going down

its going right, as it always does

the picture youre looking at is in 3D

it does not mean the x-axis is going "down"

the y-axis is not going "down" either

[subtle phoenix]

the parabola is x = y^2 right

[tulip ore]

maybe you should ignore this picture for now

it seems youre not skilled enough in mentally spinning things, we will leave that for later

[subtle phoenix]

and my confusion is how is it possible to direct it along x axis if it’s

[tulip ore]

direct it?

thats not how spinning works

[tulip ore]

ok lets pause

[tulip ore]

have you spun something before?

[subtle phoenix]

like a beyblade

[tulip ore]

thats not correct

with your hands, have you spun something with your hands before

[subtle phoenix]

what’s not correct

No I haven’t

[tulip ore]

we can make do with a beyblade, sure, but itll be harder that way

[subtle phoenix]

ok then how do we spin

[tulip ore]

because then the next question is gonna be

[subtle phoenix]

what do we spin *

[tulip ore]

its clear the thing we're spinning is going to be the thing we're spinning

it could be the beyblade or the y = sqrt(x)

that we do know

now to spin it, youre already imagining a certain way to spin it

which is around the center, right?

[subtle phoenix]

yeah

[subtle phoenix]

I get that

on xy axis

we have a parabola

[tulip ore]

no no dont bring axes into this

we're not done yet

[subtle phoenix]

that keeps continuing

[tulip ore]

hey

[subtle phoenix]

Oh

[tulip ore]

we're not done yet

[subtle phoenix]

okay

okay okay sorry

[tulip ore]

now tell me

what exactly is a center?

[subtle phoenix]

around which something spins

[tulip ore]

sure, is it a point or a line?

[subtle phoenix]

Center is a point

[tulip ore]

so for example

say I spin the beyblade sideways

thats going around the center too

but you know thats not correct

why not

[subtle phoenix]

why is it not correct

[tulip ore]

you know what I mean by spinning the beyblade sideways?

[subtle phoenix]

what

[smoky gull]

what am i seeing😭

[subtle phoenix]

ong yajant

Can u explain in Hindi

[smoky gull]

Arey how did you know mein indian hun😭😭

[subtle phoenix]

bhai

apka naan

Naam

[smoky gull]

Aisa kya hai mere naam mein😭

[subtle phoenix]

Explain kardo bhai

[smoky gull]

jee ka hai kya

[smoky gull]

ye gimme one sec

[subtle phoenix]

This is uni math

[subtle phoenix]

Thanku bhai

[smoky gull]

Ye ek baar dekho

aisa kuch visualise ho rha hai kya?

[subtle phoenix]

yes

mujhe

sirf xz samajh nhi araha

[smoky gull]

tum basically ek 2D mein parabola jiska axis is xais ko thoda ghumapge in 3D toh aisa ban jeyaga

[smoky gull]

Question ye hai kya

[subtle phoenix]

Yeh solution hai

[smoky gull]

Achaa

[subtle phoenix]

Question yeh hai

[smoky gull]

Haann smhj gaya

Trace matlab smjhte ho?

[subtle phoenix]

Yes

trace alag alag axis mein

kesi shape hoti

Hai

toh uss sab ko combine karke

But mujhse visualise nhi horaha

[smoky gull]

Acha toh basically aisa imagine karo lets say 2D ke parabola ko tumhe pi/2 se rotate kar diya

toh lets say xy plane horizontal hai, toh ab tumhara parabola xz mein aagya and vertical hogya right?

[subtle phoenix]

Tou like left side pe ek horizontal parabola ban gaya

xz mein

horizontal nhi hoga

[smoky gull]

Haan xz mein vertical hoga na

matlab mein maan ke chl rha hun ki xy is horizontal

[subtle phoenix]

Oh han

Vertical hogaya

[smoky gull]

Haan

Ab thoda visualize hua hoga like uska trace toh same hi aa rha hai

Wahi soln mein bhi bol rhe hain

[subtle phoenix]

oh y = 0 hojayega yahan phir

This stuff is so intuitive bro

[smoky gull]

Sahi bola be

ek numbar

Abhi lekin mein jee ki tayari kr rha hun😭🙏

[subtle phoenix]

😭

Acha but how do we get

x = y^2 + z^2

[smoky gull]

Arey wo xz mein parabola ki equarion kya hoi

[subtle phoenix]

x = z^2

[smoky gull]

Badhiya

Ab hum koi general angle ke liye lete hain

Since wo hamesha 0,0,0 se pass hoga hi

Hum bas simply trace ko compare karke bol skte hain ki ye jo hai paraboloid

Iska toh simply xaxis equation rahega and y and z iske cross axis honge

[subtle phoenix]

How do we know origin se pass hoga

[smoky gull]

Arey x axis ke about kiya na rotate

y = sqrt (x) toh origin se pass hota hai

And x axis ke about rotate karenge toh point of intersection with x axis change nahi hoga na

[subtle phoenix]

Oh

Han

x z , x y

[smoky gull]

yes boss

[subtle phoenix]

But why do we add y and z

[smoky gull]

kyunki jab rotate kiya x axis ke about toh phir wo 3D mein transform hua na

[subtle phoenix]

but add kyun karenge phir

[smoky gull]

Acha ruk waise samjhata hun

[subtle phoenix]

okay

Yeh dekho

[smoky gull]

Hum basically saare parabolas lenge jinka shape is similar to y = sqrt(x)

Axis ki mkc abhi

shape pe chal

Shape sabka same hai yahan par

Achaaaaa

mein samjha tera point

multiply toh hoga nahi kyunki multiply se phir hyperbola jaise hoga

[subtle phoenix]

add kyun Karen

Add kis logic se kia yaar

[smoky gull]

???

paraboloid ki general equation aise hi form mein toh hoti hai

ye padh

[subtle phoenix]

Oh

x is same

y and z are changing

jis ki wajah se

[smoky gull]

haan

x ke about rotate kiya hai na

[subtle phoenix]

Oh

Thanku bhai

Praying keh tereko JEE mein Air 1 mile

[smoky gull]

Arey tauba tauba kar diya tune toh🙏🙏

[pearl pond]

@subtle phoenix Has your question been resolved?

[pearl pond]

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[echo sandal]

I need help with this i dont know what im doing wrong

[sharp vigil]

try turning the row echelon form back into equations

[pearl pond]

@echo sandal Has your question been resolved?

[echo sandal]

maybe

[echo sandal]

yes

[pearl pond]

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[north talon]

i will translate the problem one sec

Let $a_1,a_2,..,a_{2565}$ be positive integers for which [ \frac{1}{\sqrt{1+a_1}} + \frac{1}{\sqrt{1+a_2}} + ... + \frac{1}{\sqrt{1+a_{2565}}} = 100 ] Prove that there exists $i,j\in { 1,2,...,2565 }$ such that $a_i=a_j$

[jolly parrot]

Copter

[north talon]

i'm stuck like halfway here

well if for contradiction all ai are distinct then a_j >= j for all j ≤ 2565

and it remains to show $\sum_{i=2}^{2565} \frac{1}{\sqrt i} < 100$

how do i show that?

[split void]

have you tried AM-GM? It looks suspiciously like that.

[jolly parrot]

Copter

[north talon]

oh yea i changed some things mb

[north talon]

wouldnt i get >= with am gm?

[dim frost]

for that you have to pre assume all are equal

[north talon]

also am gm is for real numbers so theres probably and easier way

[split void]

denominator so $\frac{1}{\sqrt{1+a_i}}\le \frac{1}{\sqrt{2\sqrt{a_i}}}$ if im not wrong.

[jolly parrot]

Annie Maqionde

[simple trench]

Hint: bound it using calculus

[rustic gate]

try evaluating [ \sum_{n = 1}^{2565} \f 1 {\s {1 + n}} \le \sum_{n = 1}^{2565} \f 2 {\s {1 + n} + \s n} ]

[north talon]

oh yea

wait why is that ineq true

nvm nvm i got it now

thanks!

[dim frost]

how did u even think of this tho

[north talon]

i mean its a pretty well known bounding for 1/sqrt n

but i forgor

[dim frost]

oh

[rustic gate]

it's obvious

[dim frost]

i see

[rustic gate]

im kidding

but it's pretty common

[north talon]

i was looking for it lmao

anyways thanks

.close

[pearl pond]

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[sweet lintel]

Hey, I am trying to prove all three possible implications of the following statement:

Let G be an undirected graph on n nodes. Any two of the following statements imply the third:

• G is connected.
• G does not contain a cycle.
• G has n – 1 edges

[sweet lintel]

I proved the 'if G is connected and has no cycles, it must have n - 1 edges' by induction and 'if G is connected and has n - 1 edges, it must not have a cycle' by contradiction

I am trying to do 'if G does not have a cycle and has n - 1 edges, it must be connected'

I am not sure how to do this one

[pearl pond]

@sweet lintel Has your question been resolved?

[pearl pond]

@sweet lintel Has your question been resolved?

[foggy scarab]

<@&268886789983436800>

[cursive wraith]

So you could try by contradiction (or direct depending on how you see it). Assume G has k connected components G_1 ,..., G_k for some k >= 1 (k > 1 if you're doing contradiction)

Every G_i can be applied a property that you've already proven...

[bitter herald]

Thanks!

.coose

.close

[pearl pond]

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[wispy rapids]

how can we solve this system

[wispy rapids]

no clue where to start from

[reef venture]

Tu'es en Terminale ?

[supple hill]

What ideas you have and what have you tried?

[wispy rapids]

oui bac

[wispy rapids]

nothing

[somber night]

i would lwky just try random numbers

[grave mirage]

What's that V supposed to mean?

[wispy rapids]

cmon man

[reef venture]

quel bac ?

[wispy rapids]

smallest commun mulyiple

[wispy rapids]

maroc

[grave mirage]

LCM alr

[wispy rapids]

lwhats the L?

[grave mirage]

Lowest

[wispy rapids]

aaaa ok tnx

[reef venture]

SM,

?

[grave mirage]

Can you figure out what form a and b must have seeing the lcm?

[wispy rapids]

oui l'arithmetique est exclusive au sm

[wispy rapids]

what..?

try what

[grave mirage]

Prime factorisation of a and b

[reef venture]

PGCD = 12

[grave mirage]

180=(2^2)×(3^2)x(5)

[reef venture]

Hint, use this formula:

[wispy rapids]

gauss?

[grave mirage]

Since 96 is a 12 multiple, both have atleast a 4 and a 3 multiple, but one of them has a 5 extra and one of them has a 3 extra

[reef venture]

no

[reef venture]

In what city you live though in morrocoo?

[wispy rapids]

bro wants me to dox myslef💔

dm me

[reef venture]

just tell (I am morrocan too)

[wispy rapids]

why is gcd 12?

[grave mirage]

Nvm that just solves it

[wispy rapids]

tanger

[grave mirage]

Alr then

[wispy rapids]

why is the gcd 12 💔

[reef venture]

wait

[supple hill]

Using the formula: #help-39 message. 🙂

[wispy rapids]

i dont see a ^ sign to apply it maaaan

[reef venture]

$\begin{aligned}
&\text{Recall } a \vee b = \frac{ab}{\gcd(a,b)}. \text{ Let } d=\gcd(a,b) \text{ with } a=dx, , b=dy \text{ and } \gcd(x,y)=1. \[6pt]
&\text{Then } a \vee b = dxy. \text{ The system becomes:} \
&\begin{cases} d(x+y)=96 \ dxy=180 \end{cases} \[6pt]
&\implies d \mid \gcd(96,180)=12. \[6pt]
&\text{Hence } x+y=\frac{96}{d} \text{ and } xy=\frac{180}{d}. \[4pt]
&\text{Testing } d\in{1,2,3,4,6,12}, \text{ only } d=12 \text{ yields coprime integers:} \[4pt]
&d=12 \implies x+y=8, ; xy=15 \implies {x,y}={3,5}. \[6pt]
&\text{Therefore:} \
&\boxed{(a,b)=(36,60) \text{ or } (60,36)}
\end{aligned}$

[jolly parrot]

YRIP_You

[wispy rapids]

got it i think

thanks guys

[reef venture]

gl I am from Marrakech though

[wispy rapids]

prv

.close

[pearl pond]

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[supple hill]

You are welcome!

[pearl pond]

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[lost zodiac]

Hello

[rough forge]

hi, do you have a maths question

[lost zodiac]

Trying to solve this

The first pic is the triangle I formulated