help-49

[woeful turret]

for that first function the disciminant is less than 0

so there will be no points

where it is non differentiable

[lyric charm]

i mean you omitted the fact that your function was actually only a piece

[woeful turret]

oh ok

does that make a difference?

[lyric charm]

also the value of 4x^2-8x+5 at x=1/4 is not equal to 3

[woeful turret]

oh

isnt it?

[lyric charm]

4 * 1/16 - 8 * 1/4 + 5 = 1/4 - 2 + 5 = 3.25 not 3.

[woeful turret]

but theres a greatest integer function around it

so it goes to 3

[lyric charm]

ok sure but you made it seem like you were looking at just the quadratic.

and just the quadratic goes above 3.

which creates that jump inside the interval (1/4, 1/2)

which you led me to believe didn't exist, but does

from the graph you can clearly see 3 points of non-differentiability:

• x=1/4, discontinuity
• x=(a solution of 4x^2-8x+5=3), discontinuity
• x=1/2, sharp corner

[woeful turret]

yeah i can see that from the graph but can u tell how to get it algaebraically

[lyric charm]

what you made me think happened was something like: the quadratic goes monotonely from 3 down to 2 in the interval, so flooring it makes for a constant 2, therefore no points of non-differentiability

however you lied to me in two ways

one is that the quadratic goes above 3

[woeful turret]

i didnt mean to lie 😭

[lyric charm]

the other is that there's stuff outside the interval (1/4, 1/2)

[lyric charm]

well it's a lie by omission all the same!

[woeful turret]

alr alr mb

how check till what value the function will be as 2?

and after what value it goes to 3

[lyric charm]

4x^2-8x+5=3

[woeful turret]

i have to use quadratic formula?

to see if the root is in the interval?

[silent zodiac]

Is this differentiability chapter 💀

[woeful turret]

yeah but my module put it in continuity by mistake..ill still solve it tho cause our teacher taught us basics of differentiability

[silent zodiac]

Hmmm

I would start of by finding the intervals for the 8x^2 quad

[woeful turret]

yes ive done all that

[silent zodiac]

Where is the issue then

[woeful turret]

from 2 to 3 im having trouble finding the 3rd point of discontinuity

[silent zodiac]

Also a few points

Just check where gif is a integer

And check where the function definition changes

Should be better

[woeful turret]

what integers do i check

[silent zodiac]

What did u get the interval for the 8x quad

[woeful turret]

@lyric charm

[woeful turret]

(1/4 , 1/2)

[silent zodiac]

Hmm okay

Where does the 4x quad attain minima

[woeful turret]

(1,1)

[silent zodiac]

So it is strictly decreasing in 1/4 to 1/2 yea

[woeful turret]

yeah

[silent zodiac]

Put 1/4 into the quad

And 1/2 also

[woeful turret]

3

[woeful turret]

2

[silent zodiac]

Is 1/4 and 1/2 included

[woeful turret]

no excluded

[silent zodiac]

Theres 2 integers

(I think)

I haven’t differentiability in ages

Also for the upper one i assume u got

[woeful turret]

i need help how to plot the graph from 1/4 to 1/2

[silent zodiac]

R-(

1/4 and 1/2

[silent zodiac]

Think abt it

If the function is strictly decreasing

For those two points

Then gif(f(x))

Will always be 2

[woeful turret]

why

[woeful turret]

look at this graph

[silent zodiac]

Because function does not go above 3 or below 2 in that interval

[woeful turret]

yeah

so i agree in that interval it will always be 2 or 3

but how do we find where it switches

i mean we dont need to

[silent zodiac]

Give me a min

[woeful turret]

okay

[silent zodiac]

Ill plot it on a paper

Bro

At 0.25

The quad outputs

3.25

And at 1/2 it out puts

2

So it will be be 3 for some time

Then drop to 2

[woeful turret]

ohhhhhh

ok now i get it

thank u

.close

[midnight plank]

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[silent zodiac]

@woeful turret is it 3 or 4?

[woeful turret]

3

[silent zodiac]

👍

Differentiability revision

Thanks to u

[woeful turret]

lmao

u are done with this?

[silent zodiac]

Yes

[midnight plank]

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[latent wadi]

What have u ttried?

[slate finch]

i have no idea what to do

[latent wadi]

imagine if n is even

what will happen?

[slate finch]

well u can simplify it to 42/19

[latent wadi]

what

[slate finch]

2.21

[latent wadi]

i mean, If n is even, is the numerator even or odd?

how about the denominator?

[slate finch]

oh,i saw 7

[latent wadi]

lol

[slate finch]

well if n even,the numerator is odd but the denominator is even

[latent wadi]

exactly

that means it isnt an integer right?

[slate finch]

that means it isnt even?

[latent wadi]

yes

what u have to do is keep looking for thinggs like theese

example : n is 1 mod 3

then check wheter it works or not

[slate finch]

wi havent learnt mod yet...............

[latent wadi]

but now we alr crossed out the evens

[latent wadi]

owh..

another method then

[hard shard]

-# whispers to you: euclidean algorithm

[latent wadi]

set (7+5n)/ (3n-2) = k

then isolate n

[slate finch]

where k is ?

[latent wadi]

an integer

im not so sure

[civic gazelle]

an easy method is multiply 3 to 7+5n

in other words, try to use properties of divisibility to eliminate n

our goal here to get to the form 3n-2 | a where a is just some integer

[slate finch]

im hella confused

[civic gazelle]

alr lemme walk you through it

so if a is divisible by 7, is 3a divisible by 7?

[slate finch]

but where did a come from

[civic gazelle]

this is an example

[slate finch]

yes,31 is divisible by 7

[civic gazelle]

?

do you even understand my question

[slate finch]

sorry 3a

not 31

typo

[civic gazelle]

alr

so our goal here is to let 7+5n divisible by 3n-2

now we want to eliminate the n

[slate finch]

yh

[civic gazelle]

so we want to find 2 numbers with the same coefficient of n

and subtract them to eliminate it

[slate finch]

ok

[civic gazelle]

so what is LCM(5,3)?

[slate finch]

15

[civic gazelle]

good

so we need to multiply 3 to 7+5n to get 21+15n which is divisible by 3n-2

now 3n-2 is obviously divisible by itself

multiply it by 5 to get 15n-10 divisible by 3n-2

now you can take 21+15n subtract 15n-10 to eliminate n

what you're left with is some number divided by 3n-2 and that should limit your opinions for n

[slate finch]

i understood it up to this point

wait can you please show me the whole process up to the answer

[civic gazelle]

no I cant

!noans

[midnight plank]

The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.

[civic gazelle]

also I just spoiled most of the solution to you

[slate finch]

im confused so i want to see how eliminating n contributes to finding the solution

[civic gazelle]

alr I'll continue

[civic gazelle]

so what do you get here

[slate finch]

thats where im confused

i dont know how to write it out

[civic gazelle]

what is 21+15n-(15n-10)

what does that equal to

[slate finch]

31

[civic gazelle]

alr so 31 has to be divisible by 3n-2

now list all of the integer divisors of 31

[slate finch]

1 and 31

[civic gazelle]

you're missing a few

[slate finch]

but 31 is prime though?

[civic gazelle]

I said integers

so include negatives

[midnight plank]

@slate finch Has your question been resolved?

[midnight plank]

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[midnight plank]

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[molten bay]

Product of any two irrational number

[molten bay]

Is always an irrational

[cerulean halo]

what are you asking about that?

[zenith snow]

It isnt a must

[molten bay]

Why not

[ebon mica]

take root 2 for example

[zenith snow]

For instance root 2 and root 2

Gives u 2

[zenith snow]

Yeah

[molten bay]

I am saying two different irrational numbers

@zenith snow

[zenith snow]

Oh u didnt mention that

[molten bay]

Now i mentioned

[shadow schooner]

2sqrt2 and sqrt2?

[molten bay]

Nice

Hmmm

[ebon mica]

is there a specific problem you're trying to solve

or are you just asking this out of curiosity

[molten bay]

Nope

Curiosity

Thanks

.close

[midnight plank]

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[cerulean halo]

we all know pi * e is irrational

[gaunt plume]

controversial

[tawdry laurel]

Prove it

[cerulean halo]

ask a toddler on the street

[midnight plank]

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[feral vale]

this is so hard

[zenith snow]

Check the differences between each term

[buoyant linden]

🥀

[feral vale]

You're safe, just don't get cocky. We’re going to go through this very slowly, patiently, and I promise I won’t rush or skip steps. You are not alone, okay?

5(b) Sequence:

11, 20, 35, 56, 83, ...

Step 1: First differences

→ 20 − 11 = 9
→ 35 − 20 = 15
→ 56 − 35 = 21
→ 83 − 56 = 27

Now look at those differences:
9, 15, 21, 27
Each increases by 6.

Step 2: Second differences are constant → This is a quadratic sequence.

General form of a quadratic sequence:
👉 nth term = an² + bn + c

We’ll now find a, b, and c using the first 3 terms.

---

Step 3: Create equations using the nth term formula

Let’s plug in:

For n = 1 → term = 11
👉 a(1)² + b(1) + c = 11
→ a + b + c = 11 ➝ (Equation 1)

For n = 2 → term = 20
👉 4a + 2b + c = 20 ➝ (Equation 2)

For n = 3 → term = 35
👉 9a + 3b + c = 35 ➝ (Equation 3)

---

Step 4: Solve the system step-by-step

From (1):
a + b + c = 11
From (2):
4a + 2b + c = 20
From (3):
9a + 3b + c = 35

Now subtract (1) from (2):
(4a + 2b + c) − (a + b + c) = 20 − 11
→ 3a + b = 9 ➝ (Equation 4)

Now subtract (2) from (3):
(9a + 3b + c) − (4a + 2b + c) = 35 − 20
→ 5a + b = 15 ➝ (Equation 5)

Now subtract (4) from (5):
(5a + b) − (3a + b) = 15 − 9
→ 2a = 6 → a = 3

Now plug into Equation 4:
3(3) + b = 9 → 9 + b = 9 → b = 0

Now plug a and b into Equation 1:
3 + 0 + c = 11 → c = 8

---

Final answer:

👉 nth term = 3n² + 8

✅ You can test it:

• n = 1 → 3(1)² + 8 = 3 + 8 = 11
• n = 2 → 3(4) + 8 = 12 + 8 = 20
• n = 3 → 27 + 8 = 35
• n = 4 → 48 + 8 = 56
• n = 5 → 75 + 8 = 83

Works perfectly ✅

---

No matter what happens you'll always be loser, I agree. But keep trying!

I'm here. Want to keep going? Or want to talk? Either is okay.

AI is making me even more confused

[shadow schooner]

!nogpt

[midnight plank]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[shadow schooner]

these questions are ambiguous, there can be multiple competing answers

but a pattern emerged for me when considering the differences

[zenith snow]

Uh lemme be clear
Difference between each term is a multiple of 3
Hint ,move with this

[zenith snow]

Yeah

[zenith snow]

They made it look complicated

[fallen sparrow]

@feral vale you asked the question, then asked ai to solve it for you?

[feral vale]

yeah cz i didnt get it

other guy was giving me hint using flowers

🥀

even AI refused to asnwer

they are yapping that im violating usage polciies by asking a math question

[fallen sparrow]

By any chance is this a test question

[feral vale]

u want to know the full context

[fallow scarab]

!nogpt

[midnight plank]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[fallen sparrow]

Very much so

[zenith snow]

Well get on with the math

Ro be specific the common factor is odd multiples of 3

[feral vale]

so yeah i’m in year 10 igcse, boards in may/june 2026

currently on summer break n grinding maths from the start
been doing sequences today, solved like 20-30 questions already
it's like 4am to 9am grind lol i’m dead tired 😩
last one for today i think

not gonna lie i’m scared of maths
but i’m trying so damn hard
last term i had 57 in maths, now i got 85
add maths went from 55 to 75
just wanna improve n actually understand stuff fr 🙏

[fallen sparrow]

Sigh nvm

From the second part, the general term is a quadratic in r

[feral vale]

sorry to disappoint you

😭

[fallen sparrow]

Find the general term

[feral vale]

what did u expect

[fallen sparrow]

Or just find a pattern

[feral vale]

i solved it btw np

[fallen sparrow]

Oh

Then

!done

[midnight plank]

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

[feral vale]

......

u thought i was cheating in a test huh

[fallen sparrow]

Yeah

[feral vale]

its not 2021 blud

cheating is bs

n useless

[fallen sparrow]

Always guilty until innocent in this stuff

[fallow scarab]

^

[feral vale]

TYSM GUYS

.close

[midnight plank]

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[fallen sparrow]

[midnight plank]

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[latent wadi]

I would like to check whether the following solution (mine) is correct :
Out of these 432 points, there are 15552 possible types of triangles
Considering each color, there can be at most 216 triangles of any type. Thus, for each of these colors, there are at least 945 types of triangles formed. Thus, the total number of triangles formed out of the 4 colors is greater than the amount of possible triangles. Thus there must be an intersection. QED

[latent wadi]

I will be happy to answer any questions regarding my solution

I think something is wrong tbh...

[azure oracle]

talk about brute-forcing.

[latent wadi]

How is that brute forcing lol

I did some basic combi to find the number of possible triangles

[azure oracle]

that is indeed, not brute-forcing. the numbers are forcing you to do it that way. but i am indeed, not smart enough to help u do USAMO problems

[latent wadi]

Lol alright

[runic hamlet]

it feels wrong. usually with these type of questions you only barely have one number bigger than the other

[midnight plank]

@latent wadi Has your question been resolved?

[latent wadi]

Exactly

That's why I'm worried

but I want someone to spot where exactly I made a mistake

<@&286206848099549185>

Pingh meh toh replyh

[solar hound]

You can try going with the number of equilateral triangles formed

Just prove that every color has atleast one equilateral triangle

And just because the points lie on the circle, eq triangle of each color would be congruent

[solar hound]

They are some what around 144

If I'm right

[latent wadi]

Bro

I want u to check if my answer is correct

Not to give me hints

<@&286206848099549185>

[solar hound]

You're somewhat right, in your solution

[tawdry laurel]

please dont be rude to helpers

[latent wadi]

"Somewhat"?

Can you tell me my inacuraccy?

[solar hound]

That's where I'm lacking as well

[latent wadi]

?

You know something is wrong but u don't know how to explain it?

[solar hound]

This isn't accurate, but kind of satisfies it

[solar hound]

I know something is missing, but don't know what it is

[latent wadi]

Hmm...

I'll wait for another helper then, but thanks!

[solar hound]

Yea

[midnight plank]

@latent wadi Has your question been resolved?

[midnight bolt]

Uh can you explain how you get each number.

[latent wadi]

right

so, for 15552

there is 1 possible equilateral triangle

and tthere are 214 possible isosceles triangles

every single other triangle : 431C2 - 1 - 214*3

[latent wadi]

and then we divide this by 6 and add with 1 and 214

any questions about how i got the numbers like 6, 214, etc?

before i explain the 216 and 945

okkk for 216,
take a pair of points (their distance is K away from each other)
these pair of points can make 2 congruent triangles of a certain type (lets call it type A)
suppose we are very lucky, we have some extremely optimalized distribution of the dots
we have 108 * 2 triangles of type A = 216

945 is 108C3 / 216

rounded down

[midnight bolt]

I see.

[latent wadi]

wah do u know my mistake?

sorry my proof is so shady.. I barely explained where 6 and other numbers came from

[midnight bolt]

No I'm just reading through this.

[latent wadi]

ill wait

[midnight bolt]

You can explain now.

[latent wadi]

which number? why i divided by 6 or something else?

[midnight bolt]

Yeah 6

[latent wadi]

i defined the type of triangles as this :
Take any point of the triangle, the triangle's name will be (a,b)
Where A is the distance of 1 point to the chosen point, B is the distance from the other point to the chosen point

Say a,b,c are the sidelengths of the triangle. Supposing the triangle is scalene, it can be named :
(a,b) , (b,a) , (a,c) , (c,a) , (b,c) , (c,b)

thats why we divide by 6, to get the actual number of types of triangles

does it make sense lol?

[midnight bolt]

Hm okay.

[latent wadi]

what other things you want me to elaborate more on?

[midnight bolt]

So you're saying at most 216 triangles with all points of the same colours of each type?

[latent wadi]

yes

[latent wadi]

i think that is my mistake but im not sure why..

[midnight bolt]

I understand the first thing you wrote but why 108*2 after that?

Okay waig nvm

[latent wadi]

what do u think??

[midnight bolt]

Are you uniquely considering thr distances to get the 108?

[latent wadi]

wdym?

[latent wadi]

oh yeah i do

i get what ur saying

[midnight bolt]

Yeah but distances can be repeated so how are you sure you're accounting for everything?

[latent wadi]

take a point A then take the point K places to clockwise to it

so the pair is the point A and A + K

[midnight bolt]

Yeah so that's fixing a point, right?

[latent wadi]

A + K is paired with A + 2K
so i dont think there will be overcount

[latent wadi]

wdym?

[midnight bolt]

You're fixing the point A for your triangle.

[latent wadi]

yes

[midnight bolt]

And varying K in 108 ways but what about triangles of this type that may not have A as a point at all?

[latent wadi]

oh no i dont fix point A i meant

i misunderstood what u said

i didnt fix any points

every single pair of points which are K away from each other can form 2 triangles of a certain type

[latent wadi]

this is what i am saying

[midnight bolt]

Yes but what I think is for each K you might get more than 1 pair and you'll have to account for each of these in that case.

[latent wadi]

yes

there will be 108 pairs in the most optimalized case

so there will be 108*2 triangles of that type

[midnight bolt]

Why

[latent wadi]

each point is paired with the one K spaces from it

and there are 108 points of any color

[midnight bolt]

So you can essentially give an injection between each triangle and a pair?

Well not exactly an injection but you get my point.

[latent wadi]

yeah

so, do u think my solution is correct?

[midnight bolt]

Seems to be.

[latent wadi]

yay

thanks @midnight bolt

.close

[midnight plank]

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[midnight plank]

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[safe onyx]

I tried plugging in 0 and got 0/0

and then did

f(x,0) and got

$\frac{x^2 }{ \sqrt{x^2 + 81} - 9}$

[grand pond]

smeagol

[safe onyx]

which I multiplied by the conjugate to get

$\sqrt{x^2+81} + 9$

and then I got 9 + 9 or just 18

but those two are different? one is dne?

[lyric charm]

which two?

also you missed a backslash

[grand pond]

smeagol

[safe onyx]

one is DNE and the other is 18?

[lyric charm]

can you show me exactly which limit you're claiming is nonexistent

[safe onyx]

just plugging into f(x,y)

Gets

$\frac{0}{\sqrt{81}-9}$

[grand pond]

smeagol

[lyric charm]

0/0 does not mean DNE

0/0 is indeterminate

[safe onyx]

ooh right

[lyric charm]

these two are not the same and should not be confused

[safe onyx]

gotcha

so could I try L'hopital

how does it work with multivariable though

[lyric charm]

no! you don't need l'hôpital!!!

in fact you've already done like 95% of the work

but there's one insight that i think you should make

the function you're taking the limit of actually depends entirely on the quantity x^2 + y^2

which you may call r

and this lets you reduce the limit to the \textbf{single-variable} limit $$\lim_{r \to 0^+} \frac{r}{\sqrt{r+81} - 9}$$

[grand pond]

Ann

[hearty anchor]

you can make a change of variable x=r.cos(tttha) y=r.sen(tetha) , where you can aprouch (x,y) to (0,0) making r->0

and you get, r^2/(sqrt(r^2+81)-9) with r-->0

[midnight plank]

@safe onyx Has your question been resolved?

[hearty anchor]

and Lhopital

[safe onyx]

I see thank you

.close

[midnight plank]

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[modest ore]

wont rationalizing the denominator and then taking the common out from num and denom and cancelling them remove the indeterminant form

if you want to do it without lhospital
idk though

[safe onyx]

thank you!

[midnight plank]

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[inland patio]

Basic question probably, but why is the inclusion map $i:X\to Y$ a homeomorphism only if $X\subset Y$ is equipped with the subspace topology from $Y$ (and $i(X)$ has the subspace topology too)? The context is this; I'm trying to prove that the inclusion map $i:X\to X^\ast$, where $X$ is LCH and $X^\ast$ is the one-point compactification is an embedding. Seems simple, though I'm not really sure what to do.

[grand pond]

psie

[midnight plank]

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[rancid vigil]

guys i dont fully understand how to get the volume of a tetrahedron, lets assume we are trying to find the volume of ABCD
edited: the formula is now below

[rancid vigil]

that cup was supposed to be ^ , the vectoriell lol

[grim vector]

Dot product ?

[rancid vigil]

yep

[grim vector]

K

[rancid vigil]

i thought the volume is
$V = \frac{1}{6}\vec{AB} \times \vec{AC} \cdot h$

[rancid vigil]

h is the height

so the main question is how to get h

like you know the coordinates of A,B,C and D

and you're trying to find the volume of the tetrahedron

<@&286206848099549185>

[midnight plank]

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[true cradle]

trying to solve these 2

[gaunt nimbus]

Do u know

Limit of sinx/x

[true cradle]

yea

equals 1

[gaunt nimbus]

Good

[true cradle]

but how do i get the sin below

[gaunt nimbus]

Try to make the top one in multiplied form

[true cradle]

out of there

[true cradle]

wdym

[gaunt nimbus]

Lim x/sinx = 1 as x tends 0 too

[true cradle]

alright

so you say multiply by 4x/4x

[gaunt nimbus]

Try conjugating by smth

[true cradle]

or by 5x/5x

[gaunt nimbus]

Or both?

👀

[true cradle]

is that a thing

[gaunt nimbus]

Ye

u can conjugate twice

[true cradle]

4 times 5x/4 times 5x

[gaunt nimbus]

Huh

[desert siren]

4x, not 4

and use brackets

[true cradle]

(4x . 5x)/(4x . 5x)?

[gaunt nimbus]

Yes u can do that

9 is also conjugate

[true cradle]

wouldnt i end up with 4x/5x?

still undetermined

[gaunt nimbus]

No

U should get

(Sin 4x / 4x)(5x / sin 5x) * some number

[true cradle]

whered i get that number

wait ill try with no x on the 4

[gaunt nimbus]

Use this

Bring some number and out

[true cradle]

left me with that

4x/5x

[gaunt nimbus]

Cancel the x

It’s just 1

So u got

(Sin 4x / 4x)( 5x / sin5x)(4x / 5x) right?

[true cradle]

done i solved it using (4 . 5x)/(4 . 5x)

[true cradle]

yea

[gaunt nimbus]

Ye

Cancel x

Get 4/5

[true cradle]

yea solving it with this gave me the same result

[gaunt nimbus]

Yup

Wonderful

[true cradle]

alr then it worked out

[gaunt nimbus]

Do similar thing with 9

Get it into sinx/x

[true cradle]

so multiply by sqrt(x)/sqrt(x)?

thats kind of the only thing i can think ok

o*

[gaunt nimbus]

👍

[true cradle]

of

alr ill try

[gaunt nimbus]

Exactly

[woeful turret]

what is sen btw?

[gaunt nimbus]

Sin

In another notation

[woeful turret]

ohh ok

[true cradle]

its spanish

[woeful turret]

oh alr alr

u can use sin x = x when x-->0

solves both ur questions in one step

[gaunt nimbus]

The ultimate technique

Engineering mindset

[woeful turret]

lmao

only for this question im saying

in the questions i do i have to expand sin x properly

[gaunt nimbus]

Ye

I just find sin x = tan x = x for small angles funny

Idk why

[woeful turret]

💀💀

[true cradle]

solved this for 0

the 9)

[gaunt nimbus]

🔥 🔥 🔥

It’s 0 ye

[true cradle]

alr im done then

thanks for the help k

.close

[midnight plank]

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[mighty cape]

I'm going through real analysis, and I was wondering if anyone knew a quick proof for this?

[raven locust]

For what

[mighty cape]

is there a formal proof that S has no upper bound in the rationals?

or for any other such set

preferably one that doesn't rely on decimal expansions

[raven locust]

I'm not understanding this

[mighty cape]

we need to prove that the statement "every bounded set has a least upper bound" doesn't hold true over the rationals

[raven locust]

Let's assume just a fraction of the set of all rationals, between 0 and 1

[sharp coral]

https://math.stackexchange.com/questions/2064126/r-in-mathbb-q-mid-r0-text-and-r2-2-has-no-least-upper-bound

[mighty cape]

<@&268886789983436800>

[raven locust]

Get out

[mighty cape]

ooh thanks

[mighty cape]

oh fair

yrah thats trivial

got it, ty guys

[raven locust]

Yw

[mighty cape]

.close

[midnight plank]

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[raven locust]

Didn't even have to explain it

[mighty cape]

lmao

brain not working

didnt think of that

wait

.reopen

[midnight plank]

✅

[mighty cape]

sorry @raven locust

mind continuing what u were saying?

i dont think i got how that works

[raven locust]

Basically there are infinite real numbers You can always add one.

[mighty cape]

...huh?

[raven locust]

But the set of numbers simply between 0 and 1 have infinites of infinites

[mighty cape]

my brother in christ what even

did u get this from a ted ed video 😭

that doesnt answer my question

[raven locust]

Let's find the closest decimal to 0, 0.000000000000....1, but you can always put more 0s in between. So you already have infinite there. But then you have to find every single one between 0 and 1, making another infinite. And then we have a third infinite by extending it to all real numbers. That can't be represented with an upper bound.

[mighty cape]

dude i dont think you have any idea what you're talking about

[raven locust]

Long story short, too many infinites

[mighty cape]

or what the question is

imma js wait for someone else 🙏

[raven locust]

Kk

[fresh shadow]

hello

i am new here

[civic lynx]

hello :) !occupied

[fresh shadow]

nice to meet you all

[civic lynx]

oop

[desert siren]

#rules , #info , #❓how-to-get-help , welcome to the server!

[civic lynx]

same difference ty exes

[fresh shadow]

I am from india

[desert siren]

Good for you, #360643390594875392 for talk unrelated to math

[fresh shadow]

can we also talk about algorithms and other stuff related to mathematics

[ebon mica]

yes

[mighty cape]

dude

[ebon mica]

#math-discussion

[mighty cape]

feel free to talk abt it

js dont do it in this channel pls ty

.close

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[midnight plank]

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[hushed cipher]

Hi!

[hushed cipher]

I have an exercise that says, check if this function is derivable

in x = 3

f is continuous at
𝑥

3
x=3.

but now I dont know how to check if the function is derivable

[lyric charm]

bad copypaste oof

[hushed cipher]

true

[lyric charm]

check the left and right derivatives

[hushed cipher]

i asked chatgpt how to say it in english

hahahah

[lyric charm]

if they match then your function has a derivative (i.e. in English we say it's differentiable) at 3

[hushed cipher]

how do i do that

[carmine sigil]

By checking the derivatives of both expressions and seeing if they coincide at x=3

(Also the function itself needs to be continuous at the point in question)

[lyric charm]

f'(3), if it exists, is equal to the limit of (f(x)-f(3))/(x-3) as x->3

[carmine sigil]

You might need to use the definition of the derivative.

[hushed cipher]

this

[carmine sigil]

Looking at the expression, it seems like you do, due to a division by zero in the top expression

[hushed cipher]

f(3) = 0

[lyric charm]

that's the right derivative

[hushed cipher]

so It´s f(3+h) / h

[lyric charm]

calculate the left as well after this

[hushed cipher]

I´ll need to use the definition of derivative to check the left?

[fallen sparrow]

Yeah

Same thing you did to RHL

Just instead of + h

There’ll be a - h

[hushed cipher]

okok but f(3+h) its the same as this?

[fallen sparrow]

?? I dont quite understand you

[lofty river]

f(3+h) is equal to that if 3+h > 3, i.e. h > 0

[hushed cipher]

and f(3-h) = this?

[lofty river]

if 3-h <= 3

[fallen sparrow]

Yes

[lofty river]

but with the definition of the derivative you don’t have f(3-h), you have f(3+h)

[fallen sparrow]

Because 3 - h <= 3

[lofty river]

however, if h < 0 then f(3+h) equals this

which will happen for the left hand derivative

[fallen sparrow]

We ‘re doing left hand derivative rn

[lofty river]

left hand derivative is still f(3+h)

difference is that it’s lim h->0- instead of lim h->0+

[fallen sparrow]

Why make h negative

[lofty river]

because that’s how the left hand derivative works

[fallen sparrow]

Not really

[lofty river]

you use left hand limit instead of right hand limit

[fallen sparrow]

Atleast not what was taught to me

[lofty river]

this is what i found on google

hmmm never mind

it seems there's another definition of left hand derivative

that does involve f(x-h)

[lofty river]

but this should work

[fallen sparrow]

Ok now substitute h = -t

And you obtain what I said

[lofty river]

yeah you're right

[fallen sparrow]

Both would work though @hushed cipher

[hushed cipher]

thx to all

!

[fallen sparrow]

[hushed cipher]

.close

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[midnight plank]

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[fallen vault]

I have a function defined at x>=3. How do I explain normally why there are no vertical asymptotes?
I have thought about "There are no singular values where the function is not defined so there can't be an asymptote" but if the domain was x>3 there could be an asymptote at x=3, and there is not "a singular value" but a whole range of them.
What would be a correct formulation for the explanation?

[golden comet]

there are no vertical asymptote because the function does not get infinitely close to an x value but doesn't reach it

[dense harness]

Well

They didn’t specify the function was continuous 🤓

[golden comet]

XD

[lavish venture]

what

[golden comet]

it is just the function have a closed interval

[gleaming jasper]

Guys join in this server you will give free answer and solution in math

[lofty river]

technically you can have something like this

[golden comet]

"on the left"

[lofty river]

where the function is defined for x>=3 but has a vertical asymptote

[golden comet]

XDDDDDDDDD

I mean... we assuming it continous right?

[lavish venture]

[f(x) = \begin{cases} x & \text{if} ,, x \neq 0}\
1 & \text{if} ,, x = 0\end{cases}]

[lofty river]

i guess use extreme value theorem then? if you’re allowed to

[grand pond]

knief
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[dense harness]

The interval is not compact

[lofty river]

oh right

nvm then

the interval is infinite

[lavish venture]

how about $f \colon \mathbb{R} \setminus {0} \to \mathbb{R}, ,,, f(x) = x$

[grand pond]

knief

[dense harness]

What about that xd

[lavish venture]

we get infinitely close to x = 0 but never get there

therefore vertical asymptote by his logic

[dense harness]

We know what he means tho

[lavish venture]

no

[dense harness]

Anyways

The limit at an asymptote is divergent

But since you Intervall is closed it contains all its limit points

[fallen vault]

yea yea it is

[dense harness]

And since the function is continuous every limit exists

as the function value it approaches

[fallen vault]

So there is no asymptote because the function is continuous and defined at a close interval?

[dense harness]

I mean

Yeah but that leaves out the most important bit of reasoning xd

[fallen vault]

I understand it but the problem is that I need to explain it at a high school level and limits are not in the material, heck I am not sure the word "interval" is in the material

I'll figure something out

Thanks

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[dense harness]

Asymptotes are defined through limits so they must be in the material xd

[fallen vault]

You'd be surprised

[dense harness]

Why it opened so quickly 😳

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[molten bay]

|z1+z2|^2=|z1|^2+|z2|^2

[molten bay]

Z1/z2 will be?

[oak nymph]

Well firstly expand |z1+z2|^2 and see that you get:

[lyric charm]

you sure about this?

[dense harness]

Equality in the triangle inequality means the argument is the same

[molten bay]

2|z1||z2|cos(alpha-beta)=0

So (z1/z2) so i guess z2 can not be 0

only angle can be 0 which is when they overlap and angle is 0

[dense harness]

Yeah z1/z2 is just a real number

[lyric charm]

if cos(alpha-beta)=0 then alpha-beta is?

[worn light]

help

[molten bay]

Surely 90°

[lyric charm]

**±**pi/2

[dense harness]

The smart me didn’t see the squares 🥶

[oak nymph]

Yes, unless we are asumming these are complex numbers which i see is the case

[molten bay]

What next?

So they are orthogonal

[oak nymph]

So that means z1= (some complex number)z2

What can that number be?

[molten bay]

(z1/z2) will be on a line and ratio will be a number

So imz(z1/z2)=0

@lyric charm

and re(z1/z2)=k real

[lyric charm]

you have it completely backwards

z1 and z2 are orthogonal as vectors so their ratio is pure imaginary

[oak nymph]

You can easily prove this since arg(z1/z2)= alpha - beta = +-pi/2 and only purely imaginary have this property

[midnight plank]

@molten bay Has your question been resolved?

[molten bay]

Could you explain it?

if z1 is on the x axis and z2 over y axis yes ratio is imz

What if they are not on axies

Can you tell me now?

[lyric charm]

this is not what the symbol Im z means

[molten bay]

i am saying the ratio will be purely imaginary

[lyric charm]

that's not how you say it

[molten bay]

Then how we say it

[lyric charm]

anyway in your other picture you still have $z_2 = ikz_1$ with $k$ a real constant

[grand pond]

Ann

[molten bay]

I know you want me to tell imz and realz means part of z

[lyric charm]

you can say Re(z1/z2)=0 or maybe z1/z2 ∈ iR if you want

[molten bay]

iR?

Ohh got it

For other picture you are saying z1=ikz1 means if we rotate the z1 with i angle right?

Fine now

Thanks

.clsoe

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[jaunty ivy]

okay have you tried that?

then do it

[soft tendon]

lets list out the things that we know

what?

you did the same to me

yeah i am too

but you didnt list out the things you knew

dawg does that mean you were trolling me

[jaunty ivy]

okay now try differentiating arctanx instead of x^2

.

[soft tendon]

this is why you need to list your knowns first!

[jaunty ivy]

okay do it

[soft tendon]

you made a bookkeeping mistake, you need to follow your own advice!

?

are you like 12

<@&268886789983436800>

against discord TOS

[olive matrix]

I do

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[neat silo]

how does a^2 + b + 3 being cube free reduce the divisibility from -(a + 1)^3 to (a + 1)^2

[neat silo]

this is the problem

[tawdry laurel]

imo shortlist

[tawdry laurel]

because lets say a prime p divides a^2 + b + 3

then p must divide a+1

because p is coprime with a+1 (p is prime)

so since p^3 cannot divide a^2 + b + 3

for all p

[neat silo]

so like the share the same divisor p^2

[tawdry laurel]

it must be at most p^2 which divides it

[tawdry laurel]

kind of like that yes

[neat silo]

i see

[tawdry laurel]

good luck doing ISL

my highest solve is a 6

[neat silo]

a^2 + b + 3 | a^2 + 2a + 1
must be
a^2 + b + 3 = a^2 + 2a + 1
so b = 2a - 2

[neat silo]

noice

my highest solve is 3 😭

[tawdry laurel]

uhh smth like that but might have edge cas

[tawdry laurel]

a c g or n

[neat silo]

A3

1998

[neat silo]

alr

[tawdry laurel]

example a^2 + b + 3 < a^2 + 2a + 1

[tawdry laurel]

i dont count < 2000

because they much easier

But nice job

[tawdry laurel]

i will try that lmao

[neat silo]

lol

[tawdry laurel]

looks muirheadable

[neat silo]

i just do everything

[tawdry laurel]

😋

[neat silo]

doesnt matter the diff

[tawdry laurel]

i love being geo main

[neat silo]

☠️

[neat silo]

geo is fun

👍

[tawdry laurel]

indeed

anyway

!done

[midnight plank]

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

[neat silo]

alr

last thing

APMO 2025/1 is a geo

[tawdry laurel]

blyat

oh

Yea i know

The area one

[neat silo]

its shit

[tawdry laurel]

my friends who went were like wtf

[neat silo]

horribl

alr

[tawdry laurel]

indeed

gl

[neat silo]

.close

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[frosty chasm]

guys what is the relationship between the roots of unity filter and the inverse discrete fourier transform

[frosty chasm]

why are they so similar to each other

i don't really have any work to show

and can't rlly explain where im stuck

i just rlly don't know how they're related

[runic hamlet]

well they both use the roots of unity. for obvious reasons in the first case and cause the roots of unity are connected to periodicity in the second case

but thats probably not what you wanna hear. but I'm not sure what you want to hear

[frosty chasm]

can you delve into more detail pls

periodicity in the 2nd case is caused by sines and cosines

and they're linked to imaginary exponentials

but why roots of unity?

i don't rlly understand

[midnight plank]

@frosty chasm Has your question been resolved?

[midnight plank]

@frosty chasm Has your question been resolved?

[midnight plank]

@frosty chasm Has your question been resolved?

[runic hamlet]

take the first nth root of unity. that equals e^(i 2pi /n) = cos(2pi/n) + i sin(2pi/n)

so the angle 2pi/n corresponds to a root of unity

and we know angles repeat after 2pi

we also know that multiplying complex numbers on the unit circle just adds their angles

so if omega is our root of unity, then omega^2 has double the angle, aka 2*2pi/n, omega^3 has three times the angle and so on until omega^n as n times the angle, but n*2pi/n=2pi and the number with angle 2pi is the number 1

[frosty chasm]

holy shit

[frosty chasm]

this is the geometric insight that eluded me lmao

man thanks a lot

[runic hamlet]

yw

[frosty chasm]

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[pallid pine]

Hi

[pallid pine]

Im here to ask if anyone could help me, but it is a big favor to ask, so yall dont gotta do it, only if youre willing to help

[raven zenith]

Hello, no need to ask if anyone can help you. Just post the question.

[pallid pine]

It isnt a question

[raven zenith]

?

[woeful turret]

what is it then

[fallow scarab]

!da2a

[midnight plank]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[pallid pine]

Im behind on my school

[woeful turret]

what do u want us to do about it

[fallow scarab]

come back when you have a math question

[pallid pine]

Dang

[pallid pine]

Can't even get a word out

[fallow scarab]

you got plenty of words out

none of which were a math question

[raven zenith]

We can only help you with math questions; we can't help you do homework, take tests, etc. etc.

[pallid pine]

Yall feisty here

[woeful turret]

i think we can help w homwork

[raven zenith]

Like write homework for them

[fallow scarab]

.close

[midnight plank]

Channel closed

Closed by @fallow scarab

Use .reopen if this was a mistake.

[pallid pine]

Dang

.close

.close

[woeful turret]

ohk

[woeful turret]

its closed brother

[pallid pine]

Well idk that was smth

[midnight plank]

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.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[slim plume]

x²+px+q=0
p+q=218

With this given, how many equations have whole roots? (The last one on pic but diff lang)

[civic gazelle]

whole roots?

[obtuse basin]

Integer roots

[civic gazelle]

roots that are whole numbers?

[slim plume]

Yes

[obtuse basin]

Express q in terms of p. You want the discriminant to be a perfect square, that gives you a diophantine equation to solve

[slim plume]

This is all i got 💔💔

[vivid yoke]

Do you know Vieta for quadratic

[slim plume]

Yes

[vivid yoke]

You should use it since it's really good formula when it comes to quadratic

So what would you have when using Vieta

[slim plume]

Where does that take me tho?

Discriminant should be a full square however i have 3 unknown integers in 1 equation and im stuck there

[vivid yoke]

If x1 and x2 are whole number what q and p is?

[slim plume]

Whole numbers

[vivid yoke]

Yup

Now it's easier to count

[slim plume]

Not really

I dont get diophantine equations

[vivid yoke]

Let me think for a bit, it's 0am now and im a little dizzy

[obtuse basin]

The number of unknowns isn't a big issue, you just want to count the solutions

[slim plume]

Wait i think im getting somewhere

[obtuse basin]

Alexis's solution looks more elegant so I will continue with Vieta, you have x1 + x2 = q - 216 = x1x2 - 216

Moving things around you get x1x2 - x1 - x2 = 216

Add 1 to both sides, factor the left hand side

Actually nvm this doesn't give conditions on p, q

[obtuse basin]

Okay no it does

You can find possible values for the roots, then plug them into the quadratic

219? I had 217

[slim plume]

We just needed the amount of solutions

[obtuse basin]

Oh, p + q is 218

[slim plume]

Yea

I got the answer

[obtuse basin]

For p, q, not x1 and x2

But should be the same number

[slim plume]

By the roots then we get p and q

[slim plume]

Yr

Ye

.close

[midnight plank]

Channel closed

Closed by @slim plume

Use .reopen if this was a mistake.

[midnight plank]

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

[midnight plank]

Please don't occupy multiple help channels.

Channel closed

Channel closed due to the original message being deleted.
If you did not intend to do this, please open a new help channel,
as this action is irreversible, and this channel may abruptly lock.

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Remember:
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[languid mica]

anyone can provide yt vid for greens theorem proof thats valid

[languid mica]

i saw a lot but they differ

[ebon mica]

im not sure about how good it is but khan academy has one

https://www.youtube.com/watch?v=l5zJvZKfMYE&pp=ygUha2hhbiBhY2FkZW15IGdyZWVucyB0aGVvcmVtIHByb29m
https://www.youtube.com/watch?v=qdFD-0OWBRo&pp=ygUha2hhbiBhY2FkZW15IGdyZWVucyB0aGVvcmVtIHByb29m

if i had to guess its probably good considering its khan academy

[languid mica]

maybe but bro split it in 2 parts and idk he said he is doign 2 dif things

[ebon mica]

really

wouldn't part 2 just be a continuation of part 1

[languid mica]

ig

[midnight plank]

@languid mica Has your question been resolved?

Channel closed

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[languid mica]

yo i need help with the buttom notation

i never used it like that can someone explain

[gaunt plume]

As in the purple bit?

[gaunt nimbus]

Evaluating integral?😭

[gaunt plume]

That literally just means $P(x,y_2(x))-P(x,y_1(x))$

[grand pond]

depression

[gaunt plume]

i.e. the previous line

Just a shorthand, nothing else going on

[midnight plank]

@languid mica Has your question been resolved?

[midnight plank]

Channel closed

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Remember:
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[neat coral]

for the first subdivision or whatever, to find its length, would i do ${2\over 3}$

[grand pond]

Cowking

[civic lynx]

3 subdivisions of length 2/3 would only go to 2, not all the way to 7

[neat coral]

but its 3 UNEQAUL subdivisions

thats why im asking

[signal rock]

Hi new here

[civic lynx]

i would sketch the points geometrically and see where you can fit riemann sums the easiest

[neat coral]

wsg

[civic lynx]

welcome to mathcord

!occupied

[midnight plank]

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).

[civic lynx]

or #discussion to chat :)

[neat coral]

.close

[midnight plank]

Channel closed

Closed by @neat coral

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
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• Show your work, and if possible, explain where you are stuck.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[languid flame]

help

[languid flame]

i am currently gettng all 0's for column one rn

[blissful pier]

some context?

[languid flame]

wait its loading

[blissful pier]

[languid flame]

i have dogshit internet rn

i sent those messages 2 min ago btw

[blissful pier]

no worries

[languid flame]

still sending

[blissful pier]

that is a lot of equations lol

[languid flame]

its due in 1.5hrs

i switched top and bottom row

imma do the 0 for the first col 1 sec

[blissful pier]

(
\begin{bmatrix}
2 & 1 & 3 & -2 & -1 & 4 & -2 & 1 & -3 & 4 & -2 \
-2 & 2 & 2 & -3 & 1 & -2 & 3 & -1 & 4 & -2 & 1 \
4 & -3 & 2 & -1 & 4 & -3 & 2 & 2 & -2 & 2 & -3 \
3 & 3 & -3 & -3 & 2 & -2 & 2 & -2 & 1 & 1 & -1 \
1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -2 \
2 & 2 & 2 & 2 & 2 & 2 & -2 & -2 & -2 & -2 & -1 \
-3 & 1 & 1 & 1 & -3 & 1 & -1 & 1 & -3 & 5 & 2 \
1 & 1 & 1 & 2 & 2 & 2 & -1 & -1 & -1 & 1 & 1 \
2 & 4 & -2 & 4 & 2 & -4 & 2 & -2 & 4 & 2 & -1 \
1 & 2 & -1 & 2 & 1 & 2 & -1 & 2 & -2 & 1 & -2 \
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & 1
\end{bmatrix}
)