#help-49
sharp ember
midnight plankBOT
@glad merlin Has your question been resolved?
small jasper
!occupied
midnight plankBOT
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frosty terrace
the digits must follow a zizag pattern, which is determined from your first 2 digits
let the digits be d1, d2, .., dn
eoither you have d1 < d2, d2 > d3, d3< d4, ...
or d1>d2, d2<d3, d3>d4, ... and so on
midnight plankBOT
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twilit totem
@glad merlin !status
glad merlin
.close
midnight plankBOT
midnight plankBOT
Available help channel!
tidal turret
midnight plankBOT
tidal turret
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
Let $\mathcal{F} = {f : {1,2,3,4,5} \to {1,2,3,4,5,6,7,8,9} }$.
a) Determine how many functions $f \in \mathcal{F}$ satisfy $#{x \in \operatorname{Dom}(f) \mid f(x) = 9 } = 2$.
b) Determine how many functions $f \in \mathcal{F}$ satisfy $# \operatorname{Im}(f) = 4$.
grand pondBOT
Renato
polar mortar
do you know combinations?
tidal turret
just the basics
polar mortar
that should be all you need
For a), you'd choose two elements of {1,2,3,4,5} to go to 9 and then the other three elements can go to any of {1,2,3,4,5,6,7,8} (with possible repetition)
tidal turret
P(5,2) × C(9,?)?
polar mortar
The first wouldn''t be a permutation, since order won't matter
choosing 5 and 4 to go to 9 is the same as choosing 4 and 5
The second part isn't a combination or permutation, since repetition is allowed
tidal turret
care to elaborate?
polar mortar
on which part?
polar mortar
In a permutation, order matters. So choosing 5 then 4 would result in a different function than choosing 4 then 5. But it doesn't, they both just go to 9.
So you can't count it with a permutation.
You can count it with a combination since you are choosing 2 of 5 elements, without repetition.
tidal turret
wait
tidal turret
For a), you'd choose two elements of {1,2,3,4,5} to go to 9 and then the other t...
so its a permutation because repetition is allowed?
tidal turret
In a permutation, order matters. So choosing 5 then 4 would result in a differen...
i didnt understood this part
polar mortar
You see in the image, (a)(b) and (b)(a) are both counted as distinct things in the permutation. Order matters.
In the combination there is only (a)(b) because order does no matter. So (a)(b) and (b)(a) are the same thing
tidal turret
2 out of 5 go the f(x) = 9
tidal turret
You see in the image, (a)(b) and (b)(a) are both counted as distinct things in t...
yeah, but how does it relate to our problem specifically
the permutation vs combination thingy
polar mortar
is the function
1->5
2->9
3->9
different from the function
1->5
3->9
2->9
?
polar mortar
so the order you choose elements that map to 9 does not matter
you're just choosing 2 of 5 elements to go to 9
so it is not a permutation, it is a combination
tidal turret
so it is not a permutation, it is a combination
the other way around, is a permutation
polar mortar
no. In a permutation, order matters
tidal turret
is the function
1->5
2->9
3->9
different from the function
1->5
3->9
2->9
?
ah right my bad with permutation we would be counting this twice
we need combination, my bad
polar mortar
yes
so you'd use a C(5,2) for the elements going to 9.
then there's 8 options for the other three elements to go to, with repetition. So it is neither a combination nor a permutation, it just multiplication rule.
polar mortar
each remaining element can go to any number except 9
polar mortar
so you'd use a C(5,2) for the elements going to 9.
then there's 8 options for t...
it's neither a permutation or combination. Just multiplication rule.
tidal turret
care to elaborate?
polar mortar
Say we choose 4 and 5 go to 9, for the sake of example.
how many numbers can 1 go to?
how many numbers can 2 go to?
how many numbers can 3 go to?
tidal turret
8x7x6
tidal turret
8^3
polar mortar
it definitely takes a while to get used to
viral coral
combi is so hard dude
nah
polar mortar
It's not just C(9,4) because you can have different functions within that
1->2
2->2
3->3
4->4
5->5
and
1->5
2->2
3->3
4->4
5->2
for example
polar mortar
but i gtg for a bit and get some work done
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
@tidal turret Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
burnt wraith
. @lean shale post it here
lean shale
Alright
lean shale
How do i go about these questions
The example 27
pearl hull
forest iron
Do you know those trigonometric identities?
lean shale
Do you know those trigonometric identities?
I only know these three
sin²A+ cos²A
1+tan²
1+cot²
lean shale
Like sin^2(x)+cos^2(x)=1
Yes
lean shale
But how am I supposed to know
Which identity is to be used when
last slate
You don't.
lean shale
Damn
lethal path
How do i go about these questions
The example 27
try $\sin \theta = (\sqrt{2} - 1) \cos \theta$ and $\sin^2 \theta + \cos^2 \theta = 1$
grand pondBOT
south
forest iron
Yea
lethal path
there are probably quicker ways but it's not useful to introduce tricks
Udon says they only know those 3 identities above
modern sapphire
tbf you dont need any identity, you can just substitute the first equation into the LHS of the second, and you arrive at the RHS of the second equation
lethal path
I'd love to be able to use $\cos(-\theta) + \sin(-\theta) = \sqrt{2} \cos(-\theta)$ for instance
grand pondBOT
south
forest iron
Lol
lethal path
tbf you dont need any identity, you can just substitute the first equation into ...
wait, like $2 \cos \theta - \sqrt{2} \cos \theta$?
modern sapphire
No, like $\sin \theta = (\sqrt{2} - 1) \cos \theta$
grand pondBOT
Βαχτερ10Φρ4γ
lethal path
oh right, and then you just find cos theta - sin theta, sqrt(2) sin theta separately
show those are equal
omg
modern sapphire
yea, no identity needed
lethal path
oh lmao
vivid yoke
Βαχτερ10Φρ4γ
can we just like, multiply both side by sqrt(2)+1
modern sapphire
just sqrt2 is enough
lean shale
can we just like, multiply both side by sqrt(2)+1
Now that was wonderful
Got this one
modern sapphire
can we just like, multiply both side by sqrt(2)+1
oh you meant in the original question... mbmb
lean shale
How do i go about these questions
The example 27
Now how do i go about the (ii) part
lean shale
Now how do i go about the (ii) part
Of 27
lethal path
Now how do i go about the (ii) part
I guess if you let cos(t) - 2 sin(t) = k, for some value of k
2 sin(t) + 4 cos(t) = 2, and you can add this to cos(t) - 2 sin(t) = k to get 5 cos(t) = 2 + k
similarly, try to isolate sin(t), and then cos^2 t + sin^2 t = 1 gives a quadratic in k
or actually, quicker is $(\sin t + 2 \cos t)^2 + (\cos t - 2 \sin t)^2 = 1^2 + k^2$
grand pondBOT
south
lethal path
or actually, quicker is $(\sin t + 2 \cos t)^2 + (\cos t - 2 \sin t)^2 = 1^2 + k...
yeah this is better
midnight plankBOT
@lean shale Has your question been resolved?
supple wing
How would I solve this?
$x^3+2x+1$
midnight plankBOT
How would I solve this?
$x^3+2x+1$
grand pondBOT
Renna Tempest
sick fossil
How would I solve this?
$x^3+2x+1$
is that $x^3 + 2x + 1 = 0$?
grand pondBOT
soup_norm
supple wing
Yes I forgot the zero
sick fossil
oh
can you send a photo of the original question? there might be parts to it that can help you with this
supple wing
There is no other parts
smoky seal
I would assume they want you to use the quadratic formula to solve
supple wing
Yea but this is a cubed
fallen violet
Yea but this is a cubed
use the cubic formula
supple wing
I thought up to this problem but didn’t know how to solve it
sick fossil
There is no other parts
still though, could you send a picture
supple wing
use the cubic formula
Bruh ain’t no ways there’s a cubic formula
sick fossil
Bruh ain’t no ways there’s a cubic formula
there is one
fallen violet
Bruh ain’t no ways there’s a cubic formula
there is
sick fossil
I thought up to this problem but didn’t know how to solve it
so you came up with it yourself?
supple wing
Yea I was just thinking if I saw that problem I the wild
sick fossil
ahh
fallen violet
Is there one for every dimension then?
there is only up to quartic
supple wing
But it may be above what I’m currently learning
When would you learn the cubic formula?
fallen violet
When would you learn the cubic formula?
never
hard shard
in a curriculum? never
sick fossil
Yea I was just thinking if I saw that problem I the wild
well there's a systematic way to solve cubics like this, but it isn't taught in high school usually
supple wing
So then how is one supposed to solve that problem? Or do they just never give you that
sick fossil
How would I solve this?
$x^3+2x+1$
this is called a depressed cubic since there isn't a x^2 term
hard shard
So then how is one supposed to solve that problem? Or do they just never give yo...
they wouldnt give you one unless it was easy to solve
sick fossil
So then how is one supposed to solve that problem? Or do they just never give yo...
you could use numerical approximation
hard shard
there are tricks that can be useful for solving higher degree polynomials, but those polynomials are special cases
supple wing
This looks like a simple equation tho?
hard shard
unfortunately its not
sick fossil
So then how is one supposed to solve that problem? Or do they just never give yo...
there's a method given here https://en.m.wikipedia.org/wiki/Cubic_equation#Depressed_cubic, we could try it here
hard shard
,w solve x^3+2x+1=0
grand pondBOT
hard shard
see
hard shard
however, if you changed the first plus to a minus, the equation x^3-2x+1=0 has an easy root
supple wing
But the equation would be so much simpler to solve without the +1
mortal falcon
why do u wanna solve that
hard shard
But the equation would be so much simpler to solve without the +1
yes, that is an example of a special case
fallen vigil
This is not that tedious to solve
sick fossil
But the equation would be so much simpler to solve without the +1
yeah, putting a +1 can make equations that much harder
supple wing
why do u wanna solve that
Just thought about a Random problem but didn’t know how to solve it
Do you know how to solve depressed cubes?
fallen vigil
You can factor $x^3+2x+1$ by setting $x^3+2x+1=(x+a)(x+b)(x+c)$ and expanding the right side to get some equations
grand pondBOT
purururuuriuruin
mortal falcon
that’s fair, cubics and beyond kind of suck in general but depressed cubics aren’t too bad
Look up cardano’s fornula
sick fossil
Do you know how to solve depressed cubes?
you can find a method here https://en.m.wikipedia.org/wiki/Cubic_equation
lyric charm
You can factor $x^3+2x+1$ by setting $x^3+2x+1=(x+a)(x+b)(x+c)$ and expanding th...
is that actually gonna cook tho
supple wing
I hate how I’ve never heard of this like it’s just a 3rd degree but they don’t even acknowledge it
mortal falcon
https://www.desmos.com/calculator/139357dd90
I also made a little desmos thing a while ago that lets you see how the roots of a cubic change as you vary the coefficients :D and in the process it shows how to analytically get the roots
lyric charm
hold up actually. is this school or uni
fallen vigil
is that actually gonna cook tho
Hmm I actually started with (x+a)(x^2+bx+c) but you can get to this by expanding 2 of the terms and then setting the constant and first degree parts to new constants
sick fossil
I hate how I’ve never heard of this like it’s just a 3rd degree but they don’t e...
well that's because the solutions can get hard to find and you don't generally need these in practice
fallen vigil
and it works
sick fossil
and it works
lyric charm
you couldn't have gotten a rational value for a tho could you...
supple wing
https://www.desmos.com/calculator/139357dd90
I also made a little desmos thing ...
I love me some Desmos
mortal falcon
yesss I love desmos
supple wing
Desmond is my chosen coding language
mortal falcon
it’s so fun making simulations
supple wing
Anyways thanks for solving my problem!
mortal falcon
I love seeing real roots collide and turn into complex ones
supple wing
.close
languid mica
I can not see how the Fact at the buttom was concluded can someone help me understand it
midnight plankBOT
I can not see how the Fact at the buttom was concluded can someone help me under...
subtle blaze
did you try writing out what L(f * g)(s) and F(s), G(s) might be?
languid mica
did you try writing out what L(f * g)(s) and F(s), G(s) might be?
im confused on this notation
does (f*g)(t)
mean like f times g of t
is equal to that
subtle blaze
convolute
languid mica
i didnt understand the left side
subtle blaze
write out what $\mathcal L{f*g}(s)$, $F(s)$, and $G(s)$ are
grand pondBOT
frosst
subtle blaze
$(f*g)(t) \coloneq \int_0^tf(t-\tau)g(\tau),d\tau$ and $(fg)(t) = f(t)g(t)$
grand pondBOT
frosst
languid mica
like its just a symbolation for that integral
subtle blaze
yes
it's a new symbol
it's called a convolution
you "convolute f and g together"
"the convolution of f and g"
languid mica
idk what that means but ok
subtle blaze
"f convolute g"
subtle blaze
i didnt understand the left side
it means you do this
f * g is a new function, and if you want to know what this function does you need to know how it acts on an input
languid mica
convolution in my language straight up translation doesnt make much sense
subtle blaze
does not matter it's just a combination of letters that make a word
languid mica
in this context
ok
yea nvm then
wait but whats the point of this
isnt this like already knonw
subtle blaze
no!!
it's saying that if you need to convolute 2 functions together
you can first take their laplace transforms
then just multiply the results
languid mica
oh wait F stands as L{f}
subtle blaze
then take the inverse
yes
do you know any probability theory?
like what a PDF is
languid mica
ok ill write it out to see how this is true
languid mica
like what a PDF is
the file formation?
subtle blaze
nah probability density functions
subtle blaze
like $\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$
languid mica
i keep hearing many types of theories but idk if thats a course or idk yea
grand pondBOT
frosst
subtle blaze
have u seen that before
languid mica
related to gaussian integral maybe
subtle blaze
it is
subtle blaze
anyhow if you dont know about random variables then this doesn't matter
subtle blaze
i was just going to give you an example where convolutions are used
languid mica
i was just going to give you an example where convolutions are used
no i see how its useful
ill check examples after i understand it though
wanna see why this is correct
subtle blaze
yeah i dont really know why it's called convolution either, that's just the word they use so i use it
yeah usually writing everything out will help
pearl mist
have u seen that before
is this related to those gambling machines which use tiny metal bearings which fall down in columns and they form the gaussian function pattern?
subtle blaze
open your own channel if you want to ask about it, this is taebek's channel i dont want to have random clutter here
languid mica
ok i wrote it out
pearl mist
yeah sorry i just asked bc ive seen it somewhere
languid mica
$$
\int_{0}^{\infty} \int_{-\infty}^{t} f(t-\tau),g(\tau),d\tau , e^{-st},dt
= \int_{0}^{\infty} f(t),e^{-st},dt ;; \int_{0}^{\infty} g(t),e^{-st},dt
$$
grand pondBOT
Taebek
languid mica
yea its definetely not trivial
languid mica
what
subtle blaze
yeah but im looking at a multidimensional one
languid mica
what i dont understand is
how did someone know this would be useful and defined it then proved this
languid mica
you have any context to what this was defined for
subtle blaze
you dont know anything about random variables?
languid mica
you dont know anything about random variables?
wdym
subtle blaze
like you havne't studied probability before?
languid mica
there is probability inside math
isnt that just like what maths is used on
not core math
subtle blaze
probability theory is an area of maths that deals with randomness
languid mica
i mean i have done stuff like this
subtle blaze
here i'll give you a quick overview
subtle blaze
consider a fair 6 sided dice
now consider a rigged 6 sided dice
they are different, how do we describe that they are different?
subtle blaze
meaning each side does not have equal chance to show up
languid mica
they are different, how do we describe that they are different?
well one dice has some device inside it
subtle blaze
we're mathematicians we dont care how you physically do it
languid mica
stupid answer but yea i dont understand
subtle blaze
we just care that they are different
subtle blaze
okay let me use an simpler example, coins (coins only have 2 outcomes so it's easier to list them out)
subtle blaze
a fair coins has this property: P(X = heads) = P(X = tails) = 1/2
languid mica
one has a prob 1/6 in getting a number
the other has 100%
and 0 on the rest of the numbers
subtle blaze
a rigged coin will be something like P(X = heads) = 1/3 and P(X = tails) = 2/3
subtle blaze
i can do something similar with a dice by listing out P(X = 1) = ... P(X = 2) = ... etc
languid mica
ok whats this gotta do with these integrals
subtle blaze
im getting there
subtle blaze
i can do something similar with a dice by listing out P(X = 1) = ... P(X = 2) = ...
so here im explicitly prescribing exactly how likely each scenario would be
but what if i want to describe something that's continuous?
how do i describe "pick a random number uniformly between 0 and 1"
because i could say "pick a random number between 0 and 1, but be biased towards bigger numbers"
languid mica
you just pick one
subtle blaze
this is the continuous version of fair vs rigged coins/dice
languid mica
is it not possible to have the random aspect
what is dnd
is it like a board game or some
subtle blaze
dungeons and dragons yeah
there's a mechanic called "advantage" where you roll 2 dice and pick the higher one
languid mica
idk it
subtle blaze
clearly the random numbers coming out of an "advantage" roll is going to be higher than just 1 dice
cos in a way you get 2 chances
languid mica
why
higher chance doesnt mean its gonna occur
you could get 6 first try and 1 and 2
subtle blaze
do you not intuitively see that if you roll 2 dice and pick the highest then on average you'll roll higher?
languid mica
oh on average
subtle blaze
i can do something similar with a dice by listing out P(X = 1) = ... P(X = 2) = ...
this is how we prescribe probability
languid mica
but isnt it relevant to the amount of examples you do
subtle blaze
we need to explicitly write out what the probabilities are for each event that can occur
languid mica
if i only do 5 scenarios its not certain it will be close to the odds
subtle blaze
so if i say:
Dice 1
P(X = 1) = 1/6
P(X = 2) = 1/6
P(X = 3) = 1/6
P(X = 4) = 1/6
P(X = 5) = 1/6
P(X = 6) = 1/6
Dice 2
P(X = 1) = 1/12
P(X = 2) = 1/12
P(X = 3) = 1/12
P(X = 4) = 1/12
P(X = 5) = 1/3
P(X = 6) = 1/3
can you see that dice 2 is biased towards bigger numbers?
subtle blaze
okay that's what we mean by a fair vs rigged dice
languid mica
how do we define probability
languid mica
if i have a coin and i flip it a million times doesnt mean half will be heads
same if it was a billion
and so on
languid mica
then if we flip it infinite times
subtle blaze
we dont do experimental trials in probability, we dont sample things
subtle blaze
if you go and start rolling the dices, flipping coins, and recording the result you are doing statistics
languid mica
ig i can understand what you are saying intuitively
but i can not formally write it down on paper
subtle blaze
so how do we define probability
we're going very far from how these integrals are useful
languid mica
ok
subtle blaze
you can but i'll need like another 3 hours to define everything properly
subtle blaze
so if i say:
## Dice 1
P(X = 1) = 1/6
P(X = 2) = 1/6
P(X = 3) = 1/6
P(X = 4) = 1...
so much like how this dice can be "rigged"
i can say, pick numbers between 0 and 1 but rig it towards bigger numbers
subtle blaze
and somehow this is different to "pick numbers uniformly between 0 and 1"
subtle blaze
who is making the choice
im making the choices here
subtle blaze
im just trying to explain what uniform randomness means
languid mica
ok
languid mica
what is uniformly
subtle blaze
vs uniformly pick a number between 0 and 1, then square it
do you see that i will on average pick smaller numbers in the square case
because for any 0 < x < 1, x^2 < x
languid mica
but doesnt saying average contain statistics?
subtle blaze
not really
languid mica
idk why you are refering to average
average is the average of countably many examples/samples
subtle blaze
$f_X(x) = \begin{cases} 1 & 0< x<1 \ 0 & \text{else}\end{cases}$
grand pondBOT
frosst
subtle blaze
and $P(a < X < b) = \int_\bR f_X(x),dx$
grand pondBOT
frosst
subtle blaze
no that X is called a random variable
subtle blaze
frosst
if you want to ask what is the probability that my random number lands between 0.2 and 0.5, then take this function
and integrate it between 0.2 and 0.5
the result will be the probability X lands between 0.2 and 0.5
subtle blaze
frosst
this particular "PDF" (probability density function) tells us that the probability that X lands between a and b is just b - a
subtle blaze
yeah the 0 else part is for "X cannot be outside of 0 and 1"
languid mica
ok
and if it was rigged towards bigger numbers
it would be some like f(x)= x
in (0,1)
languid mica
wdym close
languid mica
f(x) = x is biased towards higher numbers
subtle blaze
so the total probability is 0.5 that is not right
you need to scale it so the total integral is 1
subtle blaze
it'd be something like f(x) = 2x
so this will work
subtle blaze
okay
ya ya that's cos the psi there is a density or something
some phsyics thing
no it's the wave function or something idk
i dont do physics
OKAY NOT THE POINT
languid mica
its in quantum mechanics
subtle blaze
point let's say X ~ U(0, 1)
languid mica
its the probability of an electron being found in dV
subtle blaze
frosst
this is the probability theorist way to say X has a density like this
it's uniform between 0 and 1
languid mica
why not just say
subtle blaze
what is X thoguh
to explain this i will need to either abuse and handwave the explanation
or spend 3 hours starting at measure theory
languid mica
P(a<c<b) = int f(x) dx
subtle blaze
well because X ~ U(0, 1) is easier to write
languid mica
where c is the number we care about
subtle blaze
so the problem is this
languid mica
and whats U(0,1)?
languid mica
what is that saying
subtle blaze
they both have densities like that
languid mica
who are both
subtle blaze
oh sorry i haven't written that part yet
if X and Y are both (independent) U(0, 1) random variables
what is X + Y?
the answer is, you can take their so called "MGFs" and then just multiply them
now what the heck did i just say
subtle blaze
what is X + Y?
$f_{X+Y}(t) = (f_X * f_Y)(t)$
grand pondBOT
frosst
subtle blaze
this is the same * that you're seeing
now sometimes this * makes it very hard (to solve the integral)
so maybe we want to stick f_X * f_Y through a laplace transform
then we can just multiply them together
and then we might even just recognise $\mathcal L{f_X * f_Y}$ as the transform of some other known random variable
grand pondBOT
frosst
languid mica
wait so
subtle blaze
so we dont even need to inverse it
languid mica
~~the answer is, you can take their so called "MGFs" and then just multiply them...
im still stuck here
subtle blaze
okay just ignore that part
subtle blaze
yes
(technically though we're using the bilaplace transform in probability, here you talk about the one-sided transform, but for motivation it doesnt matter all that much)
subtle blaze
Yeah idk about that it looks hard
languid mica
Yeah idk about that it looks hard
im guessing it needs polar cordinates
subtle blaze
https://proofwiki.org/wiki/Convolution_Theorem
.
subtle blaze
oh it looks terribly long to derive
languid mica
how was it found
like how did someone think of laplace transforms when seeing this double integral is there any context
midnight plankBOT
@languid mica Has your question been resolved?
tidal turret
midnight plankBOT
hushed fox
stuck at the n = k + 1 step xD
Can you think of a way to get sum of 6/i^3 from i=2 to k+2 in terms of sum of 6/i^3 from i=2 to k+1
tidal turret
undone parcel
try showing that 3-3/(k+1)^2+6/(k+2)^3 < 3-3/(k+2)^2
because youre trying to prove sum of 6/i^3 from i=2 to k+2 < 3-3/(k+2)^2 right
merry kindle
bread thank you
tidal turret
try showing that 3-3/(k+1)^2+6/(k+2)^3 < 3-3/(k+2)^2
how
undone parcel
move stuff around and simplify
tidal turret
care to elaborate?
tidal turret
6/(k+2)^3 < 3/(k+1)^2 - 3/(k+2)^2
is not necessarily true
suppose k is negative number
waxen musk
but you also have to prove it
tidal turret
can someone elaborate
tidal turret
i got stuck here
waxen musk
6/(k+2)^3 < 3/(k+1)^2 - 3/(k+2)^2
we cant assume this statement, it has to follow from the assumption that k is greater than the base case
waxen musk
but i thought htat was implied by induction
it will be, i was responding to renatos request to elaborate, not correcting you
undone parcel
mb
undone parcel
induction is usually you have a base case and you build off of it
tidal turret
undone parcel
your base case is n=1 here
and you have to prove that for all k >= 1, the inequality holds for k+1
which would prove the inequality
this means all ks are positive
undone parcel
so back to this
youre trying to prove that sum of i=2 to k+2 of 6/i^3 < 3-3/(k+2)^2 right
you already know that the sum is < 3-3/(k+1)^2 + 6/(k+2)^3
tidal turret
ye
undone parcel
so now if you prove that 3-3/(k+1)^2 + 6/(k+2)^3 < 3-3/(k+2)^2 for all positive k
undone parcel
and you have to prove that for all k >= 1, the inequality holds for k+1
you finish this
tidal turret
so now if you prove that 3-3/(k+1)^2 + 6/(k+2)^3 < 3-3/(k+2)^2 for all positive ...
how to prove it dawg
undone parcel
algebra
waxen musk
how to prove it dawg
the easiest way to to work it down to k >= 1, and then rewrite bottom to top, just verify it holds
tidal turret
6/(k+2)^3 < 3/(k+1)^2 - 3/(k+2)^2
where is this coming from
waxen musk
it will always hold if you use equivalence rules
tidal turret
the easiest way to to work it down to k >= 1, and then rewrite bottom to top, ju...
is hard
undone parcel
where is this coming from
3-3/(k+1)^2 + 6/(k+2)^3 < 3-3/(k+2)^2 but you move the 3-3/(k+1)^2 to the right side
tidal turret
3-3/(k+1)^2 + 6/(k+2)^3 < 3-3/(k+2)^2 but you move the 3-3/(k+1)^2 to the right ...
when you are setting this inequality together you are assuming it's true
waxen musk
when you are setting this inequality together you are assuming it's true
well, we need to prove it from k >= 1, so its implied by the induction hypothesis
undone parcel
flip the signs and divide by 3
then expand by multiplying everything by (k+2)^3(k+1)^2
undone parcel
uhh i dont agree with the last step that overcomplicates things
multiplying by (k+2)^3(k+1)^2 should get you (k+2)^3 > (k+1)^2(k+2) + 2(k+1)^2
then expand
undone parcel
like k^3+6k^2+12k+8 > ...
tidal turret
like k^3+6k^2+12k+8 > ...
this is the simple route?
undone parcel
probably
tidal turret
ok
tidal turret
fuuuck its 4k
tidal turret
right
undone parcel
still slightly wrong
it should be k^3+4k^2+4k+2k^2+8k+8 = k^3+6k^2+12k+8
but you get the idea
tidal turret
it should be k^3+4k^2+4k+2k^2+8k+8 = k^3+6k^2+12k+8
fuck my life
undone parcel
and bc you know that k >= 1, it follows that k > any negative number
so you proved the case k+1 for all k >= 1
and you already did the case for n=1 so youre induction proof is done
tidal turret
tidal turret
@undone parcel thanks
undone parcel
np
midnight plankBOT
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Available help channel!
viral dagger
given prime $p\equiv 1 \mod 6$, prove that $127(2^p-1)\mid 2^{2^p-2}-1$
midnight plankBOT
given prime $p\equiv 1 \mod 6$, prove that $127(2^p-1)\mid 2^{2^p-2}-1$
grand pondBOT
ihave<skissue>
viral dagger
yeahh idk
i was thinking to take care of 2^p-1|(the right) then 127|(the left), but theres overlap on 2^p-1 and 127 so we need to take care of that
viral dagger
$2^{p+7}-2^7-2^p+1$
grand pondBOT
ihave<skissue>
viral dagger
undone parcel
wait i mightve done smth wrong
err
i think you need to use difference of squares on the right side
viral dagger
$127(2^p-1)\mid (2^{2^{p-1}-1}-1)(2^{2^{p-1}-1}+1)$
grand pondBOT
ihave<skissue>
viral dagger
the left bit takes care of the 127 right?
undone parcel
or smth
viral dagger
i mean thats just clearely wrong?
viral dagger
the left bit takes care of the 127 right?
nvm it doesent
viral dagger
ihave<skissue>
actually proving 127| is p easy
undone parcel
err
i lied
$2^p\equiv 2\mod p \rightarrow 2^{2^p-2} = 2^{kp}\equiv 1\mod {2^p-1}$ bc $2^p\equiv 1\mod {2^p-1}$
grand pondBOT
bread
undone parcel
i think
midnight plankBOT
@viral dagger Has your question been resolved?
viral dagger
interesting
undone parcel
so that deals with the 2^p-1 part
right
not sure about the overlap
<@&286206848099549185>
wait i think i got it
undone parcel
bread
with a similar idea to this you can prove $127\nmid {2^p-1}$ for all p > 7 and $127\mid 2^{2^p-2}-1$
grand pondBOT
bread
midnight plankBOT
@viral dagger Has your question been resolved?
undone parcel
why
viral dagger
i still dont get how you can prove it tho
fleet plume
given prime $p\equiv 1 \mod 6$, prove that $127(2^p-1)\mid 2^{2^p-2}-1$
you can use fermats little theorem on the RHS i think
oh wait nvm thats been mentioned
fleet plume
i still dont get how you can prove it tho
the only time 2^x-1 is divisible by 2^7-1 is when x=0mod7
p is prime though so it can't be =0mod7 unless p is 7
viral dagger
ohh wait i got it
mm all thats left is proving 127^2|2^126-1 which should be easy?
ok am i silly or i that just not true
forest iron
that is true
viral dagger
🤔
forest iron
hmm really
viral dagger
ooh wait the question says p>7 oopssss
forest iron
my calculator gave me a full number
viral dagger
doubt its accurate as 2^126 is massive
viral dagger
what calculator are you using?
forest iron
apple
viral dagger
but it still doesent change the fact that it doesent divide
viral dagger
and i think the problem here is that you wrote 16.129 which is not 16129
they are the same?
forest iron
no
viral dagger
ofc its not comma
forest iron
16.129=16129/1000
viral dagger
no\
forest iron
yes
viral dagger
diffrent placews diffrent standards
forest iron
at least in usa i think
viral dagger
this website doesent follow
forest iron
in hungary we use comma too but in us i think its .
viral dagger
ok legit this discussion is pointless
fleet plume
if you test it with difference of squares and cubes, then yeah it doesnt divide
forest iron
try putting it in without comma
viral dagger
you cant
lte also says you vant
v_127(2^126-1)=v_127(128^18-1^18)=v_127(127)+v_127(18)=1 and lhs is 2
fleet plume
2^126-1=(2^63-1)(2^63+1)
=(2^21-1)(2^42+2^21+1)(2^63+1)
=(2^7-1)(2^14+2^7+1)(2^42+2^21+1)(2^63+1)
you cant divide this by 127^2
fleet plume
you can split the last term
2^7n=1mod127
it would be a waste of time since the whole thing of the last term will be 2mod127
forest iron
i know it can because my calc showed the exact amount
viral dagger
.solved thanks guys!
midnight plankBOT
midnight plankBOT
Available help channel!
leaden seal
midnight plankBOT
nova yoke
is this the same I on the left hand side? why is it called I_2?
leaden seal
is this the same `I` on the left hand side? why is it called `I_2`?
Because I've used IBP twice
So it's not the same I
runic hamlet
nova yoke
notice that the integral on the right hand side of the final expression is just I
forest bolt
Que?
nova yoke
I've got I2
now plug this back into the equation that contained the I2 integral
leaden seal
So it's I2=sin(x)e^x - integral of I
nova yoke
you have I = cos(x)e^x + I_2
and you have an expression for I_2
plug that expression in
nova yoke
missing a - I at the end
leaden seal
Why?
grand pondBOT
Bungo
nova yoke
so plug the second expression into the first
leaden seal
I thought I=integral of cos(x)e^x
grand pondBOT
Bungo
nova yoke
fixed sign in the first one
you never explicitly said what $I_2$ is, but I assume it's $\int \sin(x) e^x,dx$
grand pondBOT
Bungo
nova yoke
show your final step?
nova yoke
this was ok until the last line
if the left hand side is still 2Ithen the right hand side shouldn't be divided by 2
or equivalently, if you have divided by 2 then the left hand side should just be I
flat spire
bungoo
nova yoke
hi ωβ!
nova yoke
yea this is ok, just add the final closing paren at the end of the numerator and don't forget +C
midnight plankBOT
@leaden seal Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
turbid kayak
Can anybody help me make the diagram?
midnight plankBOT
Can anybody help me make the diagram?
turbid kayak
Of example 6 -
bold peak
Gimme a moment
turbid kayak
Ok
bold peak
deft folio
Btw what is ray?
a line that starts from a single point and has no end point
bold peak
Line with one endpoint
turbid kayak
And angle ACB will be 70° (alternative angles)
bold peak
No AX isn't parallel to BC
turbid kayak
a line that starts from a single point and has no end point
Alright, so can I draw a arrow. Like this -
bold peak
Yes
bold peak
Cool
turbid kayak
Can I write angle B + angle C = angle XAC or angle XAD or angle XAC + angle XAD?
Which one would be correct?
bold peak
The last one
turbid kayak
The last one
Hmmm
bold peak
Exterior angles of a triangle
Are equal to the sum of the two non-adjacent interior angles
turbid kayak
What do you mean by "non-adjacent interior angles"?
Angle B and angle C are adjacent
Are they?
<@&286206848099549185>
fallen barn
In Triangle all angles are adjacent
To get a non adjacent angle there needs to be more and 3 sides in a polygon
fallen barn
What do you mean by "non-adjacent interior angles"?
Non-adjacent interior angles refer to, for example, In a Rectangle ABCD : angle A and angle C are Non-adjacent interior angles
Adjacent angles are angles which have one arm in common.
An Arm refers to one of the two lines between which an angle is formed.
dry pollen
Non-adjacent interior angles refer to, for example, In a Rectangle ABCD : angle ...
not exactly what is probably meant by Xavier
dry pollen
Are equal to the sum of the two non-adjacent interior angles
(though I would like your confirmation so as not to put words into your mouth, if possible, and sorry for the ping)
deft folio
In Triangle all angles are adjacent
no they arent
bold peak
Sorry yeah opposite interior angles my bad
bold peak
(though I would like your confirmation so as not to put words into your mouth, i...
Thanks for understanding what I meant, and thanks for the tag! 🌺
deft folio
the 3 angles arent adjacent to each other, they are bounded by an adjacent angle
midnight plankBOT
@turbid kayak Has your question been resolved?
turbid kayak
.close
spiral cradle
Helpppaaa
spiral cradle
Q19
How to get the inequality thing for 19b
@vivid yoke Can you help
.rrcw
.rccw
grand pondBOT
vivid yoke
hmm you already got (2x-1)(x-3)=0
spiral cradle
Yes
forest bolt
1
spiral cradle
But how do I make a inequality
vivid yoke
Alright now you can do some test here
The roots of this equation are 1/2 and 3 right?
spiral cradle
Yess
vivid yoke
Okay, check between these root, let say between 1/2 and 3 there is 1
vivid yoke
Plug into equation, if you got negative then it's always negative for any value between 1/2 adn 3
proven mesa
👍
proven mesa
💯
vivid yoke
If x>a and x>b (b>a) then we can reduce to x>b
proven mesa
What is the IPO date of Merck & Co., Inc.
vivid yoke
Imagine you have more than 4 candies and more than 9 candies and both of these statement are true then you obviously have more than 9 candies
If you have 5 candies the first statement is true but the second statement false
vivid yoke
yes
spiral cradle
Is it and
vivid yoke
Not really....
spiral cradle
How?
spiral cradle
Ok
vivid yoke
Yeah that's much better
spiral cradle
Yep
vivid yoke
So we take both
vivid yoke
So it’s 1/4<x<1/2 **and** x>3
I have no idea it's and or or, for me x has to sastify both 1/4<x<1/2 and x>3 if we use and
Which is impossible
viral dagger
midnight plankBOT
viral dagger
names for the added points should be easy to undersstand
lemme write the gist of the solution (im asking about it)
gilded current
oi is this a evan chen problem>?
viral dagger
dunno but i didnt get it from there
gilded current
I guess try proving that the quadrilateral formed is cyclic
Thats how I did it then
hard shard
uncentered circles invert to uncentered circles
gilded current
wdym
viral dagger
since AE=AR=AS then A is the center of (RES), and this extends for the rest of the verticies. invert on E to get the second diagram. notice that w_a* is perp to AE (and the rest). proving PQRS is concyclic is equivalent to proving P*Q*R*S* being concyclic. R is in w_a so R* is also in w_a*, analog R* is in w_d*, since R=/=E then R*=/=E is the intersection of w_a* and w_d* (also for other verticies). then you can get P*Q*R*S* is a rectangle so its concyclic
gilded current
dont send the entire soln
viral dagger
what i dont understand is how w_a* perp AE
viral dagger
dont send the entire soln
why?
vivid yoke
dont send the entire soln
he's OP lol
gilded current
oml
sorry
sorrrry
I though you were telling him
wait
telling yourself>?
mb
viral dagger
i mean i guess i am telling him :p
hard shard
if you reflect the triangles i think you can angle chase opposite angles summing to 180
viral dagger
yeah but im learning inversion rn
actually does that really work?
reflect AEB BEC CED DEA right
hard shard
ok youre doing inversion
last slate
I think it gets solved easily using coordinate
hard shard
uncentered circles invert to uncentered circles
use this
last slate
[
\textbf{Claim.}\quad
\text{If }AC\perp BD\text{ at }E,\text{ then the reflections of }E\text{ across }AB,BC,CD,DA
\text{ are concyclic.}
]
\paragraph{Coordinate set-up.}
Place (E=(0,0)). Rotate/translate so that (AC) is the (x)-axis and (BD) the (y)-axis.
Write
[
A=(a,0),\qquad B=(0,b),\qquad C=(c,0),\qquad D=(0,d),
]
with (a,b,c,d\in\mathbb R\setminus{0}) (convexity fixes signs but is not needed algebraically).
\paragraph{Reflections.}
The line (AB) has equation (b x+a y-ab=0). The perpendicular foot from the origin to (AB) is
[
H_{AB}=\frac{ab}{a^2+b^2}(b,a),
]
hence the reflection (P) of (E) across (AB) is
[
P=2H_{AB}=\Big(\frac{2ab^2}{a^2+b^2},\ \frac{2a^2b}{a^2+b^2}\Big).
]
By the same computation (cyclically) the reflections (Q,R,S) of (E) across (BC,CD,DA) are
[
\begin{aligned}
Q&=\Big(\frac{2b^2c}{b^2+c^2},\ \frac{2bc^2}{b^2+c^2}\Big),\
R&=\Big(\frac{2c d^2}{c^2+d^2},\ \frac{2c^2 d}{c^2+d^2}\Big),\
S&=\Big(\frac{2a d^2}{a^2+d^2},\ \frac{2a^2 d}{a^2+d^2}\Big).
\end{aligned}
]
\paragraph{Concyclicity check.}
Four planar points ((x_i,y_i)) are concyclic iff the determinant
[
\Delta:=\det\begin{pmatrix}
x_1&y_1&x_1^2+y_1^2&1\[4pt]
x_2&y_2&x_2^2+y_2^2&1\[4pt]
x_3&y_3&x_3^2+y_3^2&1\[4pt]
x_4&y_4&x_4^2+y_4^2&1
\end{pmatrix}
]
vanishes. Substitute the coordinates of (P,Q,R,S) above. A direct (but routine) algebraic simplification gives
[
\Delta\equiv0
]
as an identity in (a,b,c,d). Therefore (P,Q,R,S) are concyclic.
[
\boxed{\text{Reflections of }E\text{ across }AB,BC,CD,DA\text{ lie on one circle.}}
]
grand pondBOT
ADdisson's syndrome
LaTeX source sent via direct message.
hard shard
thanks claude
viral dagger
ew coordinate
viral dagger
what i dont understand is how w_a* perp AE
im asking about this
viral dagger
P* is the inverse (idk the word) of P with respect to a circle w with center O (can be a point as the radius is arbitrary), if and only if O,P,P* are colinear and OP\cdot OP*=r^2
hard shard
a circle going through the origin E is sin(theta)
say r is 1
then what you need is sec theta
in polar coord thats a line
viral dagger
isnt polar coordinates (r,theta)?
hard shard
r=sin theta and r=sec theta cancel
its not a proof, but it should hopefully give the intuition
or maybe you havent really thought about polar emough to understand what im saying
viral dagger
a circle going through the origin E is sin(theta)
is sin(theta) supposed to be the radius or something?
hard shard
yes
viral dagger
never really played with polar coordinates just heard of it
viral dagger
a circle going through the origin E is sin(theta)
mm so your saying if you invert this circle to a circle with center E and radius 1, you get a circle with radius sec(theta)?
hard shard
you get a curve with radius sec theta
viral dagger
curve??
hard shard
well really its a line but lets act like we dont know that
viral dagger
eh?? i dont follow
hard shard
let me gather my thoughts, i need to go to bed
viral dagger
making sure, do you invert to a circle with center E and radius 1, or just any circle with radius 1?
hard shard
ill hopefully explain it better tomorrow
hard shard
making sure, do you invert to a circle with center E and radius 1, or just any c...
doesnt have to be radius 1, but it makes the math nice
there are no lengths in the diagram anyway, so a distance of 1 is arbitrary
viral dagger
ohh wait hold on ive been graphing it wrong
hmmm
ohh wait
so your invverting r=sin(theta) to the unit circle, and by definition its sec(theta) which is a line?
hmm lets try to apply this to the problem?
the origin is E
A is (0.5,0)
hard shard
looks more believable now im sure
viral dagger
so w_a is the black circle, AE is the x axis
the inversion of w_a(cos theta) is the red line (sec theta), which is a line
(and in the polar coordinates, sec theta is a line parallel to the y axis?)
which implies the red line (inverse of w_a) is perpendicular to the x axis, aka AE!
oooo
hard shard
then you just rotate the setup for any line
viral dagger
is there a simplified/shortened form (or maybe like a property or a theorem?) for this instead of deriving it again and again
actually i swear ive seen this somewhere simmilar before uhh
so, if you have line AB that passes trough the center of a circle, the inverse of the circle torwards A is perpendicular to AB ig?
ok thanks gtg
.solved
midnight plankBOT
midnight plankBOT
Available help channel!
last slate
what are negative integers? give a n example
midnight plankBOT
what are negative integers? give a n example
graceful drum
what do you know?