if i take x as cos 2theta u mean?
#help-49
small jasper
This is one of those questions where you consider the edge case
what is the range of each side
vague wigeon
0,pi
small jasper
That’s the range of the inverse cosine function yes
small jasper
mhm
vague wigeon
lhs
as m is > o then that arc cos x will give vales from 0 to mpi
and arccos 1-x will give values from
x lies from -1 to 1
but 1-x must also lie from -1 to 1
1-x lies from [2,0]
and its outta the range for cox
so x must be 0?
small jasper
Are you trying to think about the domain? Because the left had side is defined for 1/2 too.
vague wigeon
yea
lhs is defined for 1/2?
oh ye from all values of
mb gng lemme fix a bit
so arc cox 1-x gives [1,0] and marc cos x will give again [m,0]
x lies between [0,1] only
but here given than m is always positive
and n must be either 0 or -ve
small jasper
I think you have at least some of the idea, but what I’m reading is kinda all over the place rn
vague wigeon
so to solve this
we js need to find where the values are giving us the desired -ve value
or 0 in this case
small jasper
mhm you want both sides to be zero
small jasper
yup
small jasper
can i put it here?
sure
I’m out shopping right now, so someone else will probably look at it
vague wigeon
I’m out shopping right now, so someone else will probably look at it
ye i got it
so here i was unable to grasp what the series is and how i can write it in the form of arc tan x - arc tan y
midnight plankBOT
@vague wigeon Has your question been resolved?
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vague wigeon
.reopen
midnight plankBOT
✅ Original question: #help-49 message
small jasper
$n^2 \pm n$
grand pondBOT
Civil Service Pigeon
vague wigeon
thanks
small jasper
The form they gave you is a red herring
vague wigeon
The form they gave you is a red herring
how do i catch that
it dosent follow for n=1 tho right
cuz we get 2.0
i cant find it online
small jasper
Yeah that’s why I mentioned what I did above
Albeit a bit tersely since I was just waiting for my turn in the checkout line lol
small jasper
ts not following for n=1 rite
Start it at n=2 
vague wigeon
Start it at n=2 <:sully:651816820122189834>
😮
but gng how do i think of smth like that
vague wigeon
so in this form how will the eq be represented
radiant roost
a = n(1+n) and b = n(1-n)
vague wigeon
wow
radiant roost
idk if it telescopes but it does something, at least...
vague wigeon
a = n(1+n) and b = n(1-n)
how did u get ts can u tell a lil
radiant roost
i looked at this and tried to match it to the arctan formula
radiant roost
no
radiant roost
you have to factor it differently
vague wigeon
n(n-1) one
radiant roost
yeah
radiant roost
a = n(1+n) and b = n(1-n)
^
vague wigeon
ye mb
elite knowledge req for ts
so this ones done
theres another one tricky one
nvm ill do it
gng help with ts one pls 😭
radiant roost
how do u finish it though?
noble forge
ye i got it
Do you need to show the work?
vague wigeon
then we see that the -ve and +ve terms cancel each other out
noble forge
Or you can just take the double coinflip odds
vague wigeon
then we see that the -ve and +ve terms cancel each other out
then we can see that only the 1st term and last term remain
noble forge
Can you just make an educated guess?
vague wigeon
on what
noble forge
On these exercises
noble forge
So you can just guess
vague wigeon
but -ve marking
noble forge
What?
vague wigeon
the ans is scuffed between very similar options so we ususally needa so it 100%
raw hamlet
Is this practice for competition math?
vague wigeon
ye
radiant roost
oh you use the fact that arctan is an odd function, right?
vague wigeon
oh you use the fact that arctan is an odd function, right?
u mean -f(x) = f(-x)
radiant roost
yeah
vague wigeon
ye we js needa put it in a way everything gets slimed
raw hamlet
Isn't it just comparing n^n + n and (n+1)^2 - (n+1)
vague wigeon
wait
radiant roost
ye we js needa put it in a way everything gets slimed
i got pi/2 - arctan(2)
vague wigeon
i got pi/2 - arctan(2)
yes not we use arctanx + arccotx = pi/2
so arc cotx = pi/2 - arctanx
so we get cot2
arc cot2
radiant roost
i see
radiant roost
angle difference formula + some cancellation gives (1 - cos alpha) / (1 + cos alpha)
i don't have all these formulas memorized
i guess i need half angle formulas next
raw hamlet
It's pretty much the tan^2 half angle I think
radiant roost
yeah you're right
to cancel a trig function composed with an inverse trig function (or vice versa) you can draw a triangle and label it
vague wigeon
It's pretty much the tan^2 half angle I think
can u show za working
how did u arrive
vague wigeon
to cancel a trig function composed with an inverse trig function (or vice versa)...
i triedts
i got stuck wih sinx = cosa -1/(cos^2 a +1)
radiant roost
you must have made a mistake
radiant roost
can u show za working
i think they meant that (1 - cos alpha) / (1 + cos alpha) is the square of tan(alpha / 2)
raw hamlet
Ya. I didn't work out the rest. The fact that it was so nicely the half angle leads me to believe it is correct
radiant roost
raw hamlet
I was wondering if you could "solve" this question just by picking a nice value of alpha as well
If there's a singular value of alpha that works doing that could be quicker on the real thing...
radiant roost
oh sorry i used x when i should have used a
radiant roost
I was wondering if you could "solve" this question just by picking a nice value ...
that would be nice
small jasper
arccot x + arctan x = pi/2 for x>0 
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gaunt jetty
Test
gaunt jetty
.close
hoary hamlet
I don't understand this solution, can someone explain it to me?
midnight plankBOT
I don't understand this solution, can someone explain it to me?
lyric charm
time to break it down
hold on
it's written somewhat unclearly, i'll tell you that.
the meat of it seems to be about (I)
(I) says: [for all n ∈ Z, f(n) is an integer] <=> [f(-1), f(0), f(1) all integers]
in this biconditional, the => implication is quite obvious, so the <= direction is of interest.
that make sense so far? @hoary hamlet
hoary hamlet
yeah i got it
lyric charm
you got it in that you dont need any more help?
lyric charm
ok
so then they do some algebra
f(-1) = a-b+c
f(1) = a+b+c
f(0) = c
so $c$ is an integer and thus so are $a\pm b$
grand pondBOT
Ann
lyric charm
therefore, they say, 2a and 2b are also integers.
lyric charm
c is an integer because of the <= statement assumption??
follows directly from it yes
hoary hamlet
so if a +- b are integers, then f(-1) + f(1) is also an integer
then they find that 2a and 2b are also integers from that
lyric charm
yes
lyric charm
so then they say that a and b are either integers or half-integers, but that they must be of the same type
(i.e. we can't have a half integer and b integer, or vice versa -- this would make a+b also a half-integer, which we can't have)
hoary hamlet
okay that makes sense too
lyric charm
if a and b are integers then everything is obviously good to go since an^2 + bn + c will always be an integer
thus they examine the other case: a and b both half-integers
and they prove that in fact f(n) will still be an integer even under that condition, with some algebra
starting with a = a'/2 and b = b'/2
hoary hamlet
okay..
lyric charm
further case breakdown based on the parity of n
hoary hamlet
ah okay so they factor out the half and consider even and odd cases
that makes sense
the proof of statement 2 and 3 are still not making sense to me though
and furthermore
i dont see how i couldve ever derived this in the actual exam
lyric charm
statement 2 piggybacks on the stuff derived in 1
namely that it is very much possible to have f(n) integer for all n but without the coeffs being integer themselves
as a concrete example: f(n) = (n^2+n)/2 is the function giving the n'th triangular number
thus statement 2, which says [for all n ∈ Z, f(n) is an integer] <=> [a, b, c all integers], is false
hoary hamlet
namely that it is very much possible to have f(n) integer for all n but _without...
oh and we disproved this through the a/2 and b/2 cases?
lyric charm
...yes
lyric charm
ominous how
hoary hamlet
its like uncertain
lyric charm
the solution for 3 says that 3 is actually just a dressed-up version of 1
translated back to 1 via g(x) := f(y+x)
y being a fixed real
hoary hamlet
oh that makes sense
alrighty thanks for the explanation
what would be the thought process behind this question?
for example i wouldnt even know where to start other than visualising a graph
and since this is an algebraic approach, what would give it away to think algebraically rather than visually (graphically)
midnight plankBOT
@hoary hamlet Has your question been resolved?
lyric charm
and since this is an algebraic approach, what would give it away to think algebr...
the whole thing with worrying about integers
that doesnt usually lend itself to geometry that well IME
midnight plankBOT
@hoary hamlet Has your question been resolved?
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maiden burrow
I tried to write each in the formula form and got an infinite sum that didn’t diverge
fallow scarab
I tried to write each in the formula form and got an infinite sum that didn’t di...
What is your question exactly
lyric charm
I tried to write each in the formula form and got an infinite sum that didn’t di...
didn't diverge?
so, your infinite sum converged?
@maiden burrow can you show your work
midnight plankBOT
@maiden burrow Has your question been resolved?
red lance
someone can help me with this?
midnight plankBOT
someone can help me with this?
lyric charm
this looks like extremely basic addition
maiden burrow
What is your question exactly
How to answer this
lyric charm
red lance
https://www.youtube.com/watch?v=NwJGqOMhKg4
thanks friend
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wanton spade
in the forward direction would it suffice to show if x is in only one subspace, and y is only in another, the only way x+y is in the union is if it is in the third, so x and y must be in the third and so the third subspace must contain the other two
midnight plankBOT
in the forward direction would it suffice to show if x is in only one subspace, ...
@wanton spade Has your question been resolved?
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wanton spade
<@816159049863004220> ?
.hi
wet locust
do you still need help
midnight plankBOT
do you still need help
midnight plankBOT
wanton spade
okay
wet locust
to be clear i am bad at linear algebra so
i was just checking in
i probably cant help you very much
bold peak
do you still need help
For future reference, you can start your message with a period to not claim the channel
wanton spade
mb
wet locust
For future reference, you can start your message with a period to not claim the ...
ah
this makes sense
lyric charm
.reopen
you can only reopen your own channel
only helpfuls and up can reopen other ppls channels
wanton spade
👍
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near geyser
midnight plankBOT
near geyser
In the first two lines they are saying subset...but third line they said subgroup why?
silent dock
it's just saying only when H is a subgroup do we call it a coset
wet locust
when H is subset, we define aH to be {this thing} and we do not call this anything special
when H is subgroup, we define aH to be {same thing} but we now also call it a left coset
wet locust
what is the question
near geyser
When elements are coming from the subgroup
wet locust
still unclear on your question
near geyser
I wanted to know when we are taking elements from subgroup what effects it make
wet locust
a subgroup is a special kind of subset
where the subset itself is also a group
consider, say
G = Z
wet locust
2Z is a subset of Z
it is also a subgroup of Z because 2Z is a group under addition
meanwhile {1, 2} is a subset of Z
it is not a subgroup of Z because {1, 2} is not closed under addition, nor does it have an identity, etc.
near geyser
meanwhile {1, 2} is a subset of Z
It's not a group
wet locust
yes
near geyser
And what does this mean?
.
grand pondBOT
c9
wet locust
the definition is the same for subsets H and subgroups H
are you asking why this isn't just H
wet locust
**c9**
this is not H sometimes when G and H are nonabelian
i would bet your book will probably go into what you'll use aHa^{-1} for later
considering it seems they just defined it
but like just know that aHa^{-1} is not H in certain instances
wet locust
this construction leads to certain nice properties and is interesting to consider
near geyser
What does this operation even mean?
spiral rock
specificially when talking about normal subgroups
wet locust
so i would just recommend to continue reading
because they're probably not going to define it and not come back to it
happy learning!
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past horizon
A 23 x 23 square is to be completely tiled by 1 x 1, 2 x 2, and 3 x 3 tiles. What is the minimum number of 1 x 1 tiles needed?
midnight plankBOT
A 23 x 23 square is to be completely tiled by 1 x 1, 2 x 2, and 3 x 3 tiles. Wha...
past horizon
I have shown that u cant cover the square using 3x3 and 2x2 tiles alone
so for now u need atleast one 1x1 tile
ive been drawing and placing tiles here and there to find an example
is there a better way to do this than just trial and error?
olive matrix
i can't imagine needing more than one tbh
junior flower
i have not done the problem but i would start with something "trial and error"-like
junior flower
like just try to do the best you can to start
wet locust
you would at minimum like to use 2x2s and 3x3s to tile end-to-end 23
so you want like
3 3 3 3 3 3 3 2
or 3 3 3 3 3 2 2 2 2
or 3 3 3 2 2 2 2 2 2 2
or 3 2 2 2 2 2 2 2 2 2 2
austere nest
i think the way i would approach it assuming you dont want to do trial and error
junior flower
you definitely won't need more than a few. can you do it with 1? in that case you're done. can you do it with 2? in that case can you prove you can't do it with 1?
austere nest
is start with the area of the 23x23 square and color code it into 2 colors like a chess/checker board
you have 1x1, 2x2, 3x3, of which only the 2x2 covers an equal number of both colors
past horizon
you definitely won't need more than a few. can you do it with 1? in that case yo...
yep thats the gameplan
junior flower
that would be mine too
spiral rock
you can do it with 2
past horizon
it was a headache to prove u need atleast 1
austere nest
oh wait is this a mod problem?
junior flower
it is pretty easy to do it with 2
austere nest
this should be checkable using mod
lusty python
i think
the approach is to stack 3x3s onto one row
this has a remainder of 2 tiles
which you can fill using a 2x2 and 2 1x1
austere nest
i checked with mod and i got just 1 1x1
lusty python
i checked with mod and i got just 1 1x1
hm
austere nest
hm
if you set up a checkerboard on your 23x23 grid
you'll got 1 extra white or black depending on how you order it
wet locust
im imagining a construction where the center is a 1x1 and it's surrounded by 4 2x2s?
austere nest
a 1x1 and 3x3 square cover exactly 1 more square of a color compared to the other
while a 2x2 covers an equal amount
so you should get an expression of a+b=1 (mod 2)
where a and b are # of 3x3 and 1x1 tiles respectively
meaning you have an odd number of 3x3 + 1x1 tiles
wet locust
wait i think OP already proved that you need >= 1 1x1
i think they're asking if 1 is sufficient
austere nest
then you can do 23 mod (3)
wet locust
i.e. if there exists a construction
austere nest
i think they're asking if 1 is sufficient
oh wait mb theyre asking abt constructability 😭
wet locust
all good
austere nest
hmm let me draw this and see if i can make it make sense
wet locust
ok well i managed to construct one with only one 1x1
but i have no idea how you do this in a real way
that does not involve drawing boxes for 7 minutes
austere nest
well i realized the 7x7 of 3x3 does not work
wet locust
im decently certain that to prove sufficiency you have to provide a construction so
idk
austere nest
i think if you know you have to use 1 you have to start with 529-1=528
you have to have an even number of 3x3 grids
wet locust
my idiot construction
(the other side can be mirrored over)
(180 degree rotational symmetry)
austere nest
i feel like the way im solving this is kind just mashing logic with trial and error tbh
past horizon
yh i found one too
wet locust
actually wait theres 90 degree rotational symmetry
(this is the top right corner of the 23x23)
you can just rotate this 4 times
about the center 1x1
past horizon
hmmm
viral dagger
induction maybe?
past horizon
no weve solved the problem its 1
viral dagger
actually nvm if you need atleast 1 then induction shouldnt* work
oh wait fuck why did i think it was generalizing it for nxn squares lmao
past horizon
hmmm maybe the best move is to memorize for 17x17 and try to get it to 17x17 💯
gimmick course
.close
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twilit field
I'm trying to prove this converges. Want to know if this reasoning is solid
Define $a_n = \begin{cases} \frac{1}{2^n}& \text{ n is odd } \ 0 & \text{ otherwise} \end{cases}; b_{n}= \begin{cases} 0 & \text { n is odd} \ \frac{1}{3^n} & \text { n is even } \end{cases}$.
Then as each individual series converges, and the sum is exactly this series, the original series converges too.
midnight plankBOT
I'm trying to prove this converges. Want to know if this reasoning is solid
Def...
lyric charm
some missed backslashes in the tex
anyway formally speaking you didnt define a_n and b_n correctly
though once that's fixed, yes, you can say that the sum of convergent series is convergent.
grand pondBOT
wai
lusty python
if i remember correctly, $\sum_{i=1}^{\infty} \frac{1}{2^i}$ converges and so does $\sum_{i=1}^{\infty} \frac{1}{3^i}$, so the sum converges
grand pondBOT
1 divided by 0 equals Infinity
lyric charm
im not 100% sure that interleaving any two convergent series is gonna give you sth convergent.
grim vector
It has to be absolutely converging
lyric charm
it works for sure when the series you're interleaving are each absolutely convergent (and these geometric ones are)
but i cant think of a CE with one or both of them conditionally convergent
grim vector
(-1)^n/sqrt(n) or something like this should work as a CE
unkempt sluice
Use (1+/-(-1)^n)/2
keen sinew
why even defining an and bn ?
silent dock
exactly
twilit field
rearrangements don't work for infinite series always
lyric charm
$a_{2k-1} = \frac{1}{2^k}; a_{2k} = 0$ and $b_{2k-1} = 0; b_{2k} = \frac{1}{3^k}$.
grand pondBOT
Ann
lyric charm
this is how i would define these series
lyric charm
(-1)^n/sqrt(n) or something like this should work as a CE
(-1)^n/sqrt(n) interleaved with what
quiet hinge
Lot easier would be to just compare with sum 1/(1.5)^n
midnight plankBOT
graceful drum
im not 100% sure that interleaving _any_ two convergent series is gonna give you...
doesnt wai's logic suffice for the general proof?
keen sinew
you can also compare to ∑ (5/6)^i
graceful drum
or are you looking at general cases of interleaving where its just not odd and even placement
unkempt sluice
Actually 1/2+1/3+1/2^2+1/3^2+1/2^3+1/3^3+… <=1/2+1/2+1/2^2+1/2^2+1/2^3+1/2^3… works to me
steel shore
I'm trying to prove this converges. Want to know if this reasoning is solid
Def...
Just separate out the series
½+½^2+½^3....
+
⅓+⅓^2....
keen sinew
Just separate out the series
½+½^2+½^3....
+
⅓+⅓^2....
this is hard to justify
twilit field
Just separate out the series
½+½^2+½^3....
+
⅓+⅓^2....
This is logically invalid
steel shore
We're trying to find the value of this right?
keen sinew
i believe that if you say that this is a sum of positive terms series then it works
keen sinew
We're trying to find the value of this right?
proving the convergence yes
steel shore
So you're saying we can't just separate out the geometric progressions?
twilit field
Well, we haven't proven convergence
lyric charm
.reopen
midnight plankBOT
✅ Original question: #help-49 message
lyric charm
if y'all gonna keep yapping
steel shore
if y'all gonna keep yapping
Thanks
keen sinew
So you're saying we can't just separate out the geometric progressions?
you cant always say that the sum of 2 convergent series is a convergent series
lyric charm
you can actually
keen sinew
i mean the series of the sum of principal terms
steel shore
you cant always say that the sum of 2 convergent series is a convergent series
Damn this is news to me
keen sinew
i mean the series of the sum of principal terms
sorry im french
grim vector
Don't be sorry
keen sinew
like sum u_n + sum v_n converges is not equivalent to sum (u_n + v_n) converges
lyric charm
nobody said it is equivalent
but sum u_n convergent & sum v_n convergent does IMPLY sum (u_n + v_n) convergent
grim vector
Its basically zeta(2) + zeta(3) so it converges
lyric charm
Its basically zeta(2) + zeta(3) so it converges
yaku its a suite géometrique...
keen sinew
Its basically zeta(2) + zeta(3) so it converges
?
silk pulsar
We're trying to find the value of this right?
Hi
keen sinew
but sum u_n convergent & sum v_n convergent does **IMPLY** sum (u_n + v_n) conve...
oh yes sorry i got confused
steel shore
like sum u_n + sum v_n converges is not equivalent to sum (u_n + v_n) converges
But in the question we're solving I can see that:
sum u_n + sum v_n = sum (u_n + v_n)
I'm speaking for our particular question only
keen sinew
But in the question we're solving I can see that:
sum u_n + sum v_n = sum (u_n...
yes you can say that
in that case
steel shore
So if I separate out both series and then find value of both series and then add the value then wouldn't that be considered as the original series is convergent
Because that addition of values of both series would be equal to thevalue of original series
keen sinew
yes, its just that you cant go the other way
steel shore
What do you mean by the other way
keen sinew
you cant prove that Σ u_n + Σ v_n converges by saying that Σ u_n + v_n converges
unkempt sluice
Σ (a_n+b_n) -> L
Thus sum of previous 2k terms <L+1, sum of previous 2k+1 terms also < L+1 because a_n<1
Moronically increasing and having an upper bound L+1
keen sinew
this is just the fact that ∑ 1 + Σ -1 diverges but Σ 1 + (-1) = Σ 0 converges
steel shore
this is just the fact that ∑ 1 + Σ -1 diverges but Σ 1 + (-1) = Σ 0 converges
This is because it's infinity - infinity
keen sinew
yes
steel shore
But if it's nothing indeterminate then we should be safe
steel shore
But if it's nothing indeterminate then we should be safe
Safe to say Σ u_n + Σ v_n = Σ u_n + v_n
It should always be equal as long as the range of n is the same
steel shore
Safe to say Σ u_n + Σ v_n = Σ u_n + v_n
It should always be equal as long as the...
As per my understanding
quiet hinge
Actually 1/2+1/3+1/2^2+1/3^2+1/2^3+1/3^3+… <=1/2+1/2+1/2^2+1/2^2+1/2^3+1/2^3… wo...
Same as this
#help-49 message
Any number lesser than or equal 2 greater than 1 will work for comparison
unkempt sluice
Yeah, any sequence of the form =r mod m terms sits a convergent sequence a_r_n: n works by the same logic
Upper bound L_1+…+L_m + mC for some constant C
steel shore
I think we're done
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steel basin
Prove that abscissa of a point P which is equidistant from points with coordinates A(7, 1) and B(3, 5) is 2 more than its ordinate.
Can you use midpoint formula here; if not, why? Every site online gets the ans by distance formula
midnight plankBOT
Prove that abscissa of a point P which is equidistant from points with coordinat...
wanton spade
Prove that abscissa of a point P which is equidistant from points with coordinat...
there isn't only a single point that is equidistant from both the points
steel basin
(5,3)?
steel basin
hmm okai thx
midnight plankBOT
wanton spade
Prove that abscissa of a point P which is equidistant from points with coordinat...
.you use the distance formula because it gives you an equation, and all the points satisfying that equation are equidistant from the two points you have
in this case you would get x-y=2 and you can see (5,3) works for this
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kindred veldt
So munkres defines locaclly connectedness at a point x in the following way: for every open set U containing x, there exists a connected open set contained in U which also contains x. And weakly locally connectedness at x in the following way: for every open set U containing x, there exists a connected subspace contained in U which contains an open set containing x.
First question: is the open set in the weakly locally connected case supposed to be open w.r.t. to the whole space? (since it being open w.r.t. the subspace doesnt make it interesting (take the singleton x as subspace))
Second question: if this is the case, then what is the difference between these two concepts since one could just take the open set (w.r.t. the whole space) contained in the connected subspace (weakly locally connected at x) as the connected open set in the locally connected definition making the two definitions equivalent (not only in the whole space case, but also for every single point).
midnight plankBOT
So munkres defines locaclly connectedness at a point x in the following way: for...
hushed mauve
if I understood correctly from my own studies, yes to the first, and to the second, consider that the interior of a connected set need not be connected.
this question could be a good example.
(I am still learning and thus may be wrong though, so I'd advise you to wait for second opinions!)
kindred veldt
What I mean is following: If X is weakly locally connected at x, then for every open set U containing x, there is a subspace which contains an open connected set containing x. But this open connected set containing x is by definition also in the open set U, not really making the need for this extra step (the subspace) obvious to me. (which is probably also the place where my misunderstanding lies)
https://arxiv.org/pdf/1311.5122 end of page 3 makes the difference more obvious (metric spaces here only)
ill think about it
unkempt sluice
You said in the definition, there is no open connected subset, just an open subset of a connected subspace
Any x in open U there exist x in open V in connected H in U. No mentioning V is connected
kindred veldt
ah
oh
yeah this is the reason
thank you
my misconception relied on me for some reason thinking that an open set of a connected set is connected (?)
.close
mint ravine
help
wet locust
mint ravine
i don't know how to start
oak nymph
Do you know the number of positive integral solutions to x1+x2+..+xn=m?
finite fossil
can I tell him how to start
wet locust
Do you know the number of positive integral solutions to x1+x2+..+xn=m?
aka do you know stars and bars/balls and urns/cookies and candy canes/whatever you call this thing
mint ravine
yeah i do
oak nymph
Try using that here.
mint ravine
ok
finite fossil
it's not the exact same thing with stars and bars
dusty portal
aka do you know stars and bars/balls and urns/cookies and candy canes/whatever y...
sticks and stones 😂
oak nymph
it's not the exact same thing with stars and bars
Sure, but the ideas are similar.
wet locust
i will say that there is a solution without using stars and bars
stars and bars is probably a long way of doing it
finite fossil
it just added a restriction
wet locust
there's a short way to do it
finite fossil
yeah I know
wet locust
i think i would recommend trying to do the short way
wet locust
yeah ^
wooden badger
aka do you know stars and bars/balls and urns/cookies and candy canes/whatever y...
Beggars method :>
wet locust
⁉️ wait ive never heard of this one
does it come from the version of the problem involving money
dusty portal
Beggars method :>
final form unlocked ahh
finite fossil
!?
wooden badger
Beggars and coins
keen saddle
Beggars method :>
we also call it the coin beggar method lmao
wooden badger
final form unlocked ahh
finite fossil
omg it's the wumpus man
wooden badger
Also gap method
finite fossil
can I ping 7
I want to know if he finished
bruh what
@mint ravine Did you finish the question
mint ravine
close to
finite fossil
ok nice
midnight plankBOT
mint ravine
56
mint ravine
👍
midnight plankBOT
Available help channel!
spare rover
hello I'm stuck on the question at the bottom , I don't even know if it is true for any C^infty bounded function or that it is strongly linked to (E). I think the good way to solve this is to suppose these equalities and deduce the way a and b are written but i don't know how to do it
midnight plankBOT
hello I'm stuck on the question at the bottom , I don't even know if it is true ...
fallow scarab
can't you just solve for a(x) and b(x) in terms of f(x) and f'(x)
spare rover
yes this is the way I want to get the answer but I don't know how to do it
spare rover
ok i'm blind , i missed this .................................
thank you mate
b(x)=f(x)sin(x)+f'(x)cos(x) / a(x) =f(x)cos(x)-f'(x)sin(x) ?
a,b are C^infty beacuse they are sum of C^infty functions and they are uniquely written
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midnight plankBOT
midnight plankBOT
Available help channel!
lilac finch
claim
lilac finch
[
\textbf{Question:}\quad
\begin{vmatrix}
1 & 1 & 1 \[6pt]
(x-a)^2 & (x-b)^2 & (x-c)^2 \[10pt]
(x-b)(x-c) & (x-c)(x-a) & (x-a)(x-b)
\end{vmatrix}
= 0
]
[
\begin{array}{ll}
\text{(A)} & a + b + c = 0 \[6pt]
\text{(B)} & x = \dfrac{a + b + c}{3} \[6pt]
\text{(C)} & x = \dfrac{a + b + c}{2} \[6pt]
\text{(D)} & x = a + b + c
\end{array}
]
grand pondBOT
oppenheimer
lethal path
it's not bad to expand the determinant
lilac finch
it's not bad to expand the determinant
can we do without
lethal path
idk actually
slow thorn
**oppenheimer**
multiply (x-a), (x-b) and (x-c) onto each column and balance by multiplying $\frac{1}{(x-a)(x-b)(x-c)}$ outside
grand pondBOT
midnight plankBOT
lethal path
multiply (x-a), (x-b) and (x-c) onto each column and balance by multiplying $\fr...
holy shit
slow thorn
stay tuned for funny determinant tricks
midnight plankBOT
Available help channel!
coral belfry
,tex
hello. suppose we have two sequences, that are both strictly decreasing and convergent to some number each, not necessarily the same. is it true that the supremum and infimum of the set, A, containing the numbers given by the addition of those two sequences, say $ a_n + b_m $, are sup(A)= $$ \sup(a_n) + \sup(b_m) , \inf(A) = \inf(a_n) + \inf(b_m) $$ ? If so, how do we prove it? If the proof is long i can accept a source that shows it proven there
midnight plankBOT
,tex
hello. suppose we have two sequences, that are both strictly decreasing and...
runic hamlet
ok so first we can forget the word sequence
grand pondBOT
fijokazż
runic hamlet
cause you are just treating them as sets anyway
coral belfry
alright
runic hamlet
and given two sets X,Y, it does indeed hold that sup(X+Y)=sup(X)+sup(Y)
same for inf
coral belfry
what do you mean by X+Y?
runic hamlet
you can google for "sup of sum of sets"
runic hamlet
what do you mean by X+Y?
exactly what you are doing. all sums of two elements from X and Y
coral belfry
oke i see
coral belfry
so by showing that we can just sub a sequence?
i'll try it out it seems fair
anyway thanks i got it
.close
midnight plankBOT
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Available help channel!
coral belfry
hi there. this is my proof of sup(A+B) = supA + supB. is it correct? at the end i arrived at a contradiction, and i only have left to say that supC = supA +supB. Also, ignore the thing in the box
midnight plankBOT
hi there. this is my proof of sup(A+B) = supA + supB. is it correct? at the end ...
coral belfry
just the last two lines need checking actually
wait i suddenly felt a wave of stupidity hit me
ill work on it some more
.close
midnight plankBOT
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Available help channel!
spice talon
hello just wondering chatgpt is telling me teta + alpha is 90 degrees here why ?
midnight plankBOT
hello just wondering chatgpt is telling me teta + alpha is 90 degrees here why ?
spice talon
spice talon
mhm
lilac finch
Is this related to biot savart?
spice talon
yea
lilac finch
Ic
spice talon
is it related to complemntary angles?
lilac finch
is it related to complemntary angles?
Sum of angles of a triangle
Idts complementary
Lemme see
midnight plankBOT
lilac finch
i figured out a wayy
Aight, were they?
midnight plankBOT
Available help channel!
slow thorn
$\int \frac{3-t^2}{(3+t^2)^2} dt$
midnight plankBOT
$\int \frac{3-t^2}{(3+t^2)^2} dt$
grand pondBOT
slow thorn
answer doesnt look that bad but i have no clue how to reach
dreamy lichen
kinda looks like quotient rule
or you could go the normal way and split the fractions & integrate term by term
dreamy lichen
kinda looks like quotient rule
yep, it is a simple quotient rule
slow thorn
well yeah but how would i figure the numerator even if i figured it was quotient rule
lyric charm
well if we call v := 3+t^2 (NOT a substitution, just algebraic playing-around)
dreamy lichen
well yeah but how would i figure the numerator even if i figured it was quotient...
well, we would know g for sure
slow thorn
if i assume u = kt and pray
yyeah i think thats the best i can do
thanks
.clos
.close
midnight plankBOT
dreamy lichen
<@&268886789983436800>
dreamy lichen
if i assume u = kt and pray
slightly more generally, you could assume that its a low degree polynom
that'd work more often
the low-degree comes from the fact that the RHS is 2nd degree
midnight plankBOT
Available help channel!
blissful trench
<@&268886789983436800>
viral dagger
any mistakes here?
midnight plankBOT
any mistakes here?
safe oriole
any mistakes here?
there is a mistake in the 3rd line
safe oriole
no
given a complex number z=x+iy with x,y in R, the conjugate \bar{z} of z is given by \bar{z}=x-iy
here x=-1/2 and y=1
viral dagger
like this right?
viral dagger
here x=-1/2 and y=1
uhh yea?
fallow scarab
,w 1/(-1/2 + i)
grand pondBOT
viral dagger
,w ((-1/2)-i)/(5/4)
grand pondBOT
safe oriole
uhh yea?
so the conjugate of -1/2+i is -1/2-i (which is the same as you wrote in the pic) and not 1/2+i which you suggested later
viral dagger
where did i suggest that?
safe oriole
in the second line it shoukd be (1/2)+i
ah i thought this was you suggesting that
mb lol
grand pondBOT
safe oriole
any mistakes here?
yea then this is correct
viral dagger
owh wtf wolfam can do that?
fallow scarab
,w (-1/2+i)^6 * ( 1/2 + i+ 1/(-1/2+i))^6
grand pondBOT
safe oriole
like this right?
something to note which may not be much helpful tho tbh, you couldve rewritten -1/2+i as 1/2-i and then you would be multiplying conjugates together
tho its the same thing at the end of the day
safe oriole
something to note which may not be much helpful tho tbh, you couldve rewritten -...
you can do this since you are working with an even power of -1/2+i
but yea its not a very important note, you can forget about it if you want 
midnight plankBOT
midnight plankBOT
Available help channel!
chilly cobalt
Let PQR be a triangle, and let S,V be points on side PR and RQ respectively. Let QS intersect PV at T. The segments RT and SV intersects at U. Suppose that the areas of the triangles RST, RTV, RSV are 55, 66 ,77 respectively. What is the area of PQU?
midnight plankBOT
Let PQR be a triangle, and let S,V be points on side PR and RQ respectively. Let...
chilly cobalt
I only have ideas on cevas theorem on T relative to RSV but i dont really have anything that relates the areas T-T
midnight plankBOT
@chilly cobalt Has your question been resolved?
next bobcat
chilly cobalt
I only have ideas on cevas theorem on T relative to RSV but i dont really have a...
^
so not really anything
cause cevas we get RQ/QV x VU/US x PS/RP = 1
late rover
Let PQR be a triangle, and let S,V be points on side PR and RQ respectively. Let...
This's not an olympiad like geo problem, is it?
chilly cobalt
its not
late rover
Yeah so you probably won't need Ceva's here
chilly cobalt
its entrance exam prep so i dont think the solution is too out of this world
late rover
Let TVQ=x and ST/SQ=m
Make a system of equation with these 2 terms
using ratios between 2 triangles
Like ||m(x+121)=55||
why do I feel like Menelaus' theorem is gonna pop up here
late rover
then we relate x to PVQ?
I mean you don't have to relate x to PVQ to find x and m
chilly cobalt
m = 44/(44+x) too right?
late rover
yeah that looks right
chilly cobalt
<:pandahmm:858834883760750602> why do I feel like Menelaus' theorem is gonna po...
yeah thats what i was doing with cevas😭
chilly cobalt
x = 264
chilly cobalt
and m = 1/7
late rover
I haven't checked but tbh I trust an olympiad person than myself so I'll assume it's right
chilly cobalt
and m = 1/7
i suck at addition mb😭
late rover
Oh okay
chilly cobalt
<:pandahmm:858834883760750602> now it looks like menelaus's
on SVQ?
late rover
I mean you can see how we can find PS/PR using menelaus
chilly cobalt
hmm
late rover
we could use vector
chilly cobalt
i think i got the sol
late rover
greatt!
chilly cobalt
oh wiat
i looked at the wrong thing mb 😭
i have the areas for PST and PTQ though
late rover
well ig we have to find SU/SV anyway
yeah we can't get around this
late rover
i have the areas for PST and PTQ though
We would have to find SU/SV eventually
chilly cobalt
We would have to find SU/SV eventually
my ceva idea works
late rover
or am i stupid
I dunno lol I haven't done any calculation
late rover
what does that help with finding PQU?
subtract the big triangle by 2 smal ones
chilly cobalt
how would i find the area 😭
small jasper
subtract the big triangle by 2 smal ones
hi 
late rover
Okay so PRQ-RSV-UVQ-PSU
chilly cobalt
entrance exam of doom and despair
late rover
We can relate QUV to QSV using that ratio
RSV can be easily found
we can relate PSU to PSV to PRV to the big one
late rover
entrance exam of doom and despair
Yeah I would crash out if this appear on my exam
Actually
It does
But not this tedious
late rover
why is this the first problem my teacher gave me im cooked
There might be a better way
But I'm certain that you would have to find SU/UV anyway
and some other ratios
late rover
why is this the first problem my teacher gave me im cooked
Nah it's okay, what's the exam date
It's more than 2 months from now it's alright
late rover
This month?
chilly cobalt
yeah
late rover
chilly cobalt
top 600 -> 240 to qualify for the school
anyways i think i have the solution now, thanks!
late rover
top 600 -> 240 to qualify for the school
How many % is that
late rover
anyways i think i have the solution now, thanks!
Okee
chilly cobalt
How many % is that
like half half approx
but like 30k people applied this year
.close
midnight plankBOT
midnight plankBOT
Available help channel!
vagrant warren
cldnt quite get this
midnight plankBOT
cldnt quite get this
wooden badger
cldnt quite get this
Do you have thte answer key?
lyric charm
the way it's laid out in this particular freeze-frame of the video is kinda confusing @vagrant warren
but let's say this: if a was replaced with a number, say -10, would you know how to begin solving the equation?
wanton spade
I'm guessing they're trying to make x²-x+a have exactly one zero so while completing the square they choose a=0.25 to make the root 0
midnight plankBOT
@vagrant warren Has your question been resolved?
midnight plankBOT
vagrant warren
.reopen
midnight plankBOT
✅ Original question: #help-49 message
vagrant warren
Do you have thte answer key?
yes i do
vagrant warren
but let's say this: if _a_ was replaced with a number, say -10, would you know h...
yes
lyric charm
ok right
vagrant warren
Whats the ans so i can confirm before helping😭
c
lyric charm
so you know that the left bracket always gives you x=a as a solution
thats true no matter the value of a
vagrant warren
Yes
lyric charm
the right bracket can give you:
a) 0 solutions if D < 0
b) 1 solution if D = 0
c) 2 solutions if D > 0
before accounting for a possible collision with x=a
do you follow thus far? i have more to say immediately after this.
vagrant warren
yes
lyric charm
ok so let's work out the collisions first
in other words, when (for what value/values of a) is x=a a solution of x^2 - x + a = 0?
vagrant warren
its 1/4
lyric charm
what's "it"
vagrant warren
a=1/4
lyric charm
the answer to my question, which i asked just now, is not a = 1/4
vagrant warren
when det = 0
lyric charm
that's not what i asked
vagrant warren
yes
lyric charm
which simplifies to a^2 = 0, meaning a = 0 is the only value of a that would give us a collision.
vagrant warren
and i find the values of a which are, a=0 and a=1
lyric charm
keep that in mind for later.
lyric charm
can you like not jump ahead of me
vagrant warren
thats incorrect
vagrant warren
can you like not jump ahead of me
ok
lyric charm
which simplifies to **a^2 = 0**, meaning **a = 0** is the only value of a that w...
this. stop for 10 seconds and read this. do you understand this. yes or no.
wooden badger
wdym by a collision
Only one soln for the given
lyric charm
what i'm calling a collision here is a value of x arising from both factors at once
it still counts as only one solution, and since we're interested in the solution count, that matters for us
vagrant warren
ok i see
lyric charm
ok, right.
so now let me bring back the case breakdown from before.
the right bracket can give you:
a) 0 solutions if D < 0
b) 1 solution if D = 0
c) 2 solutions if D > 0
a collision will happen if a = 0. if it does, the solution count will be reduced by 1.
case (a) gives us 1 solution in total, which is less than the 2 we want. so we don't want case (a), ever.
heady plover
What is the question?
lyric charm
see pinned msg
case (b) gives us 2 solutions in total. let's work out the value of a that makes case b happen.
D = 1 - 4a = 0 gives a = 1/4. no collision.
vagrant warren
yes
lyric charm
(also, discriminant not determinant. similar words, completely different meanings.)
ok
vagrant warren
(also, **discriminant** not determinant. similar words, completely different mea...
my fault fp3 stuck in my head
lyric charm
case (c) gives us 3 solutions in total, which is too many unless a collision knocks us down to 2.
vagrant warren
case (c) gives us 3 solutions in total, which is too many **unless a collision k...
yes
vagrant warren
yes
is that it?
not trynna be rude or anything
so thats how we get
C
so we find out if we ever get repeated solutions and that can be seen when (x-a) is a fator of x^2 -x +a
and that occours when a=0
so a solution occours when x=a
and we see when right hand bracket discriminant =0
lyric charm
is that it?
the final answer is a ∈ {0, 1/4}
vagrant warren
so we get 2 solutions from
and then we see
when disciminant is greather than 0
but since left hand bracket is always a solution
we need right hand bracket to collide
hence a repeated root
so
a=0
in that case
i see
thats gna be kinda hard to workout
i was thinking abt that
but i wasnt able to do it
i never got the a=0
midnight plankBOT
@vagrant warren Has your question been resolved?
coral belfry
hi there. I just happened upon this statement in a help channel, and while it makes sense, idk how we can justify it rigorously. is there a rigorous proof of the "equality"?
midnight plankBOT
hi there. I just happened upon this statement in a help channel, and while it ma...
silent dock
hi there. I just happened upon this statement in a help channel, and while it ma...
diamond$ds sqrt{2n+1} sim sqrt{2n} \q text{means} \q lim_{n to oo} fr{sqrt{2n+1}}{sqrt{2n}} = 1$
grand pondBOT
Oléagineux Distilliànus VIVII
silent dock
the rigorous version is usually phrased as an asymptotic equivalence
coral belfry
oh i see now
okay thats good enough, ill also go look up that asymptotic thingy
thanks cooly
.solved
midnight plankBOT
silent dock
silent bone
Hi guys
Was wondering if maths for me or not
Really love doing it but it doesn't love me back
Till high school i was very good in it. Scored 100/100 in exams while completing 2.5h paper in 1.3 hours. But faced miserable downfall after then. Want to restart, idk from where
I've realised that I can't achieve success in life while hating maths
coral belfry
u can bring your blues to the discussion channels bud
silent bone
u can bring your blues to the discussion channels bud
Sorry I'm new, do we have a general chat here
coral belfry
ill ping ya
compact copper
Maybe <#342850939306246145> and <#849776666161971240>
didn't you become green a few weeks ago
wooden badger
Also #study-discussion
silent bone
When am I supposed to use this channel
late rover
didn't you become green a few weeks ago
Nahh
wooden badger
When am I supposed to use this channel
If you have a specific math problem you need help w
late rover
When am I supposed to use this channel
You're not supposed to use this one, it's occupied
silent bone
Okayy
wooden badger
-# still hvnt read ur soln btw fionna :>
silent bone
My bad
late rover
Available one is like what cherry said
silent dock
late rover
-# still hvnt read ur soln btw fionna :>
silent dock
thanks cooly
why did blud @vale zephyr 👍 this like blud did something
midnight plankBOT
Available help channel!
silent dock
.
vale zephyr
.
lyric charm
<@&268886789983436800> scam
midnight plankBOT
buoyant linden
hi there. I just happened upon this statement in a help channel, and while it ma...
🤔
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Available help channel!
mint ravine
what is the inscribed angle theorem
midnight plankBOT
what is the inscribed angle theorem
past horizon
inscribed angle is 1/2 the central angle