woah
#help-27
toxic flower
grand edge
Such an aura farming thing to do in 1842
grand edge
Oh wait there are more of these integrals just scroll down
toxic flower
💀
grand edge
Like just how
toxic flower
dude had norhing else to do
toxic flower
Like just how
thanks for the help dude 
stoic lotus
grand edge
no worries man
grand edge
Lol
stoic lotus
Substituting tan x = u gives your exact integral
grand edge
You just pulled the answer and called it a known result
stoic lotus
You just pulled the answer and called it a known result
it's from Gradshteyn and Ryzhik
toxic flower
grand edge
it's from Gradshteyn and Ryzhik
Ik I recognize the font
But like he's trying to derive it
stoic lotus
Okay
You could also consider the function $f(z) = \frac{\ln z}{\cosh z}$ and integrate it in the upper half-plane using a contour integral 🚬
woven radishBOT
a handsome russian dude
grand edge
Cool idea but what are you gonna do about the branch cut
A keyhole contour would probably help
In fact I remember a paper which used contour integration to evaluate types of malmsten integrals
Let me see if I can find it
Yup
okay never mind, this paper is way too convoluted
stoic lotus
A keyhole contour would probably help
Ye
Just use an indented contour
I guess it doesn't make it more concise unfortunately
Cuz you'll have to use the residue theorem
grand edge
its not concise because there are infinite singularities on the imaginary axis
so probably not a keyhole then
stoic lotus
When you sum the residues for $f(z) = \frac{\ln z}{\cosh z}$ at $z_n = i\pi(n + 1/2)$, you get a series of the form $\sum (-1)^n \ln(n + 1/2)$
woven radishBOT
a handsome russian dude
stoic lotus
Which can be expressed as a derivative of the Dirichlet beta function
grand edge
lol
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Available help channel!
lilac pelican
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woven radishBOT
polar chasm
yeah, 5 seems like the minimum value
lilac pelican
Absolute?
polar chasm
yeah
tender wharf
How about in the interval [1,infinity[
same
polar chasm
Depends on the precise defn ur book is using
lilac pelican
Not absolute
polar chasm
Idk why but my teacher just put it as local
its both technically
lilac pelican
Depends on the precise defn ur book is using
F(x)>= 5 for all x
Is the condition here
In our book
polar chasm
if you have a textbook, try finding the definition of global and local minima and send a ss
lilac pelican
if you have a textbook, try finding the definition of global and local minima an...
The problem is its in arabic
Not english
polar chasm
send it anyway
lilac pelican
short hare
,rccw
woven radishBOT
short hare
the maximum local at x = c is f(c) if there was an interval containing c. so that f(c) >= f(x) for all x is a part of.... idk what that is
tender wharf
hmm..
modern lance
the maximum local at x = c is f(c) if there was an interval containing c. so tha...
-# acarvroomvroom
tender wharf
tbf a global extremum is a local too
short hare
-# acarvroomvroom
-# fionnaaaa
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thorny carbon
Solve for all real values of x:
\log_2!\big(\log_3(x-1)\big) + \log_2!\big(\log_3(9-x)\big) = 1
polar chasm
$\log_2!\big(\log_3(x-1)\big) + \log_2!\big(\log_3(9-x)\big) = 1$
woven radishBOT
MathIsAlwaysRight
thorny carbon
Please help with maths assignments
polar chasm
what even is that !
thorny carbon
Ask my teacher bruh
crystal dawn
got a pic just to confirm?
devout snowBOT
@thorny carbon Has your question been resolved?
thorny carbon
No gng🥀
thorny carbon
<@1140319119217004635> Has your question been resolved?
No
urban spindle
I'd like to help, but not sure what the "!" means. A factorial would have it afterwards... and am not familiar with other instances of "!"
Either way, looks like it uses the formula for sum of 2 logs with same base...
upbeat forge
where did you get this expression from? could you screenshot from that instead?
tender wharf
**MathIsAlwaysRight**
if we ignore the !
thorny carbon
Find all real x such that:
\log_{\log_{\log_2 x} 4}!\Big(\log_{\log_2 x} 16\Big)
;=;
\log_2!\big(\log_x 256\big)
polar chasm
where did you get the latex from?
jolly iris
lol
polar chasm
Do you type it yourself?
thorny carbon
Gng it’s my assignment
upbeat forge
screenshot, please.
thorny carbon
Am I cooked?💔
jolly iris
$\log_2!\big(\log_x 256\big)$
woven radishBOT
Aayush
jolly iris
the f
polar chasm
just screenshot the thing and dont copy the latex, i hope your teacher doesnt give u homework in raw latex
crystal dawn
OP, please consider that there are reasons people are asking for screenshots.
jolly iris
ngl the concept kinda cool
maths teacher feeds raw latex to students
tender wharf
$\log_(2!)\big(\log_3(x-1)\big) + \log_(2!)\big(\log_3(9-x)\big) = 1$
woven radishBOT
oppenheimer
thorny carbon
My head hurts
jolly iris
$\log_{2!}\big(\log_3(x-1)\big) + \log_{2!}\big(\log_3(9-x)\big) = 1$
woven radishBOT
Aayush
jolly iris
this it?
crystal dawn
My head hurts
is there something wrong with screenshotting the question?
thorny carbon
crystal dawn
polar chasm
can anyone decipher this?
thorny carbon
Sorry I was in a hurry
jolly iris
crystal dawn
is that log_{1/2}?
tender wharf
this is escalating
jolly iris
i think bro is an alien pretending to ask maths questions
urban spindle
tfw getting the source makes it even worse
thorny carbon
can anyone decipher this?
That’s the greek letter zeta
jolly iris
That’s the greek letter zeta
i personally think its a scribble
tender wharf
That’s the greek letter zeta
is it
polar chasm
and what is zeta supposed to mean in this context
silk rock
That’s the greek letter zeta
if you're gonna introduce random vars, I hope you can explain what they mean...
jolly iris
if you're gonna introduce random vars, I hope you can explain what they mean...
zeta is the amount of braincells i lose every second
thorny carbon
and what is zeta supposed to mean in this context
It’s a variable name
silk rock
It’s a variable name
what does it represent?
jolly iris
It’s a variable name
like a constant?
thorny carbon
is simply a placeholder variable, like x, a, or t.
jolly iris
is simply a placeholder variable, like x, a, or t.
parametric equation moment
polar chasm
So are we supposed to find x in terms of zeta?
thorny carbon
Yes
polar chasm
and next question, what's this? It's either tg base 2 (which makes no sense) or weirdly written log bsae 2. Or is it sth else?
tender wharf
$\log_{2}\big(\log_{\log_{2} x} 4\big) + \big(\log_{\log_{2} x} 16\big) = \log_{2}\big(\log_{x} 256\big)$
Find all real values of x.
woven radishBOT
oppenheimer
tender wharf
fuck
jolly iris
what did bro do
thorny carbon
Yes
thorny carbon
and next question, what's this? It's either tg base 2 (which makes no sense) or ...
Yeah that’s log
polar chasm
i'd probably do 2^ on both sides
thorny carbon
i'd probably do 2^ on both sides
Just be fast bro I don’t wanna get a B in math for the first time
tender wharf
$\log_{2}\big(\log_{\log_{\zeta/2} x} 4\big) + \log_{\log_{2} x} 16 = \log_{2}\big(\log_{x} 256\big)$
so this
jolly iris
$\log_z\big(\log_{\zeta}(\log_{log_{2}x}16)\big) = \log_{2}(\log_{x}256)$
woven radishBOT
Aayush
silk rock
let's not make too many demands of helpers trying to help you while understanding your question, shall we
void fox
it seems quite plausible to me that this person is trying to cheat on a test
jolly iris
Ok what’s the answer
whats not the answer
void fox
this image smells AI-generated. My guess is they did this because they obviously can't send the source image directly
jolly iris
this image smells AI-generated. My guess is they did this because they obviously...
new sense
smelling bits
jolly iris
are u in a test or smth bro
upbeat forge
!noans.
devout snowBOT
The purpose of this server is to help you learn; please don't ask for direct answers. Ask for guidance, explanations, or feedback instead.
tender wharf
i give up
thorny carbon
I’m cooked
polar chasm
I’m cooked
So is it a test or not?
jolly iris
$2+2 = \big5$
woven radishBOT
Aayush
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
polar chasm
honestly im starting to doubt its even solvable
river rune
it's weird anyway. the question asks for all real values of x. yet you have a fudge variable zeta in there that means nothing.
thorny carbon
I This is Mr. Shawn. This device has been confiscated due to a violation of the electronics policy during the assessment. I am currently reviewing this chat log to determine the extent of the academic dishonesty involved. I suggest you stop messaging this account immediately.
jolly iris
I KNEW IT
river rune
<@&268886789983436800> well...
jolly iris
HI MR SHAWN
polar chasm
it's weird anyway. the question asks for all real values of x. yet you have a fu...
its supposed to be solved in terms of zeta
river rune
its supposed to be solved in terms of zeta
normally then I'd expect the question to say something to that effect, but given that the question did not, I'm starting to agree with plante.
delicate dust
I This is Mr. Shawn. This device has been confiscated due to a violation of the ...
hi mr shawn, this is the moderation team of the mathematics discord. this account has been timed out for a week, they will be unable to send messages on the server for that duration. thank you for bringing this to our attention
polar chasm
I This is Mr. Shawn. This device has been confiscated due to a violation of the ...
I gotta admit this is pretty funny way to end a trolling session
jolly iris
Im making this an album
devout snowBOT
jolly iris
@delicate dust can we have this as a sticker plsssss
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warm drift
I need help with question b
devout snowBOT
I need help with question b
pseudo basin
see this convo from last week about quartiles
this was between you and me specifically
pseudo basin
that's 25% of the sample size, yes
warm drift
idk what to do with it 😭
pseudo basin
well, let's go from the low end of the time axis
if we take just the first bin (24 patients), is that enough to cover the 26 lowest waiting times? y/n
warm drift
no
pseudo basin
right
if we take the entire next bin as well, is that not enough, exactly enough, or too much?
warm drift
too much
warm drift
2/28
pseudo basin
right
the second bin's interval on the time axis is 30 < t ≤ 40
the core assumption behind a histogram is that we assume the data points within each bin are approximately evenly spread out
warm drift
so 2/28 times by 10?
pseudo basin
so you take the number that's-
almost but not quite
30 + 2/28 * 10
it's the number that's this fraction of the way from 30 to 40
warm drift
why have to plus 30
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pseudo basin
!occupied
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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).
pseudo basin
(sorry, this channel still has somebody's name on it. you need to take a free one)
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untold ravine
Let x1,x2,x3,x4 be reals in the interval (0,1/2]. Prove that $\frac{x_1 x_2 x_3 x_4}{(1-x_1)(1-x_2)(1-x_3)(1-x_4)} \leq \frac{x_1^4 + x_2^4 + x_3^4 + x_4^4}{(1-x_1)^4+(1-x_2)^4 + (1-x_3)^4 + (1-x_4)^4}$
devout snowBOT
Let x1,x2,x3,x4 be reals in the interval (0,1/2]. Prove that $\frac{x_1 x_2 x_3 ...
wind mason
skul
woven radishBOT
CherryMan
knotty salmon
what's your quesion
untold ravine
**CherryMan**
this one
untold ravine
doesnt matter really
untold ravine
can you tell me
grade 5
knotty salmon
holy lies
supple knot
what happened to the latex
you need to send the inequality to its own message and try using \$\$ instead of \$
sand dove
I would do a sub $a_i = \frac{x_i}{1-x_i}$
untold ravine
I would do a sub $a_i = \frac{x_i}{1-x_i}$
right am gm gives us $$s \leq \frac{1}{4} \sum \frac{x_i^4}{(1-x_i)^4}$$
woven radishBOT
CherryMan
tropic talon
wtf is this
supple knot
wtf is this
!topic
devout snowBOT
wtf is this
Please read the channel description before posting, and stay on topic.
tropic talon
Is this also abee server
untold ravine
hope that we wont have to expand (x-1)^4
untold ravine
I would do a sub $a_i = \frac{x_i}{1-x_i}$
then did you have any ideas
sand dove
Well the RHS is a weighted mean of a_i^4
So I was thinking about Jensen
But I don't have the time to think about this problem anymore unfortunately
Leaving it to other helpers
devout snowBOT
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Available help channel!
outer mauve
examine the continuity of a function
this is on my math course and i have 0 clue what to do can someone help?
uneven coral
Well we can find the continuity at x = 0 and x = 1
outer mauve
because there is a 1 and 0? what if there are different numbers like 3 and 1
uneven coral
well you need an equation just below and just above that point to know the continuity
and we have the equation for just below and just above x = 0, 1
safe jasper
you want to consider continuity on the entire domain of f. we know that the pieces that make up f are themselves continuous (they're polynomials), so the only places of concern for discontinuity are the places where you've "glued" the functions together. in this case, those numbers are 0 and 1
uneven coral
for x = 3, we only have x>=1
safe jasper
do you know how to check if a function is continuous at a point?
outer mauve
no
outer mauve
no
fierce viper
examine the continuity of a function
this is on my math course and i have 0 clu...
what is this class called?
uneven coral
Alright have you been taught this before and you just don't know it?
safe jasper
is this a standard calculus course? continuity can only be taught after limits are introduced
outer mauve
Alright have you been taught this before and you just don't know it?
my professor is awful, shes rambled on about limits but never bothered to explain much hence why i am here
uneven coral
my professor is awful, shes rambled on about limits but never bothered to explai...
Alright that makes way more sense then
okay do you know what continuity means?
outer mauve
that something goes on and on
uneven coral
that something goes on and on
then what does it mean when something is discontinuous?
outer mauve
that its got an end or a break
safe jasper
that its got an end or a break
does it make sense that the function needs to be defined at a point for it to be continuous at a point?
safe jasper
yes by "both of the points", you're probably referring to the left and right hand limits
uneven coral
okay and what do you know about limits?
outer mauve
they can end at 0, orr some can just not lead to anything like -1n^2 if i remember correctly
safe jasper
more generally, a limit is the value that a function "tends to" or "approaches"
for example, the limit as x -> 1 of x^2
as you consider x = 0.9, x = 0.99, x = 0.999, etc, you see that as you consider x values that approach 1 from the left side, your output values approach 1 too
similarly as you consider x = 1.1, x = 1.01, x = 1.001, etc, you see that as you consider x values that approach 1 from the right side, your output values also approach 1
safe jasper
as you consider x = 0.9, x = 0.99, x = 0.999, etc, you see that as you consider ...
what this means is that the left hand limit is 1
safe jasper
similarly as you consider x = 1.1, x = 1.01, x = 1.001, etc, you see that as you...
and this is called the right hand limit
outer mauve
and they are both 1 right?
safe jasper
yes
those two sided limits don't have to be the same btw, they could be different numbers depending on whcih side you approach your point from
but when they both exist and equal the same number, you say that the overall limit at 1 exists
does that definiiton make sense to you?
outer mauve
does that definiiton make sense to you?
yes beccause the limit applies to both left and right sides
safe jasper
the limit as x -> a of f(x) exists and equals L if and only if the limit as x -> a- of f(x) exists and equals L and the limit as x -> a+ of f(x) exists and equals L
where a- and a+ refer to the left and right hand limits
this is the same thing i just descirbed but using notation
still good?
safe jasper
the limit at a point exists only if the left and right hand limits exist and equal the same thing
so if your function tends to -1 from the left and 1 from the right, the limit doesn't exist
outer mauve
ok im bacck
safe jasper
you say that a function f(x) is continuous at a point x = a if lim as x -> a of f(x) exists and equals f(a)
as in, your function is continuous at a if the value that it takes on is the same as what happens when you approach a from the left and right
consider this function
at x = -2, would it be continuous or discontinuous?
outer mauve
disconinuous
safe jasper
and why is that?
outer mauve
beccausee tthe points dont meet
safe jasper
yes, the left and right limits are different
the left limit is 2, the right limit is -1
and since these aren't the same thing, the limit doesn't exist at x = -2
so it can't be continuous there
does that make sense?
outer mauve
yes
safe jasper
what about at x = 3, is the function continuous or discontinuous?
outer mauve
continuous becccause it repeats in x=-1
safe jasper
i don't really understand your reasoning, what do you mean it repeats?
outer mauve
the line intersects the x axis twice once in 3,0 and once in -1,0
safe jasper
that has nothing to do with whether or not it's continuous
remember, you're continuous at a point if the limit exists there and is the same as the value that the function outputs
you said we were discontinuous at x = -2 because the limit doesn't exist there (the left and right hand limits are different)
what about the limit here at x = 3?
safe jasper
all good, there's two things to check when checking continuity, which is the value of the limit and the value of the function
we know f(3) = -1
and the limit as x -> 3 is 0
right?
outer mauve
yes
safe jasper
since these are different numbers, it's not continuous at x = 3
what about at x = 0?
safe jasper
wdym?
outer mauve
no sure, continuity?
safe jasper
no, i mean the actual value f(0)
the output that the function gives when the input is 0
outer mauve
examine the continuity of a function
this is on my math course and i have 0 clu...
youre still refering to this right?
outer mauve
the output that the function gives when the input is 0
not sure to what i am supposed to input the zero
safe jasper
you can read it off the graph
outer mauve
so 0,1
safe jasper
yes, so f(0) = 1
what about the limit as x -> 0?
do the left and right hand limits exist and equal the same thing?
safe jasper
i'm referring to this example
you're looking at the graph? what do the output values approach as you get closer to x = 0 from the left side? what about the right?
if it helps, try visually tracing along the function graph
as you get closer to x = 0
outer mauve
we are only referring to x axisright?
safe jasper
well the limit is going to be what the function outputs approach as your input values approach x = 0
so the limits will involve the actual outputs of the function, but as your inputs get closer to 0 on the x-axis
outer mauve
so it would be =-1 on the left and 3 on the right?
safe jasper
so it would be =-1 on the left and 3 on the right?
i think what you're doing is looking at when the function outputs 0 or hits the x-axis, but we're concerned about what the function is doing when the inputs are close to 0
safe jasper
that's what these arrows are considering
as you consider inputs that move closer to x = 0 from the left, the blue arrow representing the function outputs tends to 1
on your own time btw, it might be helpful to watch a few youtube videos on limits to get caught up if you're still not understanding
this website will tell you everything you need to know, with explanations and examples:
https://tutorial.math.lamar.edu/classes/calci/limitsintro.aspx
outer mauve
on your own time btw, it might be helpful to watch a few youtube videos on limit...
i have watched a lot of yt videos and i have already learned more from our conversation
safe jasper
that's good
outer mauve
so just to make sure if i want to figure out the continuity there isnt any formula i can use to figure it out right?
i need to do f(0) and f(1)
in this case
0 would be the left -
1 would be the right +
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devout snowBOT
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white bane
if u have 31.999
how woulud u round it to the nearest hundreth
my brain is not working rn
devout snowBOT
if u have 31.999
how woulud u round it to the nearest hundreth
my brain is not w...
@white bane Has your question been resolved?
weary arch
if u have 31.999
how woulud u round it to the nearest hundreth
my brain is not w...
if it's <50 it rounds to 0, >=50 rounds to 100? (for a number in [0,100])
oh wait lmao
32.00?
same logic, <.995 rounds to .99 and >=.995 rounds to 1.00
devout snowBOT
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https://www.youtube.com/watch?v=74cOUSKXMz0&pp=ygUUSnVzdGluIHN0dWR5IHdpdGggbWU%3...
warm nymph
.close
devout snowBOT
cold haven
⁉️
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lapis obsidian
can anyone help me solve the variance for T-30P1 in part c
devout snowBOT
can anyone help me solve the variance for T-30P1 in part c
pseudo basin
but the answer i got was approx 36.8
3680% is not a valid probability
so something definitely went wrong somewhere
lapis obsidian
what?
lapis obsidian
can anyone help me solve the variance for T-30P1 in part c
no i mean, the variance
i got
was approx 36.8
pseudo basin
if you mean variance then say variance
lapis obsidian
and i dunnow how the answer got 33.6697
lapis obsidian
if you mean variance then say variance
mb ://
pseudo basin
can you show YOUR work for how you got your variance
lapis obsidian
so I used two variables
first one is T = 5C + 28S + P1
this variance i got was 0.7681
then second variable was F = T - 30P1
the variance I got was 36.7681
var (T - 30P1) = Var(T) + 30^2Var(P1)
lapis obsidian
wdym?
pseudo basin
C1+C2+...+C5 (take 5 cement bags and add their weights) doesn't follow the same distribution as 5C (take one cement bag and multiply its weight by 5)
their variances are also not the same
(even though the means are)
woven radishBOT
Ann
pseudo basin
do you understand this yes/no
lapis obsidian
ye
pseudo basin
$T < 30P_1 + 190$ rewrites as $$\underbrace{-29P_1 + C_1 + \dots + C_5 + S_1 + \dots + S_{28}}_{R :=} < 190$$
woven radishBOT
Ann
pseudo basin
do you understand this also
pseudo basin
ok
now two important properties of variance
when X and Y are independent, we have Var(X+Y) = Var(X) + Var(Y)
when X is a random variable and c is a constant, we have Var(cX) = c^2 Var(X)
do you remember and understand both of these
lapis obsidian
when X is a random variable and c is a constant, we have Var(cX) = **c^2** Var(X...
but i thought it was this?
pseudo basin
again
lapis obsidian
OH, I get it now
pseudo basin
$5C$ is \textbf{not the same} as $C_1+\dots+C_5$
woven radishBOT
Ann
lapis obsidian
right
so basically
because its 5 different samples of C, so its 5 seperate variables?
so its only times 5 not squared?
pseudo basin
5 independent samples of C
so additivity kicks in and you get Var(C) + Var(C) + Var(C) + Var(C) + Var(C)
lapis obsidian
btw, would it be possible to ask another question? (not stat)
pseudo basin
a.k.a. 5 Var(C)
lapis obsidian
so additivity kicks in and you get Var(C) + Var(C) + Var(C) + Var(C) + Var(C)
gotcha
pseudo basin
btw, would it be possible to ask another question? (not stat)
!1q
devout snowBOT
btw, would it be possible to ask another question? (not stat)
Please stick to one question at a time to help others follow the conversation better and prevent confusion/crossed answers. Once it has been answered, you may send your next question.
lapis obsidian
sure
so basically i finished part A, and got the coordinates of P n Q
but then i don't know how to form the equation for the area of the triangle for part B
Q(0, -16/3 * sintheta)
P(given)
lapis obsidian
think you need to make a new channel for that?
pseudo basin
can anyone help me with a question?
!occupied
devout snowBOT
can anyone help me with a question?
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).
pseudo basin
so basically i finished part A, and got the coordinates of P n Q
it would help to make a sketch here
lapis obsidian
I tried that too, but I still couldn't figure out how to solve it
might be kinda slow, but I don't really see it
pseudo basin
the area of triangle OMP is exactly half that of triangle OPQ
triangle OPQ might be easier to find the area of
lapis obsidian
the area of triangle OMP is exactly half that of triangle OPQ
I don't get it, why tho 🙁
or is OM perpendicular to PQ? (haven't solved it yet)
pseudo basin
OM is not known perp to PQ
but PQ = 2*PM as i hope you can see
(M is the midpt of PQ)
pseudo basin
if you draw the height from O in triangles OPM and OPQ, wherever it might land, it'll be the same for both triangles
yes
devout snowBOT
buoyant pawn
so basically i finished part A, and got the coordinates of P n Q
Is this precalc?
pseudo basin
in american terms it would probably be calculus
actually
(a) Using calculus, show that ...
buoyant pawn
They used to ask us this in elementary algebra
education is messed up where I live (Somalia)
some of the courses are way too advanced for us, while some are severely lagging
short hare
Is this precalc?
precalc is usually not calc. it includes general limits, algebra and trig
pseudo basin
They used to ask us this in elementary algebra
this is definitely not "elementary algebra"
"messed up" sounds about right
devout snowBOT
Available help channel!
rough nova
How do I even approach this question 😭
devout snowBOT
How do I even approach this question 😭
rough nova
The x^x ≡ 1 (mod p) part is diabolical
winter torrent
i don't have a clue but i would start by trying it out for small p
immediately i notice that N(p) always includes 1
also i think you can just pretend c = 1/2
well no not quite
yes quite, primes can never be squares
devout snowBOT
@rough nova Has your question been resolved?
rough nova
i don't have a clue but i would start by trying it out for small p
What does this yield btw?
winter torrent
not sure, i hadn't done it
rough nova
The problem is to show that for sufficiently large primes p, N(p) has an upper bound that is a root of p (that's how i grasp it)
winter torrent
yes you're showing that N(p) eventually stays below p^1/2
winter torrent
it feels to me like that doesn't matter but maybe it does
at any rate, oh, let's look at p-1
that is, (p-1) ^ (p-1)
when does that equal 1?
rough nova
When p=1?
winter torrent
p can't equal 1
winter torrent
in mod p ofc
winter torrent
hmm okay
it may be helpful to make like a table or something of each p and its values of x
to be clear i don't know the answer to this or whether that will actually help
rough nova
It's okay, i appreciate any ideas
rough nova
it may be helpful to make like a table or something of each p and its values of ...
x^x grows so fast, it's quite hard to calculate N(p)
rough nova
I'm assuming there aren't number theory theorems about x^x
lament kraken
the usual nt theorems didnt work?
rough nova
What's your idea?
lament kraken
this is an interesting problem
express x as g^k
g is some primitive root of p
hm
crude wasp
How do I even approach this question 😭
what's the source?
rough nova
what's the source?
A Vietnamese Math Olympiad facebook group
rough nova
yes you're showing that N(p) eventually stays below p^1/2
Yea i think you're right
polar chasm
i think i can bound it by sqrt(p)
but i have no idea how to get better than that, which is required
feral agate
Isn’t bounding by sqrt(p) the whole problem
polar chasm
not really, even if you show that it's < p^(1/2), it doesnt prove that there is a c < 1/2 which also bounds it
it would have to allow c <= 1/2
rough nova
i think i can bound it by sqrt(p)
Please show
rough nova
Isn’t bounding by sqrt(p) the whole problem
Well you need N(p)<√p, not ≤
polar chasm
in general, x^n = 1 (mod p) iff ord(x) | n
lament kraken
it would have to allow c <= 1/2
arbitarily close to 1/2
crude wasp
How do I even approach this question 😭
not a proof but a heuristic
lament kraken
/j
crude wasp
there are phi(d) elements of order d mod p
x^x = 1 mod p iff the order of x mod p divides x
so we should expect this to happen with probability 1/d
assuming the elements of order d mod p are equidistributed mod d
this gives us a heuristic of [\sum_{d \mid p-1} \frac{\phi(d)}{d}] for $N(p)$
woven radishBOT
LY
crude wasp
which can be checked to be small since it's multiplicative
rough nova
What's multiplicative? Phi?
crude wasp
What's multiplicative? Phi?
the heuristic
since it's the sum over divisors of phi(d)/d which is multiplicative
rough nova
I thought I knew English until "heuristic" and "equidistributed"
crude wasp
I thought I knew English until "heuristic" and "equidistributed"
heuristic just means like guess/estimate
what i mean by equidistributed is that the phi(d) elements of order d mod p (between 1 and p) are like roughly equally distributed mod d
polar chasm
The n.o. sols of order d is bounded by phi(d) (literally just the number of integers of order d) and also p / d (number of multiples of d, as x must be a multiple of d)
crude wasp
what i mean by equidistributed is that the phi(d) elements of order d mod p (bet...
basically i just mean that i'm assuming the number of elements of order d mod p divisible by d is roughly phi(d)/d
polar chasm
The n.o. sols of order d is bounded by phi(d) (literally just the number of inte...
phi(d) is also bounded by d
so its bounded by both d and p/d
min of those 2 will never exceed sqrt(p)
rough nova
What is d? Sorry i'm not following well
polar chasm
now comes the part where i screwed up and idk if it can be fixed
polar chasm
What is d? Sorry i'm not following well
d is order, im counting sols per order of x
polar chasm
im basically grouping all the solution x by their order d and counting them by those groups
rough nova
Mod p?
polar chasm
min of those 2 will never exceed sqrt(p)
we now get a maximum of sqrt(p) * n.o. divisors of p-1 solutions if we sum it over all possible d's
so its actually a bit worse than bound than sqrt(p)
rough nova
n.o. means non-obvious?
polar chasm
number of
rough nova
What is n.o. 
polar chasm
we now get a maximum of sqrt(p) * n.o. divisors of p-1 solutions if we sum it ov...
but yeah, i screwed up and forgot about the n.o. divisors part while i was solving it. This bound is actually quite useless
crude wasp
wait what happens if we try to sum this?
like it should be bdd by [2\sum_{d \mid x, d \leq \sqrt{x}} d] right?
woven radishBOT
LY
polar chasm
yeah
polar chasm
we now get a maximum of sqrt(p) * n.o. divisors of p-1 solutions if we sum it ov...
actually this isnt so bad, n.o. divisors is asymptotically O(n^eps) from what i've found
so we're just very slightly over sqrt
rough nova
I'm learning a lot of new words here 
rough nova
**LY**
From what I understand (which isn't much), we are considering the ord_p of the x's that satisfy x^x ≡ 1 (mod p). And x^x ≡ 1 (mod p) iff ord_p (x) | x.
For each d|p-1 there are ϕ(d) numbers of order d mod p so N(p) ≈ the arithmetic mean of ϕ(d)/d with all d (which is bounded by d)
Is this what you two have been saying, i'm confused
crude wasp
From what I understand (which isn't much), we are considering the ord_p of the x...
so that's what i initially gave as an intuition of why the statement should be true
but we're trying to see how we can actually bound it
mathisalwaysright suggested that we can bound it for each d by phi(d) or p/d but i don't think that bound is strong enough
we can get O(n^1/2+epsilon), and i bet with a bit more work we can also get O(n^1/2) but i don't think we can do O(n^1/2-epsilon)
polar chasm
just found this
apparently there are much better bounds, which isnt too surprising
rough nova
Oh so that's where this is from
polar chasm
but the simplest sol of the problem starts with O(n^(1/2 + eps)) and then improves it to 4/9 + eps
the best known bound is currently 27/82 + eps in the exponent (i.e. slightly better than 1/3 + eps)
rough nova
That link mentions the sum-product theorem, what's that?
rough nova
.close
devout snowBOT
rough nova
.reopen
devout snowBOT
✅ Original question: #help-27 message
devout snowBOT
crude wasp
Yeah too advanced
lmao open book 10 day exam is crazy
rough nova
💀
solid osprey
lmao open book 10 day exam is crazy
for something openbook like this, whats stopping contestants from working with eachother?
devout snowBOT
Available help channel!
night rune
if i am asked for a probability of a lie employee being in fact A,B,C,D , do i only find the max for the blue , or take into consideration the broader bars aswel ?
devout snowBOT
if i am asked for a probability of a lie employee being in fact A,B,C,D , do i o...
night rune
i picked B assuming the dependent probability is suggesting " we dont care about the big bars, narrow it to only the blue ones"
devout snowBOT
@night rune Has your question been resolved?
knotty salmon
so like for this question
I think you did the right answer
because if you think about it, there are 15 in fac A, 20 in fac B, 15 in fac C, and 5 in fac D. so obviously it's fac B
@night rune
glossy dew
solve for x = f(y):\\
$(1+log(x)) \frac{dx}{dy} - \frac{xlog(x)}{y} = 2y$
glossy dew
no clue
woven radishBOT
glossy dew
whoops
pseudo basin
\log
polar chasm
solve for x = f(y):\\\\
$(1+log(x)) \frac{dx}{dy} - \frac{xlog(x)}{y} = 2y$
note that 1 + logx is derivative of xlogx
and there is no other x
there is a sub you can do
hmm, itll require lambert W though
modest shadow
note that 1 + logx is derivative of xlogx
isnt x a function of y?
polar chasm
isnt x a function of y?
||(1 + log(x)) * dx/dy is derivative of xlog(x) wrt y||
devout snowBOT
rare kernel
hmm, itll require lambert W though
lambert W?? 💀
polar chasm
lambert W?? 💀
yeah, the sub will be ||u = xlogx|| and then when you solve it, youll have u = f(y) or ||xlogx = f(y)||, which requires lambert W to be solved
rare kernel
yeah, the sub will be ||u = xlogx|| and then when you solve it, youll have u = f...
oh
maybe he just wanted general soln and not x=f(y)
polar chasm
well, then you dont need lambert w
but it was stated "find x = f(y)", so i assumed its supposed to be expressed like that
rare kernel
oh alr alr
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Available help channel!
nova lake
How do i find several solutions to this recursive equation?
devout snowBOT
How do i find several solutions to this recursive equation?
nova lake
Ok
fair storm
Well, this might be an advance trick, but try transforming the sequence into another more digestible one.
I'll give you an example with the sequence you've given.
fair storm
As a solution?
nova lake
"Its given as a number sequence"
fair storm
Generally yeah, but is there anything other givens?
nova lake
K e R is a constant
fair storm
i have this one
Am I missing something here because I only see "y_{n+1} = a*y_n"
fair storm
That's it?
nova lake
oh yeah
nova lake
Sorry i wrote wrong
I am completely mixing it up
I only have to write several solutions to this, ignore the other parts
And i rewrote it to the other thing in the subtask
fair storm
Ok, I'll give you an example of that technique.
nova lake
Ok
fair storm
$y_{n+1} + \dfrac{1}{2} y_n = 0 \implies 2^{n+1} y_{n+1} + 2^n y_n = 0$
woven radishBOT
Erebus
fair storm
You basically multiplies both sides by 2^(n+1) to make it look similar
nova lake
Ok
fair storm
Here it's easy to see the 2 powers and the terms share a common thing: the term number.
We'll just make a new sequence x_n (you can name it as you like) where x_n = 2^(n) y_n
$x_n = 2^n , y_n \implies x_{n+1} + x_{n} = 0$
woven radishBOT
Erebus
fair storm
Are you processing it clearly?
fair storm
Well, because it fits the nice pattern
It might seem very random in the beginning
but if you work with it for long enough you'll start seeing the patterns
But intuitively, it's because the y_n has a 1/2
nova lake
**Erebus**
and how does this correlate to the first thing you wrote?
fair storm
Well, now the sequence is much simplier
It's just x_{n+1} + x_n = 0
No dealing with 1/2
fair storm
But i dont see why the thing with 2 added is a solution
It's just to make it look better and easier to handle.
I mean you could've solve it like y_n = -2y_{n+1}
and have the answer
nova lake
but how does half become double
fair storm
but how does half become double
It didn't turn double, we just multiplied both sides by 2^{n+1}
nova lake
And we can just do that?
fair storm
And we can just do that?
Yes
But we need to remember that x_n is not what we want
This is the case for making the general formula
It's like walking walking 2 steps forward and 2 steps back
fair storm
Just make a general sequence.
dependent on the first term only
Or the y_0 in your case
nova lake
would this be a solution, if i divide both with 2?
nova lake
I am not sure
fair storm
I just need to clarify if you want
(1) the solution for the y_n NOT in a recursive form
(2) a solution for the sequence (doesn't matter if it's recursive)
(3) No general formula, just some first few terms
fair storm
I am not sure
I see.
nova lake
Like for example, if i am in debt, and the debt doubles every year. After n years i owe y_n money, then i owe the next year y_n+1=2×y_n. If we have s as startdebt, i can find the debt for some years, without finding the first few years with the formula y_n=s×2^n
and the formula y_n=s×2^n is the solution
fair storm
Yes, that's true.
nova lake
This is what i mean by the solution
fair storm
I see.
You want the general formula then. (y_n can be made form the first term directly)
nova lake
yes
fair storm
But I do need to ask you, is it y_{n+1} + 1/2 y_n = 0 or y_{n+1} = 1/2 y_n
nova lake
y_{n+1} + 1/2 yn = 0
fair storm
Ok
nova lake
and i dont have a start value
nova lake
Yeah
fair storm
Then I'll just make a general plan to solve the solution
Make general formula? --> Make sequence look simple
nova lake
I dont get what you mean by general fomrula
nova lake
Aaaah, ok
fair storm
It's general, for all n
nova lake
I need to make a fomula, which involves the start value?
nova lake
Ok, thank you
fair storm
Wait
Let me recap
(1) Make general equation --> Make sequence or the given equation simplier
E.g. y_{n+1} + 1/2 y_n =0 --> x_{n+1} + x_n = 0 (No more x 2)
(2) You can also try manipulating the given equation
E.g. y_{n+1} + 1/2 y_n =0 --> -1/2 y_n = y_{n+1} and continue
nova lake
Would this be correct? Instead of taking the n+1 i added the start value
fair storm
No
Because you're asked to make a formula, specifically with
$y_{n+1} + \dfrac{1}{2} , y_n = 0$
woven radishBOT
Erebus
fair storm
But why did you put x instead of y?
x and y aren't the same thing
x_n is like y_n but you applied some change to it
Not related but like
You might have a sequence
and you add 1 to every term
Then it also becomes an sequence
nova lake
Ok
fair storm
So we're just trying to gang up on a smaller problem instead of fighting the large one
Pardon my english.
nova lake
So we're just trying to gang up on a smaller problem instead of fighting the lar...
Ok
But i dont see why we can just add xn
woven radishBOT
Erebus
nova lake
Ah ok
fair storm
This is like the previous example of adding 1 to every term and getting a new one, but on steriods
fair storm
Are you catching up?
nova lake
I just only caught up with the definition thing
But still not why we can just add 2 to both of them
fair storm
Add 2 to both sides of $y_{n+1} + \dfrac{1}{2} , y_n = 0$?
woven radishBOT
Erebus
nova lake
yea
fair storm
You could, but there's not much use in doing so.
$y_{n+1} + \dfrac{1}{2} , y_n + 2 = 2$
woven radishBOT
Erebus
fair storm
I'm not saying you can't do this, I'm just saying there's no clear plan.
nova lake
Ok
woven radishBOT
Erebus
fair storm
You see how the powers are of twos are muching the terms?
2^{n+1} for y_{n+1}
and 2^n for y_n
nova lake
Ok, thank you
nova lake
Thank you
devout snowBOT
@nova lake Has your question been resolved?
devout snowBOT
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thorny rune
Is this possible? I was reviewing it, and it started to look wrong.
winter patrol
woven radishBOT
ραμOmeganato5
thorny rune
Ah, thanks. Is there anyway to rewrite it or is that the simplest way to do so?
winter torrent
$a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b^1 + a^{n-3}b^2 + \dots + a^1b^{n-2} + b^{n-1})$
woven radishBOT
clumsy
winter torrent
is that better or worse, who's to say
thorny rune
So something like this?
winter torrent
yep that is correct, and you can do sum notation if you want
well
ok that isn't quite correct
this would be correct (you divided the \phi - \psi)
thorny rune
Ah, right,... So like this, and then it cancels out?
winter torrent
yar
thorny rune
How would the sum notation look?
woven radishBOT
clumsy
thorny rune
Why does the ψ not have n on top?
winter torrent
wdym
thorny rune
i understand that φ has n - k because we're counting down (a^n-1... until k), but why is ψ only raised to k?
thorny rune
**clumsy**
like it is sometimes b² here?
winter torrent
have you used sigma notation before?
winter torrent
yes. k ranges from 0 to n
inclusive
so it's $\phi^{n-\blue0}\psi^{\blue0} + \phi^{n-\blue1}\psi^{\blue1} + \dots$
woven radishBOT
clumsy
devout snowBOT
devout snowBOT
Available help channel!
ancient pollen
integrate
tanx * ( root 2+(root(4+cosx))) dx
devout snowBOT
integrate
tanx * ( root 2+(root(4+cosx))) dx
rain summit
$\int \tan x \cdot (\sqrt{2} + \sqrt{4 + \cos x}) dx$
woven radishBOT
1 divided by 0 equals Infinity
woven radishBOT
dexa.cld
rain summit
,rccw
woven radishBOT
ancient pollen
subs made,
y=cosx
y = 4sin^2z
1+cosz=u
devout snowBOT
@ancient pollen Has your question been resolved?
ancient pollen
another subs used for the same integral and got diff answer,
substituitons made,
y=cosx
v= root(4+y)
z= root(2+v)
got this
,rccw
woven radishBOT
ancient pollen
not sure which is correct
frail summit
**dexa.cld**
I found this problem on tiktok and the answer is so long💀
devout snowBOT
@ancient pollen Has your question been resolved?
deep abyss
Using exactly one multiplication, one addition, one division, and one subtraction, what is the smallest positive number you can create using the numbers 4, 5, 6, 7, and 8 once each? For instance, we can form the number ((8÷5) x 7) - (4+6)= 1.2 in this way.
devout snowBOT
Using exactly one multiplication, one addition, one division, and one subtractio...
deep abyss
Is hit nd trial the only way?
pseudo basin
i dont think theres anything more sophisticated, but dividing by a big number seems like a good option
full trellis
Get the biggest number possible in denominator and smallest possible in numerator
pseudo basin
,calc 4/((8+6)*7 - 5)
woven radishBOT
Result:
0.043010752688172
proud perch
,calc 5/((8+6)*7 - 4)
woven radishBOT
Result:
0.053191489361702
I also don't hv the answer key tho
calm blade
,calc 4/((8+7)*6 - 5)
woven radishBOT
Result:
0.047058823529412
Nice!
proud perch
Parantheses
woven radishBOT
Result:
0.0096153846153846
That's quite small
calm blade
That's quite small
Indeed
fossil locust
oh yeah I bet that's the smallest
you can't make any smaller difference between any two numbers, or by combining numbers then subtracting
cause 1 is the smallest positive integer
and then you want to make the denominator be as large as possible, so that's that
calm blade
-# cheri cheri lady going through a motion
deep abyss
Result:
```
0.0096153846153846
```
So this has to be the ans?
deep abyss
-# cheri cheri lady going through a motion
full trellis
I guess so. There's no other way to get a smaller (positive) numerator or a larger denominator given the constraints as far as I'm aware
calm blade
<:ratgrenade:1455710803113283801>
-# hehe
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idle coral
Hello, when I analyse whether or not a double limit exists, if I find a directional limit that does not exist, can I say that the double limit does not exist?
devout snowBOT
Hello, when I analyse whether or not a double limit exists, if I find a directio...
idle coral
polar chasm
Hello, when I analyse whether or not a double limit exists, if I find a directio...
Yes, if from some direction the limit doesnt exist, then the full limit doesnt exist either
idle coral
Along the path y = x diverges, so can I say that it does not exist?
idle coral
Yes, if from some direction the limit doesnt exist, then the full limit doesnt e...
Thanks! 
polar chasm
Along the path y = x diverges, so can I say that it does not exist?
if u allow limits to + inf or -inf, then you still need to keep that option in mind
oh but its gonna be +inf from one side and -inf from other side, so ye, it doesnt exist (considering y = x)
idle coral
if u allow limits to + inf or -inf, then you still need to keep that option in m...
I thought that if the limit was infinite, it did not actually exist.
polar chasm
I thought that if the limit was infinite, it did not actually exist.
depends on convention
people often write lim f(x) = +inf, even if it technically doesnt exist (diverges to inf)
timber pebble
<@&268886789983436800>
stoic lotus
<@&268886789983436800>
idle dune
how is the derivative of x^2 + y^2 giving 2xdx
young spade
thats under the supposition that y is in reality y(x)
notice. $\dv x y^2 = 2y \cdot y'$ stands for any case in reality since you can always apply chain rule.
young spade
this is also true for every other constant btw
Also, notice, in this particular case, once you fully solve the antiderivative youll get your x^2 + y^2 term back.
devout snowBOT
@idle dune Has your question been resolved?
dapper fable
Define $\lfloor x \rfloor$ as the greatest integer not exceeding $x$. Prove that $\$ (a) $$\tau (1) + \tau (2) + \cdots + \tau (n) = \sum_{i=1}^n \lfloor \frac{n}{i} \rfloor $$ Where $\tau (k)$ is the number of divisors of $k$ $\$ (b) $$\sigma (1) + \sigma (2) + \cdots + \sigma (n) = \sum_{i=1}^n i \lfloor \frac{n}{i} \rfloor$$ Where $\sigma (k)$ is the sum of the divisors of $k$.
woven radishBOT
Copter
devout snowBOT
Define $\lfloor x \rfloor$ as the greatest integer not exceeding $x$. Prove that...
dapper fable
i need help on both of these ;-;
dapper fable
is it counting pairs of integers (i,j) ij ≤ n?
stoic lotus
$$\sum_{k=1}^{n} \tau(k)$$ counts the number of ordered pairs $(i,k)$ such that $1 \le k \le n$ and $i \mid k$
woven radishBOT
a handsome russian dude
dapper fable
hmm
stoic lotus
Now what if you reverse it
dapper fable
wdym?
stoic lotus
Fix i
The integers $k \le n$ divisible by $i$ are
$[i,, 2i,, 3i,, \dots,, \left\lfloor \frac{n}{i} \right\rfloor i]$
woven radishBOT
a handsome russian dude
stoic lotus
There are exactly $\left\lfloor \frac{n}{i} \right\rfloor$ such integers
woven radishBOT
a handsome russian dude
stoic lotus
Summing over i from 1 to n, what do you get?
stoic lotus
Same idea for (b)
devout snowBOT
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Available help channel!
serene swallow
hi ineed a help
devout snowBOT
hi ineed a help
serene swallow
hello any1 there?
sand pumice
send your question
sand pumice
huh go ahead
crude niche
hi ineed a help
!da2a
devout snowBOT
hi ineed a help
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
serene swallow
ya so i am 15 years old in grade 10th and i have exams of 80 marks and since the past 5 exams in stuck in the mark bracket fo 70-72 out of 80 which is around 90-92 percent. Upon analyzing my mistakes i have come to a conclusion that i am loosing marks in non mathematical mistakes(not remembering the formula or how to do the question) but rather i loose marks in the most silliest way such as messing up a simmpe very ismpe probablity q
serene swallow
Asking the actual question right away is more likely to get responses.
Asking "...
id like to apologize for my earlier greetings
sand pumice
practice practice practice
mild anvil
do practice exams
crude niche
ya so i am 15 years old in grade 10th and i have exams of 80 marks and since the...
practice
crude niche
id like to apologize for my earlier greetings
it’s fine
mild anvil
when you do practice exams look at the hardest questions and spend the most time on them
serene swallow
the thing is that i have done 6 sample papers and im still not able to correct my mistakes
like i think i lack mindfulness but that's the only thing i focus on while doing my exam
mild anvil
read the question a bunch
serene swallow
js by reading em?
sand pumice
do you make specific mistakes like calculations
serene swallow
no calculation mistakes like for example yesterday itself i was given 2 coins are being flipped 3 times now 2^3 is 8 i read the question properly yet my stupid ass though they were flipped 2 times and i got it wrong
mild anvil
highlight necessary components if need be