-# haha i stole the moment
#help-36
radiant igloo
chilly granite
-# ~~haha i stole the moment~~
-# i feel like i stole the moment from annie š (sorry)
-# did op forget to close?
spare summit
!done?
final saddleBOT
If you are done with this channel, please mark your problem as solved by typing .close
@dusk shoal Has your question been resolved?
grave trout
<@&268886789983436800>
final saddleBOT
cunning ridge
-# sorry for ping
how do you type in that font
chilly granite
how do you type in that font
-# you simply do -# (small text)
and to type big you do # hi
u need spacce-
chilly granite
nppp!!
final saddleBOT
Available help channel!
gritty chasm
I could do the first part fairly well, but I'm stuck on the second.
Attached also is Theor. 1.19
final saddleBOT
I could do the first part fairly well, but I'm stuck on the second.
Attached als...
final saddleBOT
worldly spruce
actually nvm got it, im horribly sorry
Why, you only made the bot do work. Who doesn't get paid anyway so
ĀÆ_(ć)_/ĀÆ
final saddleBOT
Available help channel!
wintry pier
Hello. Iām not sure if this is the correct channel for this type of question, if not please let me know. Iām looking for some advice. I am a Computer Science Major and also working on a minor in Mathematics. Here is a background of the courses I have taken so far:
Calculus I
Calculus II
Linear Algebra
Discrete
Probability & Statistics (taking this over the summer)
I would like to go to grad school to study mathematics and earn a Masters degree. That said I have been feeling uncertain about the courses which I should take to elevate my application. So here are a few questions I have:
- Is it plausible for someone with a CS undergrad to pursue a masters in mathematics? That is to say, would grad schools look at my application?
- If the answer to question 1 is āyesā, the next question becomes, what courses should I take during my undergrad to āelevateā my application and make me stand out. Some people have recommended real analysis, which would require me to take calculus 3 and intro to proofs as pre-requisites.
- Am I looking at this the wrong way overall? What do you guys think?
Thank you for your time.
final saddleBOT
Hello. Iām not sure if this is the correct channel for this type of question, if...
wintry pier
I am probably gonna take calculus 3 regardless by the way, i think thatās probably a must have
loud sundial
Help channels are for specific questions, try #study-discussion, the discussion channels, etc?
wintry pier
thank you guys, i apologize
wintry pier
or <#917213985365983253>
it says no access
ornate token
Need undergrad role.
modest sequoia
it says no access
dont worry, study discussion is a good channel to go to for it
final saddleBOT
quick yew
Hu
final saddleBOT
Available help channel!
shadow marlin
let $u_0 = 4$, $u_1 = u_2 = 0$, $u_3=3$, $u_{n+4}=u_n+u_{n+1}$. show that every prime number $p$ divides $u_p$
final saddleBOT
let $u_0 = 4$, $u_1 = u_2 = 0$, $u_3=3$, $u_{n+4}=u_n+u_{n+1}$. show that every ...
soft zealotBOT
bloubbloub
shadow marlin
no idea where to start
proud jolt
I also have no ideia, but what I would probably try is suppose there is some p such that p doesnt divide u_p. Then take the smalest p such that p does not divide u_p
Now let's see
final saddleBOT
@shadow marlin Has your question been resolved?
proud jolt
Nah
shadow marlin
I'm not sure how you can use any kind of induction since p-4 and p-3 are not very well known
proud jolt
Maybe it's best too look what are primes congruent mod 4
There's a tree like structure that ends at either 4 or 3 in the end
if you write $u_n = u_{n-4} + u_{n-3}$
soft zealotBOT
tales
proud jolt
Then write $u_{n-4} = u_{n-8} + u_{n-7}$ etc
soft zealotBOT
tales
proud jolt
somewhere you'll end up at either 3 or 4
And somehow use some fact about congruence of p mod 4
Does it look like it's promising?
shadow marlin
not sure
shadow marlin
somewhere you'll end up at either 3 or 4
why is that
ok i get it
but I don't see how it could help
proud jolt
Every prime greater than 2 either has p = 1 mod 4 or p = 3 mod 4
So $u_{p - 4n}$ ends up in $u_{1}$ or $u_3$
soft zealotBOT
tales
shadow marlin
I was thinking also maybe we can count the number of u0 and u3 that appear as leaves of the "tree"
soft zealotBOT
tales
soft zealotBOT
tales
proud jolt
Write $p = 4n + r$, were $ r = 1$ or $r = 3$
soft zealotBOT
tales
proud jolt
Assume $r=1$ for simplicity
soft zealotBOT
tales
soft zealotBOT
tales
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
shadow marlin
not really i dont think so
proud jolt
$u_{13} = u_9 + u_{10} = (u_5 + u_6) + (u_6 + u_7) = ( (u_1+u_2) + (u_2 + u_3)) + ((u_2+u_3) + (u_3 + u_4))$
soft zealotBOT
tales
shadow marlin
it would be something like $\sum_{i=0}^n \binom{i+4(n-i)/3}{i}$
soft zealotBOT
bloubbloub
shadow marlin
counting the pairs i, j such that 4i + 3j = 4n
I think you could find an induction formula for this number of the form : x_n = x_{n-4} + x_{n-3}
but I don't think it's useful
I should specify that I am allowed to use abstract-ish algebra. When I asked chatgpt it tried something with an extension of Z/pZ, which I believe was false, but maybe there's something there
proud jolt
I think you could find an induction formula for this number of the form : x\_n =...
Yeah, It would involve powers of irrationals probably
final saddleBOT
@shadow marlin Has your question been resolved?
shadow marlin
Can you show me what he did?
but it's a lot of handwaving which makes my head hurt trying to understand it.
leaden moon
<@&268886789983436800> this guy is everywhere
scarlet sequoia
let $u_0 = 4$, $u_1 = u_2 = 0$, $u_3=3$, $u_{n+4}=u_n+u_{n+1}$. show that every ...
I feel like this can be solved writing u_n general formula out
you have characteristic equation r^4 = r + 1, with 4 different roots r_1,...,r_4
so $u_n = ar_1^n + br_2^n + cr_3^n + dr_4^n$
soft zealotBOT
Rafilouyear2026
scarlet sequoia
wouldn't you know it, initial conditions line up perfectly with a=b=c=d=1
and then remember that (a+b)^p = a^p + b^p mod p
shadow marlin
are the roots in C or an extension of Fp
scarlet sequoia
roots are in C
shadow marlin
and then remember that (a+b)^p = a^p + b^p mod p
so how can I use this
oh ok
uh
wait
scarlet sequoia
so how can I use this
you can work in the ring A =B/pB, where B= Z[r_1,r_2,r_3,r_4]
shadow marlin
i see
final saddleBOT
neat sail
Hi, I am starting with limits and i am js wondering whether this is correct.
Show that lim x->1 (5x-3) = 2
Heres my answer :
We can set x0 as 1 since the value of x is approaching 1. Hence 0<mod(x-1)<delta. Epsilon>mod((f(x)-L)) Therefore Epsilon>mod(5x-5) So : mod(x-1)<Epsilon/5 0<mod(x-1)<Delta Hence delta = epsilon/5.
I know its probably pretty bad but i js need some help refining it
shadow marlin
I know its probably pretty bad but i js need some help refining it
send it in an unoccupied channel if you haven't already, since this is gonna close in a few minutes
neat sail
oh i am so sorry mb
final saddleBOT
Available help channel!
vital crag
use separation of variables 
harsh stratus
we haven't been taught to do that
modest sequoia
what methods have you been taught so far?
modest sequoia
i would recommend learning separation of variables, as this is a nonlinear ODE
since you have that v^2 term
harsh stratus
hold on
hmm
the original question was to set up an equation
that describes the velocity of the object
final saddleBOT
@harsh stratus Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
haughty furnace
Hey I'm like reaaally really bad at math and I get stuck easily, but I also learn fast I need some help with this š„²
final saddleBOT
Hey I'm like reaaally really bad at math and I get stuck easily, but I also lear...
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
Well I did number 2 and that one I understand, but in 3 I don't get what Im supposed to do, like With the f(-3) and f(2), how do I use them?
patent compass
Well I did number 2 and that one I understand, but in 3 I don't get what Im supp...
Can I ask a off topic ques?
blissful meadow
Well I did number 2 and that one I understand, but in 3 I don't get what Im supp...
You're meant to find those on the graph.
The curve on the graph gives you pairs of point (x,f(x)).
haughty furnace
go ahead
blissful meadow
So for f(-3), you would look at where x=-3 is and see how high the curve is at this point.
haughty furnace
So for f(-3), you would look at where x=-3 is and see how high the curve is at t...
Ok I see that now I think, in the X axis before the -4 where it touches would be the -3?
I that what it means? š
Is*
haughty furnace
Do yk Ariis the Phonk song maker?
I have heard a song or two by him haha
blissful meadow
haughty furnace
OHH
blissful meadow
Your goal is to find how high that point is on the curve.
Which you can infer from the gridlines
blissful meadow
Yes
haughty furnace
like the 2 on the y axis?
haughty furnace
OHHH
blissful meadow
What about f(2)?
blissful meadow
Yes
haughty furnace
YAAAY
and 3b is the same?
well like searching the y value? if that makes sense
blissful meadow
Well now you're given the second coordinate essentially.
So you can do the reverse : start at y=4 on the axis, then move along that horizontal line to see where the curve is at this level.
blissful meadow
Yes
haughty furnace
so when y=-4
blissful meadow
y=4
blissful meadow
Yes
haughty furnace
okay this is so helpfull
tysm
so I think I undersand what to do in number 3
can I try and then I send my answers? to see if they are correct?
okaay im getting lost again
Im using like a base 7= a(1)+b as a base at first
idk if that is right?
haughty furnace
like -5= a(3) +b?
blissful meadow
yes
haughty furnace
cause I just stoped with 7-b=a
Oh I think I know what to do next
then I have b= -5/3 + a? or is that wrong
blissful meadow
You have 7 = a+b and and -5 = 3a + b.
blissful meadow
You may use b = 7-a and plug that for b in the other equation. Then solve for a.
blissful meadow
With b=7-a, you get -5 = 3a +(7-a).
blissful meadow
No. It's the same as -5 = 3a + 7 - a
spark agate
The question says
āFind a possible formula for the quadratic function described.ā
A) Graph has vertex (-2,4) and y-intercept -8
final saddleBOT
The question says
āFind a possible formula for the quadratic function described...
stable dew
ok so do you know basics of quadratics?
spark agate
I know I have to use the vertex formula so i got to
y=a(x+2)^2+4
But iām confused because the answer key shows that i need to get rid of the x why??
This is the answer key given to us
I just wanna understand it :,3
stable dew
ok I got no idea why they did tht
gritty chasm
This is the answer key given to us
they just substitued x = 0 and y = -8
remember that the y-intercept is -8
so the graph cuts the y-axis at (0,-8)
that's what they substituted
spark agate
Ohhhh OKAY!
Are they using ax^2+bx+c for the final formula or?
Is it just vertex cuz iām confused
gritty chasm
This is the answer key given to us
I mean they just actually substituted x = 0 and y = -8 here
they did'nt use any formula; they just used the fact that the graph cuts the y-axis at (0,-8)
as to how they got that expression, thats via vertex form I feel
spark agate
Okok that makes sense! Ty ty
gritty chasm
!done
final saddleBOT
If you are done with this channel, please mark your problem as solved by typing .close
spark agate
.close
final saddleBOT
final saddleBOT
Available help channel!
warm python
Prove simpson's 1/3 rule
Let $I_{[c,d]}$ be the integral of a function on $[c,d]$.
\begin{proof}
Let $f : [a,b] \to \mathbb{R}$. We now chose to partition $[a,b]$ using $n$ points. Thus each partition is of length $\frac{b-a}{n}$. Let the partition be ${x_0,\dots,x_{n-1},x_n}$, where $x_0=a,x_n=b$.
Then on each interval $I_{[x_{i-1},x_{i+1}]}= f(x_{i-1})+4f(x_{i})+f(x_{i+1})$.
We note that $i$ must be even in order to obtain intervals, such that , adjacent intervals' intersection only contains one common point.
We then have $I_{[a,b]}= \sum_{i=1}^{ n/2} f(x_{i-1})+4f(x_{i})+f(x_{i+1})$ where we assume $n$ to be even for simplicity.
Note $$I_{[a,b]}= \sum_{i=1}^{ n/2} f(x_{i-1})+4f(x_{i})+f(x_{i+1}) = f(x_{0})+ 4f(x_1)+ 2f(x_2)+ \dots+ f(x_n)$$ . We then get $I_{[a,b]}= \sum_{i=0}^{ n/2} = f_{x_{0}} + 4 \sum_{i=1}^{n/2} f(x_{i})+ 2 \sum_{j=1}^{n/2-1}f(x_j)+f(x_{n})$.
\end{proof}
final saddleBOT
Prove simpson's 1/3 rule
Let $I_{[c,d]}$ be the integral of a function on $[c,d...
soft zealotBOT
what a wonderful world(wai)
warm python
ykw, can I have this checked for now
If $n$ were odd replace n/2 with it's ceiling/2
soft zealotBOT
what a wonderful world(wai)
rugged merlin
where is the missing }
you are like
missing the width here
there be a delta x/3 term multiplying all those functions
also
simpson's rule requires n to be even
warm python
there be a delta x/3 term multiplying all those functions
oops
rugged merlin
you cant just ceil it
warm python
right yes just noticed that bit
warm python
you cant just ceil it
got it
rugged merlin
but also
warm python
other than that it;s fine?
rugged merlin
you need to explicitly show why the integral over a single subinterval equals that sum
warm python
I suppose I should explain where the factor of two comes from
warm python
you need to explicitly show *why* the integral over a single subinterval equals ...
as in simpson's formula for one sub interval
warm python
That would be doable using langarnge interpolation too I suppose
floral nova
I'm not sure what you are even trying to prove here.
This neither derives the formula for the parabolic caps nor does it show that this scheme converges to the integral
warm python
This neither derives the formula for the parabolic caps nor does it show that th...
the composite 1/3 rule
I missed that in the title š
floral nova
I don't see how that addresses my comment
warm python
I don't see how that addresses my comment
I meant to prove the composite 1/3 rule
loud sundial
**what a wonderful world(wai)**
.pin
final saddleBOT
**what a wonderful world(wai)**
warm python
I mistyped the heading
marsh gorge
please ping me
final saddleBOT
please ping me
warm python
!status
final saddleBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
final tangle
@marsh gorge what have you tried
marsh gorge
nothing idk anything
final tangle
do you know how to calculate the area of a sector
marsh gorge
nope
final tangle
do you know how to calculate the area of a circle
marsh gorge
i mightve forgot but it was pi and r^2 i think
soft zealotBOT
ĻĪ±Ī¼Omeganato5
marsh gorge
yep good i remmeber something for once
marsh gorge
do you know how to calculate the area of a sector
is there a formula?
final tangle
Yes
marsh gorge
what is it
final tangle
area of a sector would be a fraction of the full area,
specifically the ratio of the angle of the sector and a full revolution
marsh gorge
is there no A=......
final tangle
Wdym
chilly granite
is there no A=......
-# yes there is a direct formula but rn omega is explaining where the formula comes from so u can derive it urself i belive
marsh gorge
like for example for an area of a triangle we use a^2+b^2=c^2
final tangle
A of sector = fraction of the circle * area of full circle
radiant igloo
like for example for an area of a triangle we use a^2+b^2=c^2
fahhh
marsh gorge
so fraction/pi*r^2?
final tangle
no
final tangle
Writing that mathematically gives you $$A_{\text{sector}} = \frac{\theta^c}{2\pi} \cdot \pi r^2$$
marsh gorge
ok
soft zealotBOT
ĻĪ±Ī¼Omeganato5
marsh gorge
what does c mean
final tangle
where $\theta^c$ is the angle in radians
soft zealotBOT
ĻĪ±Ī¼Omeganato5
marsh gorge
ok
final tangle
just notation
marsh gorge
but what if i had degree would it be same formula
marsh gorge
so same formula?
chilly granite
so same formula?
-# it would be /360 instead but essential same idea
final tangle
same idea, different units
marsh gorge
-# it would be /360 instead but essential same idea
ohh ok thanks
so is that all i need to know?
final tangle
no, you also need to know how to calculate the area of a SAS triangle
marsh gorge
ik how to
final tangle
ok, then you should be able to finish the problem
marsh gorge
thank u imma try and show u if thats ok
final tangle
note that i didn't simplify the formula above
that would be an exercise for you to do
and you can check with the formula you find online / in your book
marsh gorge
note that i didn't simplify the formula above
that would be an exercise for you ...
uhh what?
final tangle
don't round too early,
also which part of what you replied to don't you understand
final tangle
you don't have to
but it's less things to write down in the future if you do
marsh gorge
well did i do it right?
final tangle
and that's what will come up when you look up area of a sector
final tangle
yes
marsh gorge
@final tangle
please tell me i cooked
also ping me so i can see ur message, while im doing other work
final saddleBOT
marsh gorge
have a good day sir š
final saddleBOT
Available help channel!
mossy chasm
BC=13, BD=10, CE=15. Every segment is an integer length. Find the perimeter of ABC.
final saddleBOT
BC=13, BD=10, CE=15. Every segment is an integer length. Find the perimeter of A...
mossy chasm
im not sure, gives menelaus and ceva vibes but im not seeing anything from it
woven ledge
do the integer segments also include bp,pd ...
mossy chasm
yes
final saddleBOT
@mossy chasm Has your question been resolved?
bronze grove
im not sure, gives menelaus and ceva vibes but im not seeing anything from it
menelaus fosho
calm storm
BC=13, BD=10, CE=15. Every segment is an integer length. Find the perimeter of A...
what have you tried?
mossy chasm
ive tried to do menelaus and ceva which im currently learning rn on abd and ace, but i dont really get anything useful
bronze grove
ive tried to do menelaus and ceva which im currently learning rn on abd and ace,...
can u show ur wokring
mossy chasm
i got AB/EB x EP/CP x CD/AC=1 and AE/BE x BP/DP x DC/AC = 1
no idea what to manipulate to get anything meaningful so im just stuck
i cant think of any other thing to use to solve this
bronze grove
oh wait nvm cevas
@mossy chasm dude
i think since it said integars
u just gotta try hit and trial
like split the ratios as they integars
final saddleBOT
final saddleBOT
Available help channel!
hard blaze
for scientific notation lets say i wanted to write 10^8 in decimals where do i start counting from? Like do i start counting from 1. and then go 8 digits to the right or start counting from 10. and go 8 digits to the right? Same thing for negative values from where do i start counting from if i wanna write it in decimals.
final saddleBOT
for scientific notation lets say i wanted to write 10^8 in decimals where do i s...
vital crag
What negative values exactly
gaunt bison
Think about scientific notation being the decimal point moving
If you multiply by 10, it moves one space the left
And the other way round
hard blaze
What negative values exactly
negative scientific notations like 10^-8, i think im using the wrong word to describe it
hard blaze
find 10^-2, 10^-3, 10^-4
I can answer those by memory but the thing is when it comes to bigger numbers I don't know where i should start counting from and I usually end up writing an extra 0 or two.
vital crag
find 10^-2, 10^-3, 10^-4
.
hard blaze
find 10^-2, 10^-3, 10^-4
10^-2 = 0.001
10^-3=0.0001
10^-4=0.00001
i feel like thats wrong tho
soft zealotBOT
Result:
0.01
where does the decimal point start
vital crag
what
vital crag
Result:
```
0.01
```
you can see this right?
hard blaze
cause for 10^-2 i know u have to shift the decimal point 2 to the left but I don't know from where I should start counting the decimal points.
hard blaze
the `.` in 0.1 is the decimal point
no but before you calculate it
I'm wording this very poorly i know
i hope u can understand my point
vital crag
find 10^-2, 10^-3, 10^-4
you're just doing this right now. you haven't even found a pattern yet
vital crag
i hope u can understand my point
not at all
hard blaze
you're just doing this right now. you haven't even found a pattern yet
I know it increases by an extra 0, every time the power goes up but that's not my issue.
thats the pattern you're talking about
vital crag
can't help if you're not listening ĀÆ_(ć)_/ĀÆ
hard blaze
can't help if you're not listening ĀÆ\_(ć)_/ĀÆ
Okay I think we got off on the wrong foot here, I know I'm not making a whole lotta sense to you but I'm having trouble understanding you as well. So can we try again?
.close
final saddleBOT
final saddleBOT
Available help channel!
final saddleBOT
polar spruce
<@&268886789983436800> another one of these goobers
final saddleBOT
final saddleBOT
Available help channel!
fathom meteor
Find the point on y=x^3-3x^2+x+3 that is closest to the point (1,5).
fathom meteor
Round to 3 decimal places as necessary in the final result.
drowsy epoch
What have you tried so far?
fathom meteor
Looking for a solution with minimal calculator work
fathom meteor
What have you tried so far?
Don't know how to approach these types of problems, just made one up and sent it here for that reason
drowsy epoch
Well what I think would be to either minimize the distance formula or intuitvely i'd say it's either the distance from the local max to the point or the around y=5
fathom meteor
Hold on I think I have an idea now
So minimize sqrt((x-1)^2+(y-5)^2) on the curve
I'll change that to (x-1)^2+(y-5)^2 for simplicity
drowsy epoch
yes
fathom meteor
Local minima occur where the derivative is zero
drowsy epoch
fathom meteor
I'll change that to (x-1)^2+(y-5)^2 for simplicity
I guess find the derivative with respect to x of this formula?
drowsy epoch
Local minima occur where the derivative is zero
yea
drowsy epoch
I guess find the derivative with respect to x of this formula?
yes
fathom meteor
I'll change that to (x-1)^2+(y-5)^2 for simplicity
So i can use 2(x-1)+2(y-5)(dy/dx)=0
dy/dx=(1-x)/(y-5) correct?
drowsy epoch
yes
drowsy epoch
This will makes sense if you plot the distance func
https://www.desmos.com/calculator/v4izsreno4
fathom meteor
Oh and then find the derivative with respect to x of the cubic
drowsy epoch
yes
fathom meteor
dy/dx=3x^2-6x+1 and then set that =(1-x)/(y-5)
3x^2-6x+1=(1-x)/(y-5)
I'm on the right track still?
drowsy epoch
you'd end up with some quintic polynomial
fathom meteor
And then set this equal to the RHS of the original cubic I suppose?
drowsy epoch
yes
final saddleBOT
final saddleBOT
Available help channel!
spark agate
Iām so confused on completing the square š hereās my work and the answer key
final saddleBOT
Iām so confused on completing the square š hereās my work and the answer key
spark agate
Can someone explain it to me? I feel like so many yt videos have different methods and rn I feel like Iām just copying what Iām seeing instead of understanding
drowsy epoch
it looks good
spark agate
But I donāt really understand it, again I feel like Iām just copying things I see yk?
drowsy epoch
\prpl You can rewrite now
[ x^2-frac{10}{3}x+frac{25}{9}=x^2-2 cdot frac{5}{3}x+lp frac{5}{3}rp^2 ]
soft zealotBOT
drowsy epoch
then apply aĀ²-2ab+bĀ²=(a-b)Ā²
spark agate
.close
final saddleBOT
drowsy epoch
ornate token
final saddleBOT
Available help channel!
golden birch
final saddleBOT
golden birch
I assume the theorem should have said k[x_1,...,x_r] ... degree <= r-1
right?
yes
.close
final saddleBOT
sacred ginkgo
k
final saddleBOT
Available help channel!
meager hamlet
how is it -ln(100-B)
final saddleBOT
how is it -ln(100-B)
echo sphinx
..
cunning ridge
echo sphinx
I m actually class 3 student
icy current
Ok, cool, did that solve your question?
cunning ridge
ideally supposed to use u right
echo sphinx
MY HEAD IS SPINNING AFTER SEEING THAT
cunning ridge
no clue
meager hamlet
yea i think i see
u can only do ln(x..) when u have 1/x or 1/u
thats why u need u sub here?
cunning ridge
thats why u need u sub here?
yeah
cunning ridge
I've got this one problem in math
theres plenty of available help channels
ornate knoll
theres plenty of available help channels
Where?
final saddleBOT
final saddleBOT
Available help channel!
thin cloud
The cells of an 8x8 board are filled with the numbers from 1 to 64, with 1 number in each cell, such that any 2 numbers with a difference of 1 are located in orthogonally adjacent cells (sharing a common edge, and the other way around doesn't have to be true ). Compute the maximum sum of numbers on the diagonal
final saddleBOT
The cells of an 8x8 board are filled with the numbers from 1 to 64, with 1 numbe...
subtle ginkgo
like 456
thin cloud
what
thin cloud
idk what to do tbh
subtle ginkgo
not even where to start?
thin cloud
we are constructing a hamilton path that's all I know
with each cell is a vertex
-# ping me if anyone has an idea
subtle ginkgo
-# ping me if anyone has an idea
my bad had to do something
anyways
this is just a chessboard
32 white 32 black
any two numbers with a difference of 1 (consecutive numbers) must be in orthogonally adjacent cells
in a chessboard coloring, orthogonally adjacent cells always have opposite colors
this means that if 1 is on a white cell, 2 must be on a black cell, 3 on a white cell, and so on
All odd numbers
$\{1, 3, 5, \dots, 63}$ are on one color.\
All even numbers
$\{2, 4, 6, \dots, 64}$ are on the other color.
soft zealotBOT
thecrumbeler2
thin cloud
okay?
subtle ginkgo
gng let me set up the problem before you okay me
in an 8x8 board, the cells on a main diagonal all have the same color
since there are 8 cells on the diagonal, and they all share the same color, the numbers occupying these cells must either all be odd or all be even
therefore, in order to maximize the sum, we want to pick the 8 largest numbers available for a single color
there are 2 cases
all evens or all odds
The even numbers are ${2, 4, 6, \dots, 64}$. The 8 largest even numbers are: 50, 52, 54, 56, 58, 60, 62, 64
soft zealotBOT
thecrumbeler2
subtle ginkgo
The odd numbers are ${1, 3, 5, \dots, 63}$. The 8 largest odd numbers are: $49, 51, 53, 55, 57, 59, 61, 63$
soft zealotBOT
thecrumbeler2
thin cloud
Okay
subtle ginkgo
when u add them all up
thin cloud
no
subtle ginkgo
For any 8x8 board, we can construct a path that visits the diagonal cells at specific times
thin cloud
subtle ginkgo
<@&268886789983436800>
thin cloud
no that doesn't seem obtainable
subtle ginkgo
give me your reason for why
thin cloud
say if we go from 1 to 64, because any 2 consecutive numbers have to be orthogonally adjacent, this means any 1->64 would be a hamilton path
say this diagonal, it divide the board into 2 parts, if you fill the diagonal then you will me stuck in one the lower part or upper part. so you would have to fill one one part first before you fill up the diagonal
Let say you fill one part before you fill a single cell in the diagonal, then the highest you can go for the lowest cell in the diagonal is (64-8)/2
or 28
which is in neither cases you stated
final saddleBOT
@thin cloud Has your question been resolved?
subtle ginkgo
this is a much harder question than i originally thought it was
each triangle has 12 white + 16 black cells, not 14+14...
the diagonal is all white
the board has 32W/32B total, so off diagonal whites = 24, which split as 12 per triangle. The other 16 cells per triangle are black
any path confined to one triangle has at most 25 cells
running out of whites
if we let t1 < t2 < āÆ < t8 be the times the path visits the diagonal
before t1 the path is stuck in one triangle cause it cant switch without crossing so t1 - 1 <= 25 meaning t1 <= 26
between t1 and t2 its still stuck so t2 - t1 - 1 <= 25 so t2 <= 52
for i >= 3 consecutive diagonal cells arent orthogonally adjacent so t_i+1 -t_i >= 2
so best possible is 26+52+54+56+58+60+62+64 = 432ā
final saddleBOT
final saddleBOT
Available help channel!
warm python
final saddleBOT
tough bloom
hello
warm python
I don't get where the Ļ'(x) comes from
tough bloom
u could use some diffrentition
final saddleBOT
radiant igloo
finally something i can understand š
final saddleBOT
Available help channel!
low flint
Anyone know a video or source for good calculation shortcut for long algebra or anything that is used in competitive exam like jee
final saddleBOT
Anyone know a video or source for good calculation shortcut for long algebra or ...
low flint
Also need help in am gm hm and rms inequality
grave trout
Also need help in am gm hm and rms inequality
What specifically
grave trout
Anyone know a video or source for good calculation shortcut for long algebra or ...
I would recommend practicing. From my experience, I always forget multiplication tricks. Maybe write hard questions on a paper and test yourself to get it faster
low flint
What specifically
For jee competitive exam for calculus(not started) algebra(quadratic sands trigo and trigonfun )
Conic section
low flint
I would recommend practicing. From my experience, I always forget multiplication...
Here whole syllabus
grave trout
If you know all the formulas , the best thing to do is to practice questions. I haven't written the JEE, but upon googling it seems like an Indian Engineering Exam
low flint
If you know all the formulas , the best thing to do is to practice questions. I ...
Ya
Bro i have practiced many questions but evey 10 to 15 questions u have to ise some trick to save time
As we have to do questions in 2 min or less
If u known a source can u tell me
grave trout
One thing that you can try, is solve a question, look at a video online about the solution, and try solve the question 3 days later. It sometimes helps me to remember tricks to solving certain questions. Also, I would recommend for any exam to solve question papers from previous years
low flint
One thing that you can try, is solve a question, look at a video online about th...
?
Can u help me solve que 2
,rccw
soft zealotBOT
upper mural
det is determinant 
sly forum
how does one do that really is the question?
low flint
Use some logic u have 2 min to do it
grave trout
So first multiply the matrices
Then with the products you can find the determinant
low flint
It can't be calculative
grave trout
Use some logic u have 2 min to do it
It doens take that long to multiply 2 matrices
low flint
Bro if i got similar questions in exam it would take atleat 4 to 5 min to solve it carefully and solve 4*4 matrix
grave trout
Just multiply it
final hedge
do you know rank of a matrix
low flint
It has something to do with rank
upper mural
Bro if i got similar questions in exam it would take atleat 4 to 5 min to solve ...
||won't it be a 3x3 matrix i havent done matrices in a while||
oh nvm it's 43 34
mb
low flint
||won't it be a 3x3 matrix i havent done matrices in a while||
Its A x B sonit would be 4x 4 matrix
sly forum
||won't it be a 3x3 matrix i havent done matrices in a while||
it is 4 x3 multiplied by 3 by 4 = 4x4
look at the ends
ornate token
So first multiply the matrices
Sounds tedious.
final hedge
you arent supposed to multiply
grave trout
Sounds tedious.
Rlly? If i got that question in an exam I would immediately multiply
sly forum
wait is not easier to first find determinant of each first
ornate token
Rlly? If i got that question in an exam I would immediately multiply
If they are 2x2.
final hedge
because jee is like a race against time kind of exam
ornate token
Rank AB is always less than or equal to min(RankA,rankB)
sly forum
det(AB) = det(A) det(B) ?
final hedge
so every q HAS to be done in 1.5 mins
grave trout
because jee is like a race against time kind of exam
Rlly? I don't know much about that exam...
low flint
How to get rank of matrix i haven't learn it
final hedge
Rlly? I don't know much about that exam...
you get approx 1.5mins per q
grave trout
you get approx 1.5mins per q
Damn
sly forum
det(AB) = det(A) det(B) ?
that sounds less tedious anyways
ornate token
Rank AB is always less than or equal to min(RankA,rankB)
^
low flint
det(AB) = det(A) det(B) ?
Its not square matric
ornate token
Well first, look at A.
final hedge
how you do this is, rank of AB will not exceed 3
ornate token
rank A = number of columns.
ornate token
No sols btw.
low flint
But why can rank exceed it
upper mural
isnt JEE mcq based asw
final hedge
No sols btw.
he had asked the same q in the morning
ornate token
But you should appeal to the fact that AB is singular.
low flint
isnt JEE mcq based asw
It has integrr 5 too and 20 mcq
ornate token
But you should appeal to the fact that AB is singular.
^
low flint
so he knows the soln
Our sir told us take we are not able to understand as grade 11 th students
final hedge
Our sir told us take we are not able to understand as grade 11 th students
our sir had taken this kind of stuff
so i know
low flint
our sir had taken this kind of stuff
In which class u learned it and how to find it btw
final hedge
In which class u learned it and how to find it btw
its a local
not big
it wont be in jee tho
its just for info
because rank isnt in the jee cirriculum
if im correct
low flint
But how to know if a thing will not exceed that rank or not
low flint
studying linear algebra
I saw whole series
warm python
or you row reduce the matrix
final hedge
But how to know if a thing will not exceed that rank or not
that question isnt from jee syllabus
ornate token
Gauss elimination?
final hedge
so you dont need to sweat it
warm python
into its RREF form
warm python
Gauss elimination?
yea
low flint
My whole mind became graph
ornate token
Again, as Velt said, it is quite tedious.
Since you are expected to spend 1 minute per question.
warm python
it is yea, but that's how you find the rank
ornate token
Well rank = number of columns.
low flint
it is yea, but that's how you find the rank
Rank of line 1 plane 2 volume 3
warm python
Well rank = number of columns.
well, linearly independent
final saddleBOT
final hedge
no
cunning ridge
Btw u gave jee?
hes in same class w me
low flint
My poor life and spending money in allen still can't make this questions
final hedge
kota?
low flint
Nah gujarat
final hedge
o
final hedge
My poor life and spending money in allen still can't make this questions
i mean
practice
ig
final saddleBOT
Available help channel!
mossy chasm
given a triangle with two cevians of different vertices which split the triangle into 3 sub triangles and a quadrilateral, is it possible to determine the area of the quadrilateral in terms of the area of the 3 sub triangles
final saddleBOT
given a triangle with two cevians of different vertices which split the triangle...
brazen breach
given a triangle with two cevians of different vertices which split the triangle...
I assume yes it is possible
mossy chasm
how so
mossy chasm
ok
brazen breach
I said I assume 
brazen breach
I assume yes it is possible
I can't remember but I'm like 99% sure there's a partly related geom theorem to this that MIGHT not be exactly what I'm remembering
mossy chasm
I can't remember but I'm like 99% sure there's a partly related geom theorem to ...
ceva?
idk
final saddleBOT
@mossy chasm Has your question been resolved?
chilly granite
given a triangle with two cevians of different vertices which split the triangle...
is there a diagram?
final saddleBOT
@mossy chasm Has your question been resolved?
final saddleBOT
@mossy chasm Has your question been resolved?
cerulean epoch
yes and here is how would i convince you without writing the entire formula down
scale the entire triangle down such that |ā³AFB| = 1/2. then perform an affine transformation T such that F lands at (0, 0), A lands at (0, 1), and B lands at (1, 0). notice that this transformation preserves the area of ā³AFB, hence preserves all areas. then the area of ā³EAF uniquely determines the line EA, and likewise with ā³DBF and the line DB. so the coordinates of point C can be written down in terms of these areas, and you can use heron's formula to find |CDFE|.
final saddleBOT
final saddleBOT
Available help channel!
tired iron
In how many ways 5 boys and 4 girls be seated ona. Bench such that the girls and boys occupy alternate seats
final saddleBOT
In how many ways 5 boys and 4 girls be seated ona. Bench such that the girls and...
ornate token
Wait, what, please re-phrase?
runic needle
BGBGBGBGB
tired iron
Wait, what, please re-phrase?
Thats what the book says
runic needle
let me leave that to someone else
honest gust
In how many ways 5 boys and 4 girls be seated ona. Bench such that the girls and...
do by cases
hasty ridge
š
ornate token
Chudcel you mean brute force?
hasty ridge
reordering - n!
runic needle
Good answer but we want to know *how* to achieve that.
i only said that to help u understand the question
honest gust
Chudcel you mean **brute force**?
yes but only to make it clearer
ornate token
Ah.
hasty ridge
now put that together
honest gust
you shouldn't brute force the entire problem when you can do combinations for the other half
ornate token
Bald guy from whiplash sorry for misunderstanding your sentence.
runic needle
np
ornate token
Yeah brute force is much tedious.
hasty ridge
brute force š lock in gang
sterile marsh
5! Ć 4!
honest gust
ok i just realized that like
i don't think the order of the boys and girls matter here
BOII
whole breach
so you guys just couldnt make it in computer science huh
hasty ridge
yes it does
ornate token
so you guys just couldnt make it in computer science huh
What?
tired iron
Gang why the whole srvr here
hasty ridge
Gang why the whole srvr here
struggling with a pre statistics question btw
tired iron
Is this topic fun
sterile marsh
Only possible arrangement is
B G B G B G B G B
Can't start with girl coz that will need 5 girls
So, potential placing of boys = 5!
For girls = 4!
Total possibilities =5! Ć4!
bleak granite
Is that because we got like 5 benches for 5 boys
there are 5 spots to put the first boy, 4 to put the second, etc... hence 5! possibilities for boys, similar logic gives 4! possibilities for girls
ornate token
!redir
final saddleBOT
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
tired iron
there are 5 spots to put the first boy, 4 to put the second, etc... hence 5! pos...
Oh alr got it
tired iron
struggling with a pre statistics question btw
Idk Npr and ncr jst hard for me
Ty guys!
.close
final saddleBOT
strange mason
hii
final saddleBOT
Available help channel!
shell plover
These are notes my teacher gave for the LCT test. It says to see if An converges/diverges then Bn must converge, but isn't rhe whole point of the test to see if Bn follows An?
final saddleBOT
These are notes my teacher gave for the LCT test. It says to see if An converges...
blissful meadow
,rccw
soft zealotBOT
shell plover
,rccw
ty
so i asked some1 and they said if the limit comp of them is > 0 then An follows Bn
but if its smaller than 0 or 0 then An doesn't follow Bn so it does the opposite
final saddleBOT
@shell plover Has your question been resolved?
loud sundial
so i asked some1 and they said if the limit comp of them is > 0 then An follows ...
This is bullshit.
First of all, the test requires positive terms, so the limit can't be less than zero
If the limit is a finite number greater than zero, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.
soft zealotBOT
Civil Service Pigeon
loud sundial
If the limit of $\frac{a_n}{b_n}$ is zero, then $\sum b_n$ converges $\implies$ $\sum a_n$ converges, but the test is inconclusive if $\sum b_n$ diverges.
soft zealotBOT
Civil Service Pigeon
loud sundial
Which is (hopefully) what the image says
looking up the statements of tests/theorems online is also a thing
shell plover
thank you
shell plover
looking up the statements of tests/theorems online is also a thing
and sorry :(
loud sundial
looking up the statements of tests/theorems online is also a thing
the tldr is that the limit comparison test is supposed to determine the behaviour of an unknown series by comparing it to an unknown series ($\sum a_n$ and $\sum b_n$, respectively)
soft zealotBOT
Civil Service Pigeon
loud sundial
I think that's everything I have to say
final saddleBOT
loud sundial
alr ty yeah that makes sense
final saddleBOT
Available help channel!
echo rune
I have been experimenting with the quadratic dynamical system
[
N = 2k+11,
\qquad
U_0 = k+1,
\qquad
U_{n+1} \equiv U_n^2-k \pmod N.
]
We say that (N) is captured at iteration (n) if
[
U_n^2 \equiv k \pmod N.
]
Equivalently,
[
U_{n+1} \equiv 0 \pmod N.
]
If the orbit enters a cycle without ever reaching this condition,
I call (N) non-captured.
Empirically, I observe the following:
For captured composite integers (N), their divisors (both prime and composite) also seem to be captured.
Example patterns suggest a possible hereditary property under divisibility.
My question is:
If an iterate satisfies
[
U_m \equiv 0 \pmod N,
]
does it follow trivially from CRT/modular projection that every divisor (d \mid N) must also be captured under the induced system modulo (d)?
Or is there some subtlety due to the dependence
[
N = 2k+11
]
and therefore the parameter (k) changing with the modulus?
final saddleBOT
I have been experimenting with the quadratic dynamical system
\[
N = 2k+11,
\qq...
soft zealotBOT
mojogi.
final saddleBOT
@echo rune Has your question been resolved?
bleak granite
of course it follows trivially:
[a\equiv b\pmod{N}\implies a\equiv b\pmod{d},\forall d\mid N]
soft zealotBOT
Ī Ī±Ļ³Ī±Ī¼Ī±ĪĪ±Ī¼Ī±ĪĪ»Ī±Ī¼Ī±
sinful oar
final saddleBOT
sinful oar
nvm ill just ai ts
vital crag
nvm ill just ai ts
!done
final saddleBOT
nvm ill just ai ts
If you are done with this channel, please mark your problem as solved by typing .close
sinful oar
!done
final saddleBOT
If you are done with this channel, please mark your problem as solved by typing .close
sinful oar
.close
final saddleBOT
slate tiger
so in a cartasian product
final saddleBOT
so in a cartasian product
slate tiger
does it have to be two non zero sets?
or i can take one null set and one non zero set
nd basically have the non zero set back
like AXB=A where B is a null set while A is a non zero set
loud sundial
so in a cartasian product
I'm assuming you're using "null set" to mean an empty set
slate tiger
I'm assuming you're using "null set" to mean an empty set
yea yea
loud sundial
in which case the cartesian product of any set $A$ and an empty set $B$ is always the empty set
soft zealotBOT
Civil Service Pigeon
loud sundial
since the empty set has no elements to begin with, meaning no elements can be formed
loud sundial
no?
That's what I just said
$A \times B$ will always be the empty set if $B$ is the empty set
soft zealotBOT
Civil Service Pigeon
final saddleBOT
final saddleBOT
Available help channel!
raw patrol
final saddleBOT
final saddleBOT
@raw patrol Has your question been resolved?
brave pivot
This feels like Physics š„
Isn't this just SHM just extended.....
ornate token
I got up to the red line
So at which part are you stuck actually?
Half way of B, or at C?
raw patrol
Half way of B, or at C?
Half way of B
raw patrol
This feels like Physics š„
Isn't this just SHM just extended.....
Err not really itās pure maths
I donāt take further mechanics
sly forum
$$T = \frac {2 \pi} {\sqrt{\frac{7}{5}}}$$
soft zealotBOT
InterGalactic
final saddleBOT
ornate token
Well you need to find the distance in 6 seconds?
If I am not wrong.
@raw patrol reopen the channel please?
final saddleBOT
Available help channel!
raw patrol
Hi
raw patrol
<@1075979065208619069> reopen the channel please?
Yes I have
topaz kite
raw patrol
Well you need to find the distance in 6 seconds?
Yeah thatās the goal
ornate token
Thanks Chiaki!
Okay, let me take a look at that problem.
Have you found the arc length?
ornate token
Recall that the arc length = $r\cdot \theta$.
soft zealotBOT
Mercury (ć¤ćć©ć)
raw patrol
I know itās r x theta
Yep
ornate token
Distance= 7 * ( ....) is doing i think
I am asking OP.
Then what is the arc length?
sly forum
I am asking OP.
-# sorry
raw patrol
Idk what theta is
ornate token
Idk what theta is
Suppose let angle just be theta, then what is the length?
In term of Theta.
raw patrol
7 x theta
ornate token
Now, what's the period?
raw patrol
**InterGalactic**
This
ornate token
I am just going to copy the image here again, to see it easier.
Then recall this, the pendulum travels 4 amplitudes.
In 1 period.
ornate token
Amplitude in terms of arc length will be?
raw patrol
4 x 7 x pi/15
ornate token
Also, the remaining time is?
raw patrol
6-5.31 =0.69
ornate token
Now, at t = 6, can you find displacement?
raw patrol
The angle is pi/15cos(root35/5 x 0.69)
raw patrol
But it says the original angle is not theta = 0
So im not sure what the displacement is
ornate token
At t = 6, $\theta(6) = \frac{\pi}{15}\cos(6\sqrt{1.4})$ rad.
soft zealotBOT
Mercury (ć¤ćć©ć)
ornate token
Right?
raw patrol
Yep
ornate token
7 x theta
But you knew this already: Which is the length of the arc.
Then arc position when t = 6 is?
raw patrol
**Mercury (ć¤ćć©ć)**
7 times this
raw patrol
Yeah
ornate token
Then the pendulum has travelled one full cycle and swinging back to its centre.
So it would be distance in 1 period + (amplitude in terms of arc length - position when t = 6).
raw patrol
I donāt understand
ornate token
Well, what did you get for amplitude in terms of arc length?
ornate token
4 x 7 x pi/15
This, right?
raw patrol
Yes
ornate token
Now, position when t = 6?
raw patrol
**Mercury (ć¤ćć©ć)**
Itās here
raw patrol
pi/15 x 4 x 7
ornate token
Correct.
Now, we knew that at t = 6, the pendulum travelled for a cycle, and going back to the centre again, right?
raw patrol
Yes
ornate token
So you want to take the full cycle length.
Add to the extra time we travel.
Because each cycle only 5. something seconds, 6 seconds is exceeding that.
So it must travelling some extra distances.
raw patrol
Yes
ornate token
So you got it now?
raw patrol
I think so
So we have the distance travelled in one period + the extra distance
= pi/15 x 4 x 7
The extra distance is
Umm
ornate token
Well, at t = 6, we found out the arc is somewhere around 0.9?
raw patrol
Itās the angular displacement from the original angle
ornate token
The arc position is somewhere around 0.9 at t = 6.
raw patrol
When time is 0 the angle is pi/15
raw patrol
The arc position is somewhere around 0.9 at t = 6.
Yes
ornate token
The arc position is somewhere around 0.9 at t = 6.
And this is measure from the equilibrium.
Or (the centre).
raw patrol
Ohhh
ornate token
But at the start of the second cycle, the height was like at 1.4 something?
raw patrol
Yeah
ornate token
No worries!