competition-math

[clever galleon]

Wait wdym?

[silver forge]

You solved my previous problem right

[clever galleon]

Yeah

[silver forge]

so would you like another one

[clever galleon]

Oh oh yeah my brain glitched lol

Yeah sure

[silver forge]

ok

[strong latch]

oh damn

[silver forge]

I dont think a lotta ppl solved this

btw i only give questions which i have solved soif u have any q i can explain

[clever galleon]

Hmm I’ll solve it later in the evening cuz I’m about to shower rn

[clever galleon]

Yeah sure I’ll ask if i have any

[silver forge]

okiee enjoy da problem

[neon summit]

nuh uh

[clever galleon]

Thanks :DD

[neon summit]

at least 12

[silver forge]

what

[neon summit]

at least 12

oh it's multiple choice

[silver forge]

r u referring to the number of ppl who solved it or the answer

[neon summit]

hmmm

[neon summit]

I'm like 67.89% sure it's perchance ||C||

I'm prolly stupid tho

hard to say I can't divide by 10 in my head

[silver forge]

Lemme find my answer

[inland totem]

hmm wait i've seen this

is it ||c||?

[neon summit]

I should rly relearn how to do stuff in my head real fast

[silver forge]

yeahs

that was quick plens

[silver forge]

when u have final answertell meh

[neon summit]

I did type it

[silver forge]

also u shud blur ur answers to not spoil it forothers

[inland totem]

ye i've done it before

think wen sent it

[silver forge]

lol cuz i sent wen it

[neon summit]

I was like 67.89% sure now I'm pretty sure it's 100%

mm

[silver forge]

yessirs corrects

how yall mental math so fast

[neon summit]

u got any comp calc problems

[silver forge]

it took me like five minutes

[silver forge]

no

but i can give u a hard one

[neon summit]

larpmaxxing

[neon summit]

== I'm so bad at normal olympiad math I just do this bc I suck

[silver forge]

tis is not intuitively obvious

[neon summit]

I can't load it

[inland totem]

calculus?

[neon summit]

ye it's not loading for me

[silver forge]

@inland totem u shud watch takopis original sin

[silver forge]

refresh

[neon summit]

DON'T

DON'T DO IT

[inland totem]

ok

[silver forge]

u watched it?

[neon summit]

yez

[silver forge]

soooo peak

[inland totem]

uhh i saw one this morning

[silver forge]

but sooooo sad

[inland totem]

it was cool

[neon summit]

I had my review on it

[neon summit]

dude I can't load da problem

[inland totem]

idk if this quals as comp calc

[silver forge]

wait ik how to do ts

[neon summit]

I can't load this either

[silver forge]

just refresh discord

[neon summit]

huh

am on phone

[inland totem]

hmm

[silver forge]

just close and reopen the app

[inland totem]

alr wait i havs smth to do rn

[silver forge]

Your phone gets tired too yk

[silver forge]

ok bye plens

[neon summit]

am I stupid

is it ||B||

it's probably intuitive but I don't know series

[inland totem]

ye ur right

[neon summit]

da hint given in the Q

[neon summit]

cuz like limit k->inf summation n=1 to k n^10/k^11, take the 1/k^11 out and u get like

uh

yeah not a pretty number

[inland totem]

ye u can turn into a riemann sum

[neon summit]

sounds made up

glad I never studied that(severe larp detected)

[inland totem]

hmm

i larp a lot of things

[silver forge]

lol larping in math is the funniest

[neon summit]

27>x>7
consider right triangle sides wherein the hypotenuse would be greater than
so we get x^2>10^2+17^2 and x^2+10^2>17^2 giving us x values of 19-27 and 9 giving us 208 AND it's not in the choices I'm so smart

I'm the goat

what is 17^2 anw

oh it's 289, not 189

289

🤦

[ruby bramble]

I'm building numberthon (numberthon.com) — what’s the best part of competition math for you guys?

[inland totem]

the geometry problems

[neon summit]

larp part

[ruby bramble]

wdym

[amber finch]

The part where you win

[silver forge]

lemme checks

[ruby bramble]

oh, ok

[neon summit]

ok so 289 389 20-27, x^2>189 so x>14 hm

looks weird

[silver forge]

thats not even an answer

[neon summit]

ah hypotenuse larger

[neon summit]

yeah it's 17 squared

[silver forge]

yessir

[inland totem]

i love bashing

[silver forge]

no

[neon summit]

8-13, 20-26
8 9 10 11 12 13 20 21 22 23 24 25 26 should be ||E or 224||

[neon summit]

289 isn't 17 squared?

[inland totem]

split into cases where x is the longest side and then 17 is the longest side

i'm pretty sure it's ||e||

[neon summit]

why can't 10 be the longest side

[inland totem]

bc 17 is alr a side

and 17 > 10

[neon summit]

they never specified euclidean geometry

[inland totem]

[neon summit]

yeah I got the karma I deserved, I was walking bear foot in the dark and stepped on a Lego 🥀

[idle aspen]

Which set

I love lego

[neon summit]

how would I know the set it just hurt like hell

[idle aspen]

Fair

[silver forge]

yes

[silver forge]

yes

[silver forge]

17 squared just isnt correct

[neon summit]

isn't 17 times 17 289

I'm confused

my answer was 208 I was wondering why I was wrong I got the wrong number and did 189-100 not 289-100

[clever galleon]

C?

[silver forge]

lemme check

[clever galleon]

Uhh my working might be hard to read

[silver forge]

spot on

corrects

[clever galleon]

Thanks 😄

[silver forge]

u shud try ts next

even nautilus got confused

[clever galleon]

For reading through that mess

[clever galleon]

Yeah I’m doing it now

[silver forge]

theres like two main equations i gotta look for lol

[clever galleon]

Ohhhh

[unreal spruce]

positive area

😭

area cant be negative anyway

so like

[silver forge]

yeah now im wondering why did they put that

kinda pointless

[unreal spruce]

exactly

[clever galleon]

For specifying it’s..oh yeah right it can’t be non degenerate

Uhh okay i got…from the triangle inequalities it’s 323

Guys I didn’t use the first 2 parts tho…

Anyone wanna read my working?

[silver forge]

ill read

[clever galleon]

Wait I’ll send

[silver forge]

ok thats wrong but ill still read

[inland totem]

||oh ye it said that the final triangle had to be obtuse||

[clever galleon]

It’s wrong?

[inland totem]

uhmm yeah

[clever galleon]

Oh yeah right.

Then I think it’s 161?

[silver forge]

nah

[clever galleon]

Uhh might not be readable. I wrote the last few steps on another page

[clever galleon]

Oh lol I’m totally wrong then

[silver forge]

ok you forgot to consider another case from what i seeing

[clever galleon]

Oh…but…wait lemme think

[rotund venture]

hI

[silver forge]

greetings

[clever galleon]

Oh yeah it’s E

22r

224*

lol i totally forgot about this case

[silver forge]

yess

[clever galleon]

Like my brain warped around “either x is opp acute angle or obtuse” and forgot to consider this

[silver forge]

lols

want another one

[clever galleon]

Sure

But no calc I’m just learning calc

[silver forge]

i havent learned calc

harharhar

[clever galleon]

Neither have i, ATLEAST not in the Olympiad way.. my coaching center has started calc but it’s like the basics

[silver forge]

The last one was kinda annoying

[silver forge]

o cool

[silver forge]

u do math coaching?

[clever galleon]

No JEE coaching. It’s like an engineering entrance in India (yes many Indian parents have a very binary view on careers- either you become a doctor or an engineer)

[silver forge]

lolsss funny

[clever galleon]

No not math specifically, but it’s included in the JEE syllabus, but like i do Olympiad math on my own

[silver forge]

Engineering coaching is a new concept for me

[inland totem]

kinda like my parents

[silver forge]

o nice

[clever galleon]

So like this foundation course is like- super intensive so they’re spoon feeding us calculus with 0 proofs

[silver forge]

xd so pure memorization?

[clever galleon]

Like mine

[clever galleon]

Basically, but i chose the Olympiad cycle, so now i have to do both- the memorisation and aops on my own so that i can give Olympiads

[silver forge]

lol thats good

[clever galleon]

It’s very stressful

To say the least

[silver forge]

hey it will pay off in the future

[clever galleon]

I guess

[silver forge]

btw TRY THESE OUT

all roads lead to one (collatz conjecture)

[clever galleon]

But like sometimes my parents try to make me abandon Olympiads…but I don’t want to so they said I’m gonna have to do both

[silver forge]

https://klipy.com/gifs/tik-tok-meme-4

[clever galleon]

OH DONT REMIND ME

[silver forge]

bruh

[neon summit]

at least 12

[silver forge]

https://klipy.com/gifs/patrick-star-dumb-10

[clever galleon]

As a 10 year old i tried solving it. Gave me mental trauma

[inland totem]

why are we trying to solve the collatz conjecture at 10 years old 😭

[silver forge]

Bru

[silver forge]

lmao

[clever galleon]

Ask 10 year old me who saw the Veritasium video

[inland totem]

oh ye i did too

[clever galleon]

I was a very delusional kid

[inland totem]

but didnt it explicitly say "do not try doing this problem, do some normal math for once" or smth

hmm

[clever galleon]

I genuinely don’t know what 10 year old me was thinking

[silver forge]

lol

[silver forge]

how did you try solving it

what would the first step even be

[clever galleon]

I tried using primes

Like seeing if there was a specific structure on how each of them come back to one

[silver forge]

Youd need statisticsl analysis and a bit of probability theory minimum

[neon summit]

circle rectangle thing, uhh probably ||D||, it's a bit shorter and there's no choice 3950
quarter circle's a nice one
don't think id's possible for the grasshopper to do that

hard to imagine on phone mm

[silver forge]

lol

[clever galleon]

Uhhh I didn’t know the difference between LCM and HCF at that time btw

[clever galleon]

Yeah uhh that’s what happens when ur neurodivergent

[neon summit]

Just a buncha number theory jargon you'll do fine

pfft

[silver forge]

wrong

[silver forge]

LMAO

[silver forge]

oo i see

[silver forge]

the grasshopper uses magic

[clever galleon]

Dude I obviously studied them later but like, yeah that was how versed i was at that time

[neon summit]

mm 10sqrt15~= to like 38.5ish times 50 hmmm

I'm lazy

[neon summit]

the grasshopper is prolly ||E|| or smth

[neon summit]

10 10sqrt5
triangle base is 10sqrt5-10
10sqrt5-20=base of really small triangle, 30-10sqrt 5 is base of the larger shaded so now u can do maht to do 100(14-6sqrt5)+50(9-4sqrt5) which I'm not calculating thx

[clever galleon]

Uhhh i think the answer is ||option d 4000|| but I’m not sure

For the first 3 circles one

[tiny eagle]

I got 3950 cuz u get the radii 5, 15, 25, and then pythag on XZ and some stuff

[clever galleon]

Mine is wrong guys just found out

[tiny eagle]

Will send sol later

[clever galleon]

||using approximation for root 15 as 38.7 (yes i used a calculator) I got this…||

[unreal spruce]

is using a claculator

liek

like*

forbidden or sum

😭

[silver forge]

naur

[silver forge]

no but most ppl dont

mental mathing is good

[unreal spruce]

u can mental math roots?

thats

intersting

smh

[stone canopy]

you don’t need to write out approximate decimal places i assume

so all it is is just dividing a number by primes until you get all primes

[unreal spruce]

oh

i didnt know

u can do that

lowk

[silver forge]

like up to three digits

babylonian method

not mental math

[unreal spruce]

?

never heard of it

lowk

[silver forge]

also i dont usually mental math roots

[unreal spruce]

i am not really into olympiad maths

anyway

[silver forge]

i mean like with addition stuff

[unreal spruce]

so most of this is just gibbrish to me

lol

[silver forge]

oof

[silver forge]

lol

[unreal spruce]

i kinda like doing it tho

like in my free time

anything thats not calc related tho

and trig

related

[silver forge]

yessir

i only let myself do it after i do my textbook work

so it feels like a reward

[unreal spruce]

noice

[inland totem]

<@&268886789983436800>

[ruby bramble]

what do you guys like abt competition math?

(I am asking bc I'm making a competitive math platform called numberthon.com)

[echo ocean]

Cuz there're many fun combi problems

Now get as many combi problem as possible into the platform, people will love it, trust me

[silver forge]

i like how i dont immediately know the solution to them

they challenge me and force me to use new methods that might not be new to the world but is new to me

[thorny tusk]

kewl

[thorny tusk]

the feeling of success yk

[clever galleon]

They asked for closest, and we know it’s less than 4, so like, so basically we know its less than 4 (so less than 4000) but not like very less, so guessing 3950 without finding root is possible. But like there’s actually a long division method to find square root (digit by digit algorithm i think) which you can do in ur head with practice

[ashen hatch]

idk i mean i like that it teaches you different stuffs than regular hs math

[clever galleon]

Uhhh the fact that it values deep thinking over rote speed and memorisation and I’m not good at speed but im better at depth…

[inland totem]

i mean some comps like mathcounts also prioritize speed

[clever galleon]

Yeah but like, the speed in those competitions is developed by pattern recognition and like to recognise the pattern u gotta understand it deeply?

I guess?

[inland totem]

hmm ye fair

[silver forge]

3950 hm

[clever galleon]

Uhhh it’s wrong? I did pythogoras on XZ like someone else messaged…

I first got 4000 but then corrected it

[silver forge]

i mean recognizing 5x12 is 60 doesnt require deep understanding just seeing it a lot engraves it in your head

[clever galleon]

Idk how I got 4000 first

[silver forge]

3950 is correct

*i think

[clever galleon]

Uhhh but like most Olympiad problems in speed based tests usually follow common Olympiad topics and methods

[clever galleon]

Yeah that’s what i got

[silver forge]

Did u do the grasshopper q

[clever galleon]

No not yet i had a test in my class for the past 2 hrs

I just got out of the test (it was online)

[silver forge]

o cool

[silver forge]

yay how did u do

[clever galleon]

It was like moderately difficult, but the results come out tomorrow afternoon after the teachers check the results (like it was McQ but there’s like a check before they publish the results) so yeah

[silver forge]

o math?

[clever galleon]

Math, physics and chem but like, it was mostly testing speed

[silver forge]

o

[neon summit]

sqrt15 and 5 aren't really 'nice'

I can do 2 digits but 3 digits is preferrable cuz of the amount of 10s

[grim brook]

its the key to larp

my friend tries to solve as many things as he can with combinatorics to maximize how sophisticated he looks

[neon summit]

combinatorics can't possibly be that hard there's only 3 formulas

3 formulas everyone refuses to learn and make sense of

[quaint nexus]

Im trying to figure a way to do these without groking it out

I know /varnull is (a + b + c)^2- 3(ab + bc + ac), but not sure that helps

[ivory ember]

Elementary symmetric polynomials my beloved

[quaint nexus]

Never heard of them

[spark scaffold]

54 use pascal triangle

[quaint nexus]

For the coefficients?

[spark scaffold]

Yes

[quaint nexus]

Im aware of the binomial theorem for the coefficient but this implies actually working it out entirely?

[spark scaffold]

That’s what I want to know

I would solve for the LHS

Then to compare

[quaint nexus]

Yeah that’s how I did it; simplify both sides a bit and it’s easy to show

Just not very elegant

Although

For context it’s from an algebra book’s chapter review on polynomials and factoring, so it’s entirely possible it’s just meant to impart a ridiculous amount of computation. Just caught my eye and I was curious

[restive anvil]

3 formulas?

Like perms combs and binomial expansion u mean?

[neon summit]

what're those

(I refused to learn them)

[neon summit]

omg calc

nabla reference

[ruby bramble]

will do

[clever galleon]

I solved 51 in case anyone wanted?

[unreal spruce]

Can u send it?

I'm quite puzzled

Idk how to solve it

[heady summit]

Does anyone have any moderately easy number theory problem? I'm getting kinda bored

[vernal axle]

Prove that 2^n-1 is not divisible by n for any integer n≥2.

[clever galleon]

Yeah sure I’ll send it in some time my notebook’s not with me rn

[unreal spruce]

Via contradiction orr?

[unreal spruce]

Bet

[vernal axle]

Yep

[inland totem]

||newton's sums probably||

[dapper meadow]

Is the -1 in the exponent or not? If it is then it is divisible by n for n = 2

[vernal axle]

Of course it is this

[dapper meadow]

my bad

[clever galleon]

I think this should be the solution to 51

[dapper meadow]

One flipped sign and it’s over

[clever galleon]

Wait it’s wrong?

[dapper meadow]

No no just in general

[clever galleon]

Oh, yeah fr

Lengthy calculations do that

[dapper meadow]

I was planning on splitting it between even and odd but then couldn’t do odd. I’m new to number theory and proofs in general, could you help me out

[vernal axle]

yep, this is kind of obvious, an odd number (2^n-1 is always odd if n>=2) can never be divisible by an even number.

[tiny eagle]

I think u have a good idea but instead of multiplying by n make it so it’s 2^(n-1) -1/n = k

Progressing from there should be easy

[unreal spruce]

i think thats correct

[inland totem]

i did this with ||fermat's little theorem||

[vernal axle]

yep, I used it too

[ruby bramble]

What do you guys think abt this competitive math website that I made?: https://numberthon.com/ Any suggestions on features to add (it aims to be the "chess.com of math")?

[inland totem]

i quite like the ui and design tbh

[ruby bramble]

Thanks!

[quaint nexus]

Yeah so this is how I did it too; I’m just curious if there’s a way to derive the rhs from the lhs or vice versa

Also, for what it’s worth, 52 is pretty easy if you just multiply out the RHS

[quaint nexus]

As far as I’ve tried, there doesn’t seem to be a way to ‘deduce’ the RHS from the LHS in 52. Best I can do is something like this (basically equates to: “oh! Notice this occurs, which I honestly think is BS when including on a problem set, but I guess it makes the authors feel cool)

Also, in my opinion, the only reason anyone would even bother trying to figure this out is because the factored form is already given. I highly doubt anyone would recognize/care for this “in the wild”

There are theorems in abstract algebra regarding symmetric polynomials, I just don’t see what the use of including them here is- especially when no mention or even reference is made to those notions/results elsewhere

[unreal spruce]

Ty!

[grim brook]

weren't lying when you said everyone then

[quaint nexus]

In particular, this is exhibiting an instance of the fundamental theorem of symmetric polynomials (with no mention of even the term symmetric polynomial)

[quaint nexus]

Yeah I mean it’s kind of upsetting to me when stuff like this comes up; like this is a result that is usually shown using galois theory; if you’re going to include these, say something about that so people aren’t rereading the chapter and rechecking their work because they’re under the impression they can’t solve it because of some fundamental misunderstanding

Why are you tagging me?

You seem to think because no one else is responding right now that no one else will respond at all

[tawdry furnace]

@silver forge pls dont make irrelevant posts in topic channels

[shy raptor]

And you can discontinue speaking for a longer time, considering your history...

[ivory ember]

Yeah this is one of those "classic" facts that's normally proven by expansion. Ig you could also say that
$$(x+y)^3=x^3+y^3+3xy(x+y) \implies x^3+y^3=(x+y)^3-3xy(x+y),$$
and so
\begin{align*}
a^3+b^3+c^3-3abc &= [(a+b)^3-3ab(a+b)]+c^3-3abc \
&= [(a+b)^3+c^3]-3ab(a+b+c) \
&= (a+b+c)((a+b)^2-(a+b)c+c^2)-3ab(a+b+c) \
&= (a+b+c)[(a+b)^2-(a+b)c+c^2-3ab] \
&= (a+b+c)(a^2+b^2+c^2-ab-bc-ca).
\end{align*}

You can also inspect that setting $a=-b-c$ forces the expression to be zero, so $a+b+c$ is a factor and then you can do division. (I've never done this, so it probably sucks.) \

Fwiw, this is precisely the determinant of a $3 \times 3$ circulant matrix:
$$\begin{vmatrix} a & b & c \ c & a & b \ b & c & a \end{vmatrix}=a^3+b^3+c^3-3abc,$$
So the factorisation makes it easier to find the eigenvalues. (Note that circulant matrices are common in digitial signal processing.) \

In a sense, you could also use this to prove the identity. Because
$$\begin{vmatrix} a & b & c \ c & a & b \ b & c & a \end{vmatrix}=\begin{vmatrix} a+b+c & b & c \ a+b+c & a & b \ a+b+c & c & a \end{vmatrix}=(a+b+c) \underbrace{\begin{vmatrix} 1 & b & c \ 1 & a & b \ 1 & c & a \end{vmatrix}}_{a^2+b^2+c^2-ab-bc-ca},$$
and so
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).$$

[gilded halo]

Civil Service Pigeon

[ivory ember]

rough first draft, so you could probably try to refine these proofs as an exercise (especially the first one)

[quaint nexus]

Yeah, great points- I missed the determinant equivalence!

I guess I was looking for a way to factor the LHSs to obtain the RHSs, which I don’t think is possible unless you’re aware of decomposing symmetric polynomials in terms of elementary symmetric polynomials- which no one reading a college algebra book to learn from would have any idea about or be able to deduce

[neon summit]

why u spoilering it like it's an answer

[neon summit]

define moderate

[inland totem]

idk it leads to the answer

[neon summit]

oh ye it's just fermat

wait

hmmm

yeah it is I'm prolly just dunv

[ivory ember]

🎁

[gilded halo]

Civil Service Pigeon

[ivory ember]

[neon summit]

this looks like someone made it with hate

243 is 3^5

I know ur in there euler I just don't wanna think about it

||what I'm thinking is that either x or y must have a common factor with 243 so basically just all numbers divisible by 3 and stuff and then the gcd is any factor of 243 prolly||

but I'm prolly approaching wrong

[ivory ember]

yeah ||common factors|| is a relevant idea

[neon summit]

247
hmm

[young halo]

||Like on rearranging we get that gcd(x,y) will divide 243 ( since gcd divides LCM always)
Idk whether this will be a relevant idea
So you can now look into possible gcds||

[sinful flicker]

ts is a fun problem but id be cooked if i had no calculator

is the answer ||2668||?

yay

[ivory ember]

wait where tf did I read ||2666||

I was like "nooooo why are you off by 2 "

[ivory ember]

yeah

[ivory ember]

[sinful flicker]

oh lmao

[ivory ember]

reading is hard

[sinful flicker]

fr

i did it by
||letting 3^a = gcd and 3^b = xy/gcd²+1 and since a + b = 5 i solved for every possibility of x and y|| is there a faster way

[ivory ember]

You can find the sum without finding the actual pairs $(x,y)$ ig

[gilded halo]

Civil Service Pigeon

[ivory ember]

that's really the only way I can think of making this more efficient

[sinful flicker]

how tho

[ivory ember]

Let $\gcd(x,y)=3^a$ and let $x=3^a u$, $y=3^a v$ for $\gcd(u,v)=1$. Then, $\operatorname{lcm}(x,y)=3^a uv$. Hence,
$$3^a uv-3^a=243 \implies 3^a (uv-1)=3^5.$$
So, $uv-1=3^{5-a}$. \

But since $u$, $v$ are coprime, every prime power in the factorisation of $3^{5-a}+1$ must go entirely to $u$ or $v$. This means the number of ordered coprime positive pairs $(u,v)$ is $2^{\omega(3^{5-a}+1)}$. At this point, you can simply run over $a=0$, $1$, $2$, $3$, $4$, $5$ by finding the pairs $(u,v)$ and multiplying them by $3^a$ since
$$x+y=3^a u+3^a v=3^a (u+v).$$

[gilded halo]

Civil Service Pigeon

[sinful flicker]

noo training my reading comprehension

ok ima read

oh wait thats what i did basically i think

cool

what is $\omega$ though

[gilded halo]

aKube

[ivory ember]

https://en.wikipedia.org/wiki/Prime_omega_function

[clever galleon]

Anyone wants a moderate geometry problem I’ve solved?

[sinful flicker]

thats cool, it looks like theres lots of greek lettered number theory related functions

or i think so im not too sure

[ivory ember]

oh yeah there's a lot lol

[ivory ember]

If you want to share it, then I'm sure someone here would take interest

[sinful flicker]

a list of these and all but two are greek

[sinful flicker]

sure

[clever galleon]

I might be busy for some time so I’ll not be replying, but i have solved it and i can help whoever needs help

[ivory ember]

[sinful flicker]

oh god theres more https://en.wikipedia.org/wiki/Arithmetic_function
and it doesnt help that the sum and product symbol are both greek too

number theory might as well have been made in greece atp

[ivory ember]

$$x+\frac{2\Delta}{x}=y+\frac{2\Delta}{y} \implies x-y=2\Delta \left(\frac{x-y}{xy} \right) \implies \Delta=\frac{1}{2}xy.$$
This implies that we have a right triangle with legs $60$, $63$, and so the longest side of the triangle has length $\sqrt{60^2+63^2}=3 \cdot 29=\boxed{87}$.

[gilded halo]

Civil Service Pigeon

[clever galleon]

Correct

[clever galleon]

Dude how’d you do it so fast lol it took me like 20 minutes