competition-math

[severe eagle]

,ti

i gotta sleep i have school tomorrow

[gilded halo]

The current time for craze.cv is 08:39 PM (+08) on Thu, 17/07/2025.

[severe eagle]

if you want to groupsolve you are welcome to, just uhh i am slightly busy

[high goblet]

,ti

[gilded halo]

The current time for lyheart is 12:39 PM (GMT) on Thu, 17/07/2025.

[severe eagle]

hm

Oh yes do you know

The montenegro #2 hard carried

@high goblet this is the British score btw

[high goblet]

yeah i saw

[severe eagle]

dominic yeo compile fast

[high goblet]

that's quite good for the UK

[severe eagle]

he taught me olympiad geom for a while btw

[high goblet]

lmao what

[ornate blade]

hey LY!!

[severe eagle]

yes

[high goblet]

hi!

[severe eagle]

he taught us how to complex bash

[high goblet]

lol how do you know him?

[severe eagle]

nice teacher

[sweet pewter]

hatewatch moment

[severe eagle]

some tuition centre posted him in our lesson

for marketing

[high goblet]

oh lol

yeah dominic runs all the UK olympiad stuff over here

[severe eagle]

pro

[high goblet]

took over from geoff smith who was busy running IMO for the last few years lol

[severe eagle]

pog

singapore has quite a few

there are 3 good schools and each school has one good math oly teacher

the rivalry is insane

of course i orz my own schools teacher

i think he is the best of the 3 in terms of personality

get ghost pinged

[simple tiger]

what

[swift imp]

I remember usa2

[severe eagle]

nothing

https://media.discordapp.net/attachments/409481404913811476/968186215734079528/speed-2.gif

gl

[simple tiger]

ong bro

[severe eagle]

what

[simple tiger]

these random ahh pings

[severe eagle]

its just a whoph whop wweowoeow eowo whopw ehop weow eowhop

[severe eagle]

mbmb

[prisma python]

huh why does this take me to the main imo page?

[high goblet]

website's currently down it seems

[swift imp]

who do u think wins

[high goblet]

surely china right?

i mean US have 0 P6 solves, china have 2 iirc

[swift imp]

how do u know

[high goblet]

live scoreboard, except it seems to be down rn

[warm rock]

team usa is js offbrand team china atp

💀

[swift imp]

no all chinese

tho

they just all asian

[warm rock]

true

i meant it more as a joke tbf

[zinc tundra]

thats like the same with like every olympiad

[urban parcel]

why is amc 10 so hard gng

how can i improve

and lock in

[near hill]

It's a contest; if it was not difficult there wouldn't be any clear winners, just a whole bunch of participants who all get perfect scores.

[torpid cairn]

just focus on solving problem sets and read/rewrite solutions until you understand them

generally you shouldn't need much theory but if you don't know like special right triangles u should focus on that lol

[subtle aspen]

Does your school have a math team?

[odd mica]

how do i start getting to a level to a point i can speedrun papers and get 90+%. (currently getting 60% on FM core pure 🙃🙃)

[subtle aspen]

what is FM?

[odd mica]

further maths

[subtle aspen]

huh

unaware of that one

[odd mica]

what country r u from?

[subtle aspen]

usa

[odd mica]

i presume américa?

got you

so there’s A-levels in UK. the 3 subjects i pick are maths, further maths and physics. Further maths is just maths on steroids. much harder. but i struggle so much with it

and i want to get to a point where im getting 90% at least on papers

[subtle aspen]

nice!

If real life was like DND I only leveled up marh

[odd mica]

but 3D vectors keep bending me over

[subtle aspen]

I only learned about 2d vectors so far

[odd mica]

and proof by induction

it’s horrible

[subtle aspen]

wht grade are you in?

[odd mica]

don’t do 3D, that shi will send you sideways for months

[odd mica]

i’m in year 12 which i THINK is grade 13

[subtle aspen]

already created a 3d replication of Pascals triangle

[odd mica]

well i’m going into the next year in september

[subtle aspen]

oops

[odd mica]

do you know about dot product?

[subtle aspen]

Yeah

[odd mica]

okok that’s used in 3D vectors often

tbf i’d like to know how the content differs from Uk to USA

[subtle aspen]

IDK im in 9th-10th grade rn

[torpid cairn]

I think it's easier in the US but I honestly don't know

[odd mica]

that’s what i’ve heard

but not too sure

[subtle aspen]

I tend to just not pay attention in math class until the day before the quiz where I see if there is anything I don't know

[torpid cairn]

gcse or whatever is definitely harder than the SAT for instance

I'm not sure about class difficulty though

[subtle aspen]

yeah the SAT looks easy

[odd mica]

yeah id agree with that

[subtle aspen]

math part at least

[odd mica]

trust me, A-levels are some different kind of gravy

[subtle aspen]

turtle are u typing a paragraph?

yeah I kind of forgot how to use derivatives a bit

[torpid cairn]

I wasn't typing lol

I tab switched it probably said I was typing

[subtle aspen]

oh

[torpid cairn]

notably nothing over here needs any calculus

[odd mica]

dang really

[torpid cairn]

yeahhh

[odd mica]

lucky smh

[torpid cairn]

hold on can u send a non-calc question

if any exist

I'm curious

[odd mica]

if i’m honest, how do i differentiate the difference between calc and non calc

[torpid cairn]

I haven't taken calc so if ur asking that it probably means they're all too hard for me but

but like

lims/derivatives/integrals

[odd mica]

calculus is integration and differentiation right

i’ll have a look

[torpid cairn]

yeah

[odd mica]

is this considered calc?

[torpid cairn]

maybe but I could solve this one

[odd mica]

there’s another

[torpid cairn]

assuming I'm allowed to use sum of consecutive squares that is

[subtle aspen]

never really knew only knew how derivatives worked, so I couldn't use the integral well

[torpid cairn]

I don't even know vectors ngl im taking precalc next year

[odd mica]

it’s proof by induction, so you treat is as if you know nothing and you prove true for letting n=1, Assume true for n=k and therefore prove true for n=k+1 by subbing into both sides and getting them equal

[torpid cairn]

I know induction i didn't realize it was required here

that makes sense tho

[odd mica]

i quite like that topic as it follows the same steps each question

that’s if it’s a series question tho, if it’s prove that something is divisible by an integer, you follow different steps

[sour light]

guys where do i go to ask for math tips

[subtle aspen]

here

one of the options

cool we both joined today

[torpid cairn]

wym divisible by an integer

can't u just plug it in

[odd mica]

no

[sour light]

okay! uh. so hi guys is it okay if i’m in middle school i’m barely gonna start 8th grade and i’m really struggling in 7th grade math. i got masters on my texas starr test for reading but did not meets on my math. any suggestions?

[odd mica]

you have to prove it

you follow the same steps by subbing in n=1 and showing that’s divisible by 5 for example for both sides. then you assume true for n=k. then for n=k+1 you set it as a function. you minus the f(k) function from f(k+1) and eventually it will be divisible by 5

[subtle aspen]

sure

[torpid cairn]

right this makes sense

[odd mica]

thank god i thought i was waffling

it’s a shame there’s no voice channels here

[torpid cairn]

and is induction required for problems like this? because I would just synthetic divide if it's polynomials or use some other method if possible

[sour light]

okay how can u help

[odd mica]

it tends to state ‘prove by induction that f(‘function’) is divisible by 5 (for example)

[subtle aspen]

with what?

[odd mica]

so then the only way would be by induction

[torpid cairn]

are u going into algebra or prealgebra

[torpid cairn]

rightt

[odd mica]

anyway i’m going to bed it’s 2:25

gn all

[torpid cairn]

gn

[subtle aspen]

lol

gn

[urban parcel]

no

[urban parcel]

bet

[subtle aspen]

If there is a problem you don't understand and it isn't an official test then spend time figuring it out.

otherwise just move on

[torpid cairn]

yeah some of ur problem solving should be focused on solving harder problems, and for the rest of it you should time yourself and treat it like a real competition

[orchid linden]

uhh I mean idk work on what you struggle with and understand concept
Try to figure out reasoning behind stuff(for longer than youre comfortable with), but if you can’t ask gpt

^ and if there’s specific stuff you dunno just send it into help channel or here and someone will explain it

[runic pasture]

Not sure what level of math you are at, but from the rough grade range I’d highly recommend going through AOPS’s Prealgebra book (and by go through I mean actually read the book like a normal book and attempt to solve all the practice questions). This will serve as a very solid foundation for late middle/early high school math

[signal rain]

Hi guys,
Are there any math olympiads for high schoolers that have past papers that I can use for practice, I'm preferably looking for a website with a lot of these olympiad-style questions,
Cheers.

[ornate blade]

Art of Problem Solving has AMC and AIME

[signal rain]

nice

thank you

[subtle aspen]

also you could try Mathiscool

[formal ice]

Anyone else struggling to load aops recently? Super duper slow

Like are we deadass

[severe eagle]

its gone now

[severe eagle]

fair

[fickle ridge]

Im an Italian 9th grader, i want to train for next year olympiads, I have some experience in the Bocconi's circuit but im not that good. Where do i start from?

[prisma python]

oh ok

also where did you get this year's IMO questions?

[severe eagle]

aops

[prisma python]

thanks

[jagged current]

Something happened with their domain

[formal ice]

oh that’s really unfortunate

[jagged current]

Yeah, I think their https expired at some point and they had to renew it

[thin stratus]

guys how old should i be before preparing for maths olympiads, and does anyone have any tips in practicing?

[feral drift]

Can someone help me with this question?-How many ways can you pick three different numbers from 1 to 1999 so that any two numbers in the chosen set are at least 9 apart?

Im getting 71732 and im not sure if this is correct

[feral drift]

Earlier, the better

[thin stratus]

ive done some thinking and i think the answer is 1980lowercase sigma + 1979lowercase sigma ... until its 1 sigma but after that im not sure how to solve it

[pallid tundra]

do lots of problems

also while the most prestigious contests tend to be high school level

if you’re in middle school I’d recommend looking around for contests at that level too

the earlier you start exposing yourself to these contest problems the better

[thin stratus]

my school isnt the greatest for that and im in uk so tutors or mainstream is practically impossible at my fianancial state

ive done lots of uk maths challenges for my age group and i typically score close to full marks

but i dont have insighful methods for harder olympiads at my age group which holds me back and im basically smarter than my maths teachers in highschool now which leads me into my dilemma

[pallid tundra]

have you looked at the contest collections on AoPS? that’ll be a huge wealth of practice material

[subtle aspen]

I doubt this is correct, as you have over 1,000 options for the third number considering the first 2 are the smallest, and another 1,000 for the other 2

so at least 1 billion

[feral drift]

Yeah i misread the question on that one, it should at the "most" 9

[subtle aspen]

ohhh

at mosr

I was thinking it was 19811980/2 + 19801979/2 +... 2*1/2

I forgot Discord doesn't like the * symbol it thinks we are italicizing

[soft vigil]

"olympiad-style"

[subtle aspen]

I got 161589, IDK if it is correct tho

[pallid tundra]

and a lot of national olympiads too

[soft vigil]

yeah

those r better if anything

[swift imp]

@subtle aspen Why u playing jtoh

a jtoh member was on team usa though

[subtle aspen]

FR?

huh

[swift imp]

popop614

[subtle aspen]

I aspire to do the smae

[swift imp]

is in team usa

[subtle aspen]

no way

[swift imp]

yea

the glowup from jtoh to imo

wild

[subtle aspen]

yeah since Aternils took kver it has improved

gtg continue in a bit

[swift imp]

this is prob the first and last time someone from jtoh will ever make it to an intl competition

ngl

[subtle aspen]

do you play JTo?

JToH?

I plan to change that, I play JToH and am great at maths

[orchid linden]

wat is jtoh

[subtle aspen]

have you heard of Tower of Hell?

the Roblox game

[swift imp]

i used to

[subtle aspen]

ah. Whats ur hardest?

[swift imp]

topz

[subtle aspen]

nice. Mine is only ToTL

planning on jumping to ToRER tho

Why did you leave, and why don't you think anyone else from JToH will be on the team?

[swift imp]

well cuz ppl in jtoh tend to not be those kinda ppl

and i quit cuz i have a life

[subtle aspen]

fair, has been a bit since I played

[misty onyx]

I play jsab :D

[subtle aspen]

whas tha

[misty onyx]

Just shapes and beats

[subtle aspen]

oh a rythm game?

[misty onyx]

Yeah

And

Fnf

[subtle aspen]

I used to play FNF

[misty onyx]

They made some nice updates

[subtle aspen]

yeah my sister played

recently

[misty onyx]

Yeah u can play as pico as well

https://tenor.com/view/pico-picosschool-fnf-friday-night-funkin-newgrounds-gif-27611251

[stark swallow]

Can you explain the solution in a simple way i couldn't understand it

[prisma python]

use divisibility rules for 9 and 11 basically

[near hill]

Alternatively: When there's an odd number of digits, reversing them moves each digit by an even number of positions.
Moving a digit d from position m to position n corresponds to adding d·(10^n-10^m).
If n>m, then 10^n-10^m written in base 10 looks like 99...9900...00 with n-m nines and m zeroes; when n-m is even this is obviously a multiple of 99.
If m>n, then subtract d·(10^m-10^n) instead, again a multiple of 99.

[drowsy sluice]

I need help

with math

[near hill]

In other words we want combinations of p and q such that p+q | 100p+q. That is the same as p+q | 99p. But if p and q are coprime, then p is also coprime to p+q, so that means p+q | 99.
We want the largest possible p, which means we ought to look for p+q=99 with q as small as possible.
q=11 doesn't work, we get p=88, not coprime to 11.
q=12 doesn't work, we get p=87, not coprime to 12 (both are divisible by 3).
...

[subtle aspen]

(100p+q)/p+q is an integer basically, each digit is 1-9

[subtle aspen]

how did you get p+q | 99p from p+q | 100p+q?

[near hill]

If 100p+q is a multiple of p+q, then 100p+q-(p+q) is also a multiple of p+q.

[torpid cairn]

what do y'all think of deepmind getting a gold

[subtle aspen]

smart

[ornate blade]

https://mathstodon.xyz/@tao/114881418225852441
so what?

Terence Tao (@[e-mail])

It is tempting to view the capability of current AI technology as a singular quantity: either a given task X is within the ability of current tools, or it is not. However, there is in fact a very wide spread in capability (several orders of magnitude) depending on what resources and assistance gives the tool, and how one reports their results.

One can illustrate this with a human metaphor. I will use the recently concluded International Mathematical Olympiad (IMO) as an example. Here, the format is that each country fields a team of six human contestants (high school students), led by a team leader (often a professional mathematician). Over the course of two days, each contestant is given four and a half hours on each day to solve three difficult mathematical problems, given only pen and paper. No communication between contestants (or with the team leader) during this period is permitted, although the contestants can ask the invigilators for clarification on the wordi…

Mathstodon · Jul 19, 2025, 6:55 PM

[stark swallow]

how did they calculate area of isosceles triangle

[ornate blade]

didn't you ask this exact question before

[stark swallow]

when?

i maybe asked

i am doing this problem sheet again

[ornate blade]

anyways because CEF is isosceles, you have FX = XE

but by similar triangles, say CX = XE

so FE = 2 * CX

area = 1/2 * CX * FE = 1/2 * CX * (2 CX) = CX^2

[pallid tundra]

cut that triangle in half and arrange the pieces to form a square of side length CX

[stark swallow]

What do i have to find in the figure, the area of pentagon or the area of circle

[misty torrent]

the maximum possible area of a circle inside the pentagon, rounded to the nearest integer

[pallid tundra]

aka literally what it says in the problem statement

[subtle aspen]

39

[stark swallow]

U guys are smart

[scarlet root]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[subtle aspen]

Oops. Should I not have done that? Was it correct?

[prisma python]

yeah i think 39 is correct

[near hill]

Certainly not correct -- the correct answer has to end up being pi times some algebraic number.

[prisma python]

you can use trig to find the radius, the rest is pretty simple

[prisma python]

rounded to the nearest integer though

[near hill]

Ah.

Dammit.

Okay, that's what I get too.

[subtle aspen]

oh. I did it algebriacly

with substitution

I guess they overlap, pythagorean theorem and memorizing the special triangles

[severe eagle]

yay i did q1 and q2 of 2012 imo in 1 hour

yayayayayayay

q1 was trivial took me 45 min

got lucky for q2

[subtle sundial]

congrats !

geometry people truly are amazing

[tacit nest]

Yessss

[eternal kelp]

Could someone offer me a hint on how to go about this

based on the question I know there's a maximum value of n after which we can always find a counterexample

so I thought of reciprocals like 1/a 1/b but that didn't lead to much

now I've not really got a clue

[subtle aspen]

wait I'm confused. Shouldn't it be infinity, because the higher n is the lower the right side of the equation is?

[eternal kelp]

that's why I thought of reciprocals

[subtle aspen]

it says integers tho

and positive

[eternal kelp]

a, b, c are simply nonnegative reals though

n is a positive integer

[subtle aspen]

yeah, lets say it is 1,1,1, then n could be literally any integer and the right side would be lower

sqrt(1+sqrt2)≥nth root of1, any root of 1 is one

or is it saying every value of a,b,c

[eternal kelp]

It is

[subtle aspen]

oooooh

I get it, the higher n is when 1≥a,b,c, tte higher the value if the right side is

its all or nothing, if n gets too high, the lowest the right side can go is 8th root, so when n increases, I think it doesn't stop

[subtle aspen]

pretty sure it is infinity, as even with the lowest value on the right (c ≥ b ≥ a), which would be sqrtc 4th root b, and 8th root c, if you had an n higher than 8, it would be smaller (without the multiplying), if we consider multiplying (abc) to a max of cubing the largest number, then n=24 would be the lowest, 24/3=8, and keep one variable, with the lowest being c with 8th root. From then on, the right side would be less than the left

[high goblet]

ayyy nice

[high goblet]

tbf IMO 2012 Q2 is ||easier the less theory you know lol||

||i mean it's just AM-GM, but the results don't seem to suggest it was that easy, so maybe ppl just wasted time trying other approaches lol||

but yeah no gg

[severe eagle]

yes i got lucky

||i was gonna sleep and just wanted some random progress so i used amgm||

💀

[tired crypt]

Anyone in here doing MAT ? UKMT ? BMO1 ?

I have like ~ 3 months before I’m fkd 🥲

[high goblet]

gl for the MAT!

[high goblet]

i mean SMC/BMO1 are just for fun so don't stress over it

[half arrow]

BMO1 and TMUA here

[tired crypt]

I’m trynna get into Oxford i need to stress for everything ngl 😭

[tired crypt]

Nice nice

[tired crypt]

SMC ?

[half arrow]

Yeah

[tired crypt]

Sent friend request we can help each other out !

[pure mauve]

SMC/BMO questions can be asked here, FWIW

I mean, this is literally what this channel is for

[coarse shuttle]

JOIN FTW GAME PLS

[celest estuary]

HIIII im looking for some help regarding a maths olympiad i am participating in and wanted u guys's helpp

[orchid linden]

? What help bru

[pure mauve]

!da2a

[scarlet root]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[eternal kelp]

MAT and hopefully BMO

[vapid stag]

SMC--->BMO1

Unlikely but bmo2 asw

[celest estuary]

basically do you guys have any idea abt ioqm

??

like ioqm is Indian Olympiad Qualifier in Mathematics which is the first stage in reaching IMO

and this is my first time giving that exam soooo i need help with materials and topics and all

[dusky oyster]

just a thought maybe we can split sqrt(x) into sqrt(x)/2 + sqrt(x)/2 so am-gm removes the sqrt

[dusky oyster]

i think by repeated am-gm we get RHS >= a/2 + b/4 + c/8

and then more am-gm gives us a/8 + a/8 + a/8 + a/8 + 4(b/16) + 8(c/32) >= 12 * 2^-16 cbrt(abc)

so then we can do some stuff

[swift imp]

Go look at past contests

And see what it’s like

[wide sequoia]

hey! Same here, if you find anything useful please do let me know

my teachers told me to mainly just focus on 12th grade chapters

that’s all

[subtle sundial]

aops has past ioqm papers

im sure theyll be useful,

[vivid dock]

how is pre college mathematics for ioqm?

[subtle sundial]

i'd say its very much doable compared to other olympiad papers

[celest estuary]

one of my friends (benchmate) has cleared ioqm in the past and he was saying that learn the basics from VOS (vedantu olympiad school) and then solve rmo pyqs because they are the most relevant for ioqm or smthing

[celest estuary]

absolutely can i dm you cuz he sent me some resources

[wide sequoia]

yes please!

[obsidian plank]

Can someone solve me the problem 8x²-y²=7 (x,y non negative integers)

[sleek ivy]

difference of squares

maybe

[still forum]

@obsidian plank 8x**2 = 64 - 7 = 57. So X would be 64 and Y would be 7 (square root of 57)

so X, Y = 64, 57 or 8x, 7

there are decimal points

but it says integers only

[obsidian plank]

No, because both factors become irrationals

[half arrow]

There’s a solution at ||x=y=1||

[still forum]

Anyways I solved P vs NP guys and sent it to clay mathematics institute and CST2025: https://www.academia.edu/130474412/Solution_to_P_vs_NP_by_John_Mathews_Hearle_12271997

pretty scary the peasants don't know this

[subtle sundial]

no way thats real lol

[half arrow]

Orz

[subtle sundial]

i call bs!

[obsidian plank]

Yeah but I need all the solutions

[still forum]

On mine????

[mighty pine]

I call

[subtle sundial]

yes

[still forum]

No it's real

[half arrow]

It’s obviously correct bro

[still forum]

its 200% correct

[sweet pewter]

1 mil dollars!!!!

[subtle sundial]

yeah sorry for doubting the guy

[still forum]

I solved all the other ones too lol

[subtle sundial]

vythan!! the absolute goat!! solving all the millenium questions

[still forum]

I mean they not hard actually

[subtle sundial]

😍

[still forum]

for example poincare

[mighty pine]

piss cannon

[still forum]

a rose ellipse does not match a 3 circle

and yetis closed and smoth

🙂

because when you rotate it 30 degrees the radius is no longer the same

so broke that one too in bout 5 minutes

[subtle sundial]

absolute stud !

[half arrow]

If a^2+b^2=c^2. Then a^2+b^2=E/m

[still forum]

E as in energy?

that's just saying E1+E2 = Etotal

But E = MC**2 is not correct because E = V*C and while the mass contains the multiplied voltage and current x mass there is a F = MA which takes movement into account for energy which is equal to voltage which is force and acceleration which is not necessarily related to current.

movement is actually related to frequency

so a large star will pull in a another star based off it's FREQUENCY not it's voltage or current

In otherwords a tiny UV radiation star will easily dominate the electromagnetic field of a low frequency star like earths sun

[sweet pewter]

<@&268886789983436800> troll

[still forum]

Not a troll, I'm just not a robot.

Unlike you, I invent like einstein, not just read einsteins math and take it for granted.

[somber belfry]

Please keep channel on topic, thanks

[sweet pewter]

generalized Pell's equation

[glossy falcon]

E = gamma mc^2

[sweet pewter]

you should be able to find 2 recursive series

[subtle sundial]

https://cdn.discordapp.com/attachments/1333005447044665366/1396620298182459613/togif.gif

[uneven spire]

who knows the rule

[tribal thistle]

How do you do this

[pallid tundra]

!status

[scarlet root]

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

[trim bramble]

kinda looks like they make that special rectangle in every row

probably something with that

[sweet pewter]

what property does the segment EF have

[stark meteor]

https://discordapp.com/channels/268882317391429632/359052581022203914/1397376598189146122

[torpid cairn]

I don't think a large improvement over public models is really indicative of much at all

there's definitely reasons to be skeptical of this result specifically though -- I saw somewhere that it's worse at non-IMO problems of a similar difficulty

looking for it

[rain edge]

theres another bot??

[stark meteor]

The exact new unreleased OpenAI's and Google's models? Would love source for that. I would update then. Afaik the only currently public results are the IMO gold medal results.

[kindred wraith]

the terence tao analogy is great

[kindred wraith]

more people should read this

[frank valve]

property of chords: the perpendicular bisector always passes through the center
thales' theorem: if two points lie on opposite ends of a diameter and a third point is chosen, the triangle connecting those three points is a right triangle

[frank valve]

I don't like these problems

pattern recognition problems are icky

not real math

[pallid tundra]

<@&268886789983436800>? can’t tell if this is just a scam but either way clearly off topic

[fast atlas]

Where can I find a list of competition inequalities with solutions (that are sorted by difficulty)?

[muted marsh]

Hello, may I get a hint for this problem? Source: problem 319 Putnam & Beyond

I have tried proving it is monotone by induction without success

[sly pawn]

Yo

Anyone have ideas on how to simplify this?

[sly pawn]

Maybe not sorted by difficulty but I have a pdf of inequalities: https://artofproblemsolving.com/articles/files/MildorfInequalities.pdf

[sly pawn]

Make the right triangle by connecting AF and then realize the answer

[sly pawn]

I know the rule

[sly pawn]

Top triangle on the 4 8 3 diagonal, bottom triangle on the 1 6 8 diagonal, top line on the 3 5 7 diagonal, bottom and top line on the 2 5 7 diagonal, side lines on the 2 5 9 diagonal, bottom triangle on the 1 5 9 diagonal

That should cover it

So 9 should be bottom triangle side lines

[prisma python]

best i can think of is making the log to log_3/2(4/sqrt2*2/3) then take out the 2/3 and rationalize the denominator to get log_3/2(2sqrt2)-1

then add 6 at the end to get 5+log_3/2(2sqrt2)

[uneven spire]

I have no idea what u mean

wana vc?

[long bobcat]

here's what i got

[sly pawn]

Number the squares 1-9 like a keypad

thats what I mean

[sly pawn]

hmm

but this is just combining it

I think there is supposed to be a way to remove the log

@unkempt dove

[long bobcat]

i don't think there is one

[sly pawn]

Yeah me neither

[unkempt dove]

The infinite square root would simplify to

X = (√4-1/√3 x )

So now we can square both sides

X²= 4-1/√3 x

X² + 1/√3x - 4 = 0

Now this is a quadratic

Idk

What to do next

Maybe solve the quadratic

But I am getting it wrong

[sly pawn]

what

[sly pawn]

wdym we already did this

We already simplified the infinite sequence

Does the solution want you to remove the log or is it good enough as just one log

oh oh oh

I know the problem

I missed something when I converted it into latex

There is another 1/3sqrt(2) on the outside of the big sqrt

Ok lets do it again then

-3sqrt(2)(18S^2-4) = S

54sqrt(2)S^2+S-12sqrt(2)=0

108S^2+sqrt(2)S-24=0

S=(-sqrt(2)+sqrt(2+96*108)/216

uhh

S = (sqrt(2)(1+sqrt(5185))/216

6+log base 3/2 (1+sqrt(5185) + log base 3/2 (sqrt(2)/216)

smth like that

10370 = 61 * 17 * 2 * 5

wait thats all wrong

18S^2 = 4-S

oops

18S^2+S-4=0

(2S +1)(9S-4)=0

S=4/9

@unkempt dove

The answer was 4

🤦‍♂️

Anyways the important part of this kind of problem is just to know how to find the answer of an infinite series

[royal zodiac]

How is this pre university 😭 what am i getting my self into

[sly pawn]

its pretty simple if you arent dumb like me

[unkempt dove]

Haha it's india

So you shouldn't be 🙀

Surprised

[unkempt dove]

Ye 😊

You're right

[sly pawn]

woah

[unkempt dove]

Thanks for the help

[sly pawn]

this question is very interesting

[unkempt dove]

From which chapter?

[sly pawn]

What is the most efficient way to break a triangle into rectangles is what its asking basically

[sly pawn]

from an imo problem

[unkempt dove]

Imo?

[sly pawn]

olympiad

[unkempt dove]

Ohh alright

If u can send me more problems

Pls do so

[sly pawn]

actually

I think this problem is too easy

[unkempt dove]

Now I've to go to sleep

By gn

Take care

[sly pawn]

bye

[unkempt dove]

God bless you

[unkempt dove]

Ye nothing is hard for you fr

[sly pawn]

for imo p6 its probably too easy

[sly pawn]

I mean in context

like p3 is a monster but this one doesnt seem as bad

[sly pawn]

wait.

this is a lot harder than I thought

because there could be a better setup than diagonal squares

yeah there definitely is because this setup only needs

wait.

its the same?

[royal zodiac]

p6?

[sly pawn]

both require 8?

[royal zodiac]

cambridge?

[sly pawn]

imo p6 2025

Wait a minute

is it always the same no matter the configuration you pick?

maybe

[royal zodiac]

im edexcel, just finished igcse's

[sly pawn]

I think it might always be n-1 * 2 for any odd sized grids

or maybe for any grids in general

no matter what squares you pick

if you use a greedy algorithm

ohhhh I figured out why

because when you pick a square, there is the section above and below, right and left

and if that section is 0, thats why its n-1 * 2 no matter which square you pick

but proving that could be hard still idk

ohhh maybe you could prove it by saying that the above below left right sections are different for every missing square so it has to be n-1 * 2 because there are only 2 0 distance outliers for each direction and they can be the same too

because if you go up down or left right rectangles or big or small it doesnt matter

[foggy python]

hi

[wanton ridge]

<@&268886789983436800> advertising

[verbal jetty]

guys can you tell me if this proof is correct

Basically I’m trying to prove the sum is equal to 2^m times n choose m

[wanton ridge]

I think you got the right idea, but it's not complete. For example, you didn't explain why you chose k out of n instead of m.

Also, in general, you didn't explain the role of n.

[sly pawn]

The first analogy doesn't make much sense

the second one does

Where is your proof of this?

actually this looks really fun to try and prove

Wow

I think I found something useful

Here is something I have proven on the way to the final conclusion

wait a minute

bruh

Just prove this one instead

ok

well

that makes it a lot simpler ig

wait I found a way easier way to prove it

proved it.

[honest ingot]

these are combinations right?

[sly pawn]

yeah

[honest ingot]

ty

[sly pawn]

it was a lot easier than I was thinking

but the problem is difficult to think about

[verbal jetty]

So I’m correct?

[honest ingot]

yes that's the joke in competition math

thinking

[sly pawn]

as soon as I realized you can just shift it up to become the same as some pascal row above it becomes trivial

[sly pawn]

yes it is correct

[verbal jetty]

let’s go

that hasn’t happened in a while 💀

[sly pawn]

if you want the proof for sum k = 0 to m of m choose k = 2^m that is pretty trivial

just use pascals to see that each row doubles the amount of 1s or smth

[sly pawn]

Yes it's a very interesting problem to prove though!

thanks for the problem

[sly pawn]

Can you explain what the sum is supposed to mean in context of a problem?

[torpid cairn]

I'm not sure how formal it is, but this can be proven combinatorically very easily

because the sum represents all possible subsets of a list with m elements

& each element can either be included or discluded

[high goblet]

wtf no??????

it's like one hardest IMO p3/p6s in the last few years

[stark meteor]

Did you find it?

[soft vigil]

This is rage bait

[high goblet]

well that's what i've heard

haven't actually attempted the problem, but like the stats show like really not that many ppl solved it

and like P4 and P5 really weren't that hard this year

[spiral atlas]

can anyone give me a roadmap for exams like imo or jee if someone know as it would be beneficial

[austere nimbus]

Prepare for jee i would suggest

Imo in india is more of a gamble

The team has like 4-5 members, and the syllabus doesnt even allign with jee

Focus on Chem, Physics and Astronomy Olympiad tbh

Zero risk High Reward

[spiral atlas]

@austere nimbus what do you study?

[austere nimbus]

Pcm

I got approx 50 in NSEP in 10th

98.6 in boards

So i g i m a reliable source

[spiral atlas]

class 11 or 12

[austere nimbus]

11😔

This year will probably clear nsec nsep and nsea

[spiral atlas]

1 and 2 chapters are easy

[austere nimbus]

Bro i have done almost full 11th in 10th💀

[spiral atlas]

i am starting 11 in 9th

[austere nimbus]

Im just practcising now and revising

[spiral atlas]

done 2 chapter and trigo

[austere nimbus]

Good move

[austere nimbus]

I would advise you to do quadratic in math aswell

And pnc

[spiral atlas]

i was doing polynomials in maths and starting linear equation in two variable but started sets and logs

that was easier

[austere nimbus]

I dont even consider linear eq in 2 var a chapter bro

Its a pre requisite

Also how much are u getting

In tests?

11th wale tests mai?

[spiral atlas]

me 9th me hu

[scarlet wedge]

Hey guys

[spiral atlas]

hello

[austere nimbus]

Bro pdhne se kuch nhi hota

Tests toh de

11th ke tests de

Kahi test series lag walo

[scarlet wedge]

What is bro studying ?

[austere nimbus]

Ya koi achi coaching join kro

Jo tests le

[spiral atlas]

konsi badhiya he?

solving books right now for practice

[scarlet wedge]

which books

[spiral atlas]

cengage and all

[austere nimbus]

Cengagae for math is good

Btw chem bhi pdhlena

[scarlet wedge]

bruhh cengage is boring asf

[austere nimbus]

Bro waise tumhe yeh sab apne teachers se ouchna chahiye

So they can see ur level and reccommend books

[austere nimbus]

I mean hes a beginner

[spiral atlas]

my teachers says ncert exampler

[austere nimbus]

Krlo phir wohi ig

9th mai hi tih ho

[scarlet wedge]

These books make Math look dull , i would recommend starting with TAACOPS by Paul Zeitz

[austere nimbus]

Woh hone lage toh aur ache level ki krlo

[spiral atlas]

i do rd

[austere nimbus]

Never heard if it so cant comment on it

Though A DAS GUPTA is a gtreat problem book

Jee adv type

[scarlet wedge]

What is your motivation for studying Maths >

[austere nimbus]

Not good for jee

[spiral atlas]

i do that for school lvl

[austere nimbus]

Yes

[scarlet wedge]

Bro why are u pushing JEE onto this guy

[austere nimbus]

Coz he was asking?

[spiral atlas]

yup

[scarlet wedge]

May i ask you why does JEE interest you ?

[spiral atlas]

maths is good at jee lvl

and i find maths fun

[scarlet wedge]

Your fav problem if any

[austere nimbus]

In which class are you in though?

[scarlet wedge]

11th

[austere nimbus]

Nice

[scarlet wedge]

you ?

[austere nimbus]

Same

[scarlet wedge]

What are your interests /hobbies ?

[austere nimbus]

Watching tv

[scarlet wedge]

Anything specific

[austere nimbus]

Nah

I do random stuff

Making chlorine gas in my house

💀

[spiral atlas]

i make crytal m**h

with walter

[austere nimbus]

I am like 90% sure you havent watched breaking bad

[spiral atlas]

i have

[austere nimbus]

Which stream though?

How much are u getting in ur coaching tests?

[spiral atlas]

boards me 98 kese aagaye?anything specific??

[austere nimbus]

Nah

Just books pdhli

Ncert

10th is not tough

Neither is 11th

[spiral atlas]

teacher to bolte 10 faadh dengi tumhari

??

like it is impossible to even get 70%

[austere nimbus]

Bro in which state are u?

[austere nimbus]

Nhi

Free ke number bat te hain

Though dont become overconfident

[spiral atlas]

me underconfident rehta hu

[scarlet wedge]

PCM

[austere nimbus]

Which state though?

[spiral atlas]

up

[austere nimbus]

Coaching ke test mai kitne aate hain?

[austere nimbus]

Hmm

Jaha pe mai rhta hun

[scarlet wedge]

Not good enough

[austere nimbus]

Waha pe facilities achi hai by gods grace, toh sabko easy lgti hai 10th

[scarlet wedge]

Which Coaching do you study in

[austere nimbus]

None

Just

Local coachings

For mainly tests

Adv level

[scarlet wedge]

Which books are you currently reading ?

[austere nimbus]

Jd lee

[scarlet wedge]

Same here

[austere nimbus]

Though thats for extra knowledge

Mainly coaching ke notes

[scarlet wedge]

I mean its too good , do u read the original one

[austere nimbus]

Yeh

Orignal hi pdhta hun

[scarlet wedge]

cool

[austere nimbus]

Reading it is more like a hobby

Ioc ke liye coaching notes are enough ig

[scarlet wedge]

Knowledge is never enough

[austere nimbus]

I mean if youu are getting low marks in tests

First thouroughly read coaching notes

And then practice

Then move on to extra books

[iron warren]

hi

me also doing jee

[scarlet wedge]

"Extra " books , books are my primary source

[austere nimbus]

Okay then thorughly follow them

Thiugh jd lee orignal isnt that syllabus oreinted

I recommend you to read sm other thing aswell

[scarlet wedge]

I am reading the Sudarshan Guha one

[austere nimbus]

Maybe jd lee sudarshan guha

[austere nimbus]

Yeh

Then its alright

[scarlet wedge]

What abt Maths and Physics'

[austere nimbus]

Math i just do my coachuong assignemenrs and black book

Phy i do my coaching suirs book

[iron warren]

which coaching

[scarlet wedge]

What is your fav subject ?

[scarlet wedge]

some local one he says

[austere nimbus]

Local

[austere nimbus]

Math

[iron warren]

ohh

same

[austere nimbus]

Nice

[scarlet wedge]

Same but lately Physics interests me more

[austere nimbus]

Though find a good local coaching

Some arent good

[austere nimbus]

Phy was my fav last year

[iron warren]

im going to pace rn

[austere nimbus]

Though it changes

[scarlet wedge]

Which coaching are u studying in ?

[iron warren]

Pace

[scarlet wedge]

nice 🙂

[austere nimbus]

Which state guys?

[iron warren]

maharashtra

[austere nimbus]

Hmm nice

[iron warren]

you?

[austere nimbus]

Haryana

[iron warren]

nice

[iron warren]

so ur prepping for jee too?

[austere nimbus]

Hmm

[iron warren]

what chapters do you like

[austere nimbus]

Rotation

Sequnce and series

General Organic Chem

[iron warren]

how do you like rotation 😭

SnS is pretty fun

goc too

[austere nimbus]

Its fun ig

[iron warren]

ok lol

[sly pawn]

Yeah the reason why it doesn’t have many people solving it is because it is so thinking based

just like last years problem 5 had an unusually low amount of solves for such an easy logical reasoning/strategic thinking problem

this kind of problem 6 relies more on visual spatial reasoning and thinking rather than the typical problem which can be solved with tons of experience in traditional competition math.

But it is a very beautiful problem for these exact reasons.

It’s the kind of problem that is likely also much harder to prove than to solve

It’s just more of the problem I think is a lot more accessible and good for the kind I like most to solve, so I think it’s easier than most other p3/p6’s

Somewhat similar in difficulty to the p3 from last year, but slightly easier I think

[strong frigate]

Aalag banda hai bhai

[winter sable]

only the prodigies remember me

[proud dove]

https://fxtwitter.com/_Dave__White_/status/1947461492783386827

Dave White (@_Dave__White_)

the openai IMO news hit me pretty heavy this weekend
︀︀
︀︀i'm still in the acute phase of the impact, i think
︀︀
︀︀i consider myself a professional mathematician (a characterization some actual professional mathematicians might take issue with, but my party my rules) and i don't think i can answer a single imo question
︀︀
︀︀ok, yes, imo is its own little athletic subsection of math for which i have not trained, etc. etc., but. if i meet someone in the wild who has an IMO gold, i immediately update to "this person is much better at math than i am"
︀︀
︀︀now a bunch of robots can do it. as someone who has a lot of their identity and their actual life built around "is good at math," it's a gut punch. it's a kind of dying.
︀︀
︀︀like, one day you discover you can talk to dogs. it's fun and interesting so you do it more, learning the intricacies of their language and their deepest customs. you learn other people are surprised by what you can d…

FxTwitter · Jul 22, 2025, 12:59 AM

i solved imo p1

I AM OFFICIALLY BETTER THAN A PROFESSIONAL MATHEMATICIAN

[signal rain]

why isnt the answer 26^2 x 3

which is 2028

[solemn cape]

that would double count words that have multiple As

it double counts the 75 words of the form AA_, A_A and _AA. and then it triple counts the single word AAA

which is why it’s off by exactly 77

[signal rain]

interesting

so I'd have to consider the cases of A's frequency separately

,calc 25^2 * 3 + 25 * 3 + 1

[gilded halo]

Result:

1951

[signal rain]

alright ty

[trim bramble]

complementary counting (which is what they did) is probably easiest bc it removes casework

bc what if it was like 7 letter words 😬

[signal rain]

yeah I just got a more lengthy question

learnt it the hard way

[sturdy knoll]

how

[soft vigil]

professional mathematician my ass

1959 imo p1 🥀

"i dont think i can answer a single imo question"

[torpid cairn]

right because professional mathematicians are expected to know every single IMO question from every year

that's on the professional mathematician entrance exam isn't it

[torpid cairn]

you see these people crying about AI and how it's gonna kill us all and then you check their bio and see they're building it lmao

[soft vigil]

yeah it is

[proud dove]

think he means recent ones

[proud dove]

i mean this guy is at paradigm so he doesn't fit that archetype but it is funny

[pulsar kettle]

what web do you use for these problems?

[agile gazelle]

But isnt the IMO for high schoolers? I tough it was easy for atleast college student lvl

[torpid cairn]

unfortunately idk what that is I just looked at their most recent rt and it was about building math ai

[near hill]

Olympiad-style problems often depend of fairly esoteric tricks that most professional mathematicians have no particular reason to know by heart.
Math knowledge areas are not linearly ordered.

[pulsar kettle]

?

[sly pawn]

Bro there is so much wrong with this post its insane

[sly pawn]

IMO and competition math require a completely different skillset and abilities than most college+ math does

[sly pawn]

Well I doubt most IMO participants know all of them either, all you really need to know is some theorems which comes with experience, and then the rest is problem solving ability

(in certain cases some tricks could still be very useful though)

[agile gazelle]

Did you guys ever compete in imo?

[agile gazelle]

Skilled or intuition

[near hill]

I didn't.

[agile gazelle]

Sadly me too

But after hearing this i think i can train for it even if i dont compete ill try atleast

[torpid cairn]

👆

[torpid cairn]

explain

[sly pawn]

What country are you from

[agile gazelle]

Morroco

[sly pawn]

That’s where you live?

[agile gazelle]

Yes

Why?

[sly pawn]

Okay then it won’t be impossible to get into the IMO like for countries like India, USA, China

But it is still very difficult

[agile gazelle]

Its too late man i just finished high school

[sly pawn]

wdym you would try then?

[torpid cairn]

I think the way it's taught in school "advanced math" is seen as higher level math but in the context of math competitions it means something completely different (ie very complex problems & solutions) so going through a college curriculum won't help enough to make IMO problems "easy"

[agile gazelle]

The only way to be chosen in my country is to be one of the 6th first in the country in national olympiad and then they can chose you but in orther to compete in naitional olympiad u need to be atleast in first or second year of high school

[sly pawn]

Math competition math is completely different from higher/advanced math

[torpid cairn]

yes

[sly pawn]

Yeah same for us too

Actually idk what the min age here is though

[agile gazelle]

Its 20 ig

[torpid cairn]

I don't think we have one

[agile gazelle]

Here its 16

[sly pawn]

what

[torpid cairn]

didn't tao compete at 11?

[sly pawn]

hes from australia

[agile gazelle]

I tough u said max

Mb

[torpid cairn]

oh I didn't know lmfao

[sly pawn]

are you in high school?

[torpid cairn]

yeah

[sly pawn]

gonna try to compete up to usamo or imo?

[torpid cairn]

just usamo lol I'm not good enough for imo

wbu?

[sly pawn]

Probably just usamo too but I can try 😂

maybe ill make it if I grind hard enough

[torpid cairn]

what grade are u in

[sly pawn]

going into 11

wbu

[torpid cairn]

going to 10th

gl bro

[sly pawn]

ok

yes I only have little time left

🙏

[crisp forum]

I really appreciated some recent videos by Timothy gowers going through some of the recent imo problems cold but using just natural problem solving ideas. I mean he is a former gold medalist and field medalist but those videos are actually really instructive on how you could in theory approach these problems. They did take him much longer like near 2 hours to solve a problem but I don't think he keeps up with any of these types of problems in his research and his approach again was using really natural approaches nothing fancy.

What I'm saying is I think a professional mathematician could solve these problems it would probably just take them longer. Research problems are way harder.

[severe eagle]

stupid combi

[maiden panther]

this is not at all related

problem 5 had few solves because it's an anti-problem and tricked many experienced contestants under time pressure

not because olympiad contestants are incapable of simple logical reasoning

[maiden panther]

this is not completely true

most top math olympiad contestants are bright enough to excel in pretty much any undergraduate math program

however the reverse statement is probably a lot less true

[stable path]

how to start adv maths

[high goblet]

tbh i thought the number of solves for P3/P6 for recent years was higher than i thought but i've just looked at it and the # of solves for P6 was quite low for last year & the year before

[high goblet]

i mean P3 last year just has more solves, but i haven't done this year P6 so i don't really wanna comment too much on the difficulty

[high goblet]

btw that's not the right answer

[signal rain]

Art of Problem Solving

Alcumus

[severe eagle]

Welcome to mathcord @signal rain

[pulsar kettle]

thanks! is it good?

[prisma python]

yeah it has a lot of resources, but it lags sometimes i think

[pulsar kettle]

it is quite fun!

[sly pawn]

Hmm ok

[sly pawn]

Yeah because the skills that competition math teaches you are very useful and applicable to a lot of things

[sly pawn]

I think competitors are getting better is the reason why

[sly pawn]

Ooh I watched the video and now I really see why this problem is so hard

it makes more sense now 😂

[prisma python]

yeah i think it's like bait because most people would directly think of a diagonal

[lament bluff]

Is it a good idea to do Uni Proof textbooks problems to practice for USAMTS? Like are they similar difficulty

[signal rain]

I'm honoured

to be associated with math

[high goblet]

as in ||that's not the right answer||

[prisma python]

i know, i just watched the video

[pure yarrow]

Hey I am really confused about consistency like I can't practiceaths daily so do you recommend any documents for the IMO

I have two years and I 'll be 19 and go to university so 2 years is the maximum timeline

[fast atlas]

I just did this in like 10-15 minutes and I'm way too proud of it

[vernal axle]

Jensen + Titu

[fast atlas]

I did it with AM-GM + Titu only

I know what it is but I haven't studied Jensen's inequality yet

[sly pawn]

good job

[sly pawn]

what is titu?

[polar tangle]

what is titu?

[vernal axle]

https://artofproblemsolving.com/wiki/index.php/Titu's_Lemma

[fast atlas]

Here's my solution by the way, I was pleasantly surprised by how smoothly it went (and how quickly I came up with it). I know nothing about LaTeX so this was a pain
||PS: The second line is what LHS becomes after my subsitutions, I applied Titu's Lemma to it, forgot to type that in the code ||

[polar tangle]

which program do you use to type this?

[fast atlas]

I'm on mobile and there's this app called VerbTex which takes in LaTeX code and outputs a MS Word file (or other files, I picked MS Word) from that code

[short kernel]

Yea it's pretty neat

[vernal axle]

Jensen + Titu seems shorter:

[vernal axle]

You can use the Mathpix service to convert handwritten math into latex pretty cleanly.

[sly pawn]

bro it took you only 10 minutes to do all of that and the expanding and stuff?

danggg

that's fast

[sly pawn]

m1, m2, ... , mk all coprime

[sly pawn]

what happened from step 2 to 3?

[vernal axle]

multiplied numerator and denominator by the same number

[sly pawn]

Okay

thanks

and then used titus?

[vernal axle]

yep

[sly pawn]

nice

[scenic hazel]

i have a idea

can we design a evolved machine

[flat harbor]

hi

[river radish]

for this problem I first noticed that whatever position you start with, if you consider the pivot colorless, then the number of points on each side of the line stays the same, then I figured that okay this line will certainly rotate a full 360 degrees, switching pivots during this rotation but in 'the lab frame', it will rotate 360 degrees.

Now each pivot will be 'responsible' for a certain fraction fo this full rotation, for example if you had 3 points placed as if they were vertices of a triangle, and the original position of the line was such that it goes through the triangle but only through one vertex (so it has one point on either side), that 'state' will stay the same, and as the process occurs, certain vertexes will 'own' certain parts of the 360 degree rotation.

(img 2)

So P1 might be the pivot for angles (measured from the -ve vertical) 30 to 150 then if we switch over to P2 it will cover some angle and so on, and none of these will overlap (if you draw a line for every point from P1 to PN on a number line that goes from 0 to 360 that represents how much of the number line it covers, these wont overlap (except for I guess when they switch the pivot from one point to another for like a really tiny instant, but we'll ignore that).

Now if there is no overlap that means that for every pivot there is a unique range of angles that only it can cover, given an initial state ( a points on one side of the line and b points on the other side). Okay. Now we have to prove that the windmill uses each point an infinite number of times. I guess if you prove that it hits every point at least once and eventually returns to the original point after hitting them at least once, then it will hit them infinitely (process will loop).

now just consider the fact that this line will rotate a full 360 degrees, so if you imagine it like a mop sweeper, it will cover the entire plane, of which S is a subset of. so isn't that enough to imply it will hit every point at least once?

like the only case where it will leave some area out would be if like it rotates and changes the pivot every time it is tangent to the vertices on the 'perimeter' of the polygon formed by the points (e.g. start with say a triangle and the initial position is such that it doesn't pass through the triangle, only touches one vertex, then it will rotate 'around' that triangle), but even then it touches every point, besides we have freedom over the initial condition so we can just make sure it goes through the outermost polygon formed by the vertices or something (this is handwaving ik, im just hoping someone will correct me here if they notice something)

if not, then why, but if the above is true:

Now I said that there is a unique range of angles corresponding to each pivot, which it 'owns'. So if we pause the process at some moment, and then translate the line so that it's pivot goes from P to P' (but the lines orientation is the same, since we are translating) then this new position will not obey our state invariant condition, so it will be invalid. This means that once we are done with the 360 degree rotation, and we arrive at 0 degrees. We must be at the same point that we were on the previous run, so we are again at that same instant and the process will loop infinitely so it hits every point infinitely

[fast atlas]

It's cyclic so I didn't have to expand them all, that made it a lot easier

[fast atlas]

Thanks, i'll try that

[fast atlas]

Nice solution. It's a middleschool math problem so I did not think about Jensen's. Also I only learned the special case of Jensen's when the weights are all equal to 1. Reading this solution opened my eyes to new approaches

[prisma python]

do you have an explanation of this theorem? when i searched it up i only found the jensen's inequality for graphs i think

oh wait nvm it's just that the process of finding if a function is convex or concave takes a long time right?

[vernal axle]

Not really. You just take the second derivative and check whether it's positive (or negative) on the interval in question.

[prisma python]

positive is concave?

i think i was just looking at the proof then

[vernal axle]

positive means that the set above the graph is convex

[maiden panther]

p3 from last year had 12 solves and p6 from this year had 8 solves

it's not that different

although fundamentally i would say they are difficult for different reasons, since p3 from last year had a lot of details/small steps while p6 from this year has one large central idea

[prisma python]

the set above the graph?

[maiden panther]

but it's hard to actually compare the difficulty even though they are both combinatorics

[vernal axle]

yep

[prisma python]

could you explain what that means?

[vernal axle]

The set of points on the coordinate plane lying above the graph of the function f(x). That is, the set of all points (x,y) such that y>=f(x).

[maiden panther]

ngl, this is a roundabout way of saying that any segment between two points on the function lies above it

i suppose it does show where the term "convex" comes from which is pretty neat

but for beginners it's probably not the best way to explain it

[vernal axle]

I think this is an easy way to memorize which is convex and which is concave. And the definition is ok, but actually we check if the function is convex by computing its second derivative which is not obviously connected to the definition with segments.

[prisma python]

i'm kind of getting it now though
but how about for functions like x^3 where f''(x)=6x? is it convex for x>0 and concave for x<0?

[vernal axle]

exactly

[prisma python]

how would it affect the inequality? is it just considered convex since the weights a_i>0?

[vernal axle]

convex or concave changes the sign of the inequality.

[prisma python]

yeah, but i mean how would you use x^3 for that inequality?

[vernal axle]

you take f(x)=x^3, weights are 1/3, 1/3, 1/3. and points say x1, x2, x3>0. then we have 1/3*(f(x1)+f(x2)+f(x3))>=f((x1+x2+x3)/3), since f(x)=x^3 is convex for x>0. That is, (x1^3+x2^3+x3^3)/3>=((x1+x2+x3)/3)^3.

[prisma python]

so the only restrictions are x1, x2, and x3 have to be greater than 0? and if x1, x2, x3<0 then 1/3*(f(x1)+f(x2)+f(x3))<=f((x1+x2+x3)/3) ?

[vernal axle]

yes

[prisma python]

oh ok, i get it now, thanks

[vernal axle]

Here are some more problems that can be (more or less) quickly solved using Jensen's inequality.

[prisma python]

wow, thanks for the resources. i'll try solving them

[sly pawn]

ooh I remember this problem, it was very fun

[sly pawn]

what is cyclic? 🤔

[prisma python]

basically symmetric for all variables i think

like xy+xz+yz=0 is cyclic but x+y^2+z^3=0 is not

[fast atlas]

Cyclic and symmetric are very different

ab+bc+cd+da is cyclic, ab+ac+bc+cd+bd+da is symmetric

[sly pawn]

ah I see

[sly pawn]

ok

[prisma python]

how do I solve the first one? using Jensen's and AM-GM I only simplified to $\begin{aligned} & \frac{1}{1+\frac{1}{n} \sum_{i=1}^n x_i} \leq \frac{1}{n}\left(\sum_{i=1}^n \frac{1}{1+x_i}\right) \ & \frac{n}{1+\frac{1}{n} \sum_{i=1}^n x_i} \leq \sum_{i=1}^n \frac{1}{1+x_i} \ & \sum_{i=1}^n \frac{1}{1+x_i} \geq \frac{n}{1+\frac{1}{n} \sum_{i=1}^n x_i} \leq \frac{n}{1+\sqrt[n]{x_1 x_2 \cdots x_n}}\end{aligned}$

[gilded halo]

MarvinA

[vernal axle]

You need to choose an appropriate function. You see, there is a product there. And Jensen's inequality is about some sums. So, you should choose something that transforms products into sums or backwards. For example, you can prove AM-GM with Jensen: choose f(x)=ln(x), and weights 1/n,...1/n. Then
ln((x_1+...x_n)/n)>=(ln(x_1)+...+ln(x_n))/n
which is exactly AM_GM if you apply exp on both sides.

[prisma python]

could you give me another clue? i tried $f(x) =\ln \left(\frac {1}{1+x} \right) but i got \ln \left (\frac {1}{1+\sqrt[n]{x_1x_2 \cdots x_n} \leq \sqrt[n]{\frac {1}{(1+x_1)(1+x_2) \cdots (1+x_n)$ and I don't know if I'm going in the right path

[vernal axle]

try ||f(t)=1/(1+e^t) with t_i=ln(x_i) and weights 1/n||

[prisma python]

whoops uh idk how to fix this latex thing but thanks for the clue

[prisma python]

oh, with that clue seems like i get the answer basically immediately, which is that ||the inequality was false||
but also, the x_i in [0,1] wasn't used for anything right?

[vernal axle]

nope, the inequality is always true for 0<=x_i<=1 🙂