A MEAN Problem from India [ 2016 RMO Mathematical Olympiad ]

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Today we'll be using the AM GM inequality to find out an upper bound for the product of 3 unknows given a fractal equation :) Enjoy! =D Video sponsored by Brilliant btw :3
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[@kathanshah8305]

Thumbnail and title suggests that fresh toadwalker has inspired you

[@PapaFlammy69]

nah

[@3ckitani]

I think it's from Michael Penn

[@mikaeelshah7205]

FReSh ToAdWalKeR: The answer is TOO good. a GOLDEN puzzle. TrAp (Trapezium)

[@kathanshah8305]

@@3ckitani yeah mixed it up

[@kathanshah8305]

@@PapaFlammy69 yeah it’s inspired by micheal penn not mindyourdecision

[@AndrewDotsonvideos]

This changes everything.

[@kshitijgarg1470]

Sir that move of dividing the both sides by 4 was dope 🔥. Love from India.

[@TechToppers]

You don't need that... AM GM destroys it on the spot.

[@kshitijgarg1470]

@@TechToppers Ya that's the same thing

[@rajnikantsharma4054]

@@TechToppers the main matter was that this man has to show power of Indian competitions

[@kathanshah8305]

Don’t lie you forgot password of flammable maths 2 so you posted this video here

[@JonathonRiddell]

Embarrassed to say I did not know the AM GM inequality

[@bsmith6276]

Embarassed to say I do know AM/GM inequality but still forget to ever use it.

[@adityaruplaha]

@@bsmith6276 same man

[@devjitghosh550]

so do i bro

[@1987Videolover]

AM GM i think never told to a reguler student...

[@muahmuah4135]

There are not much people who love mathematics and usually dedicate an entire channel and talk about it with such passion...... except for those teachers who are OP because it really hard and honestly nobody give a shit... so i really appreciate guys like you who has great passion for the subject

[@prashant2650]

Ur Indian impression is like what we do for Bri-ish peeps

[@alokmishra1414]

7:33 yeah the education system really sucks here..it never forced our brain to think out of the box..instead always to remember tricks and formula on which we rely..i remember solving these questions not even knowing what i am actually doing..just do it man..just do it don't ask why and how..😓

[@PapaFlammy69]

exactly

[@supercuntt5876]

Is this sad that I was taught this kinda trick when learning about inequalities? Lol

[@sohamvirkar802]

Honest replies are deleted here!😂

[@sohamvirkar802]

Time to find new teachers bro😂 dont blame the whole education system just because YOU are taught that way

[@supercuntt5876]

@@sohamvirkar802 lol ok

[@theblinkingbrownie4654]

Apparently you're 'better' than other countries' people if you can memorize and solve equations, even if you don't get any of the intuition, I live close to India and hate it when my friends brag about 'knowing' math. :/

[@PapaFlammy69]

...

[@adityaruplaha]

As an Indian, here competition is tough enough that it's the only way you can get an advantage.

[@einzelganger7744]

This is the first level examination. Even IMO problems involve AM GM and other inequalities. I've seen level 1 question papers of most countries and tbh they are all kinda same. Try INMO problems.

[@theblinkingbrownie4654]

@@einzelganger7744 you talking to me?

[@AlphaCurveMath]

As an Indian I can only warn you against the sheer horrors of the JEE Advanced content train. The audience is half a billion whiners who will hear nothing but praise for India and how the rest of the world would be stumped by JEE. It's only a downward spiral from there which, sadly enough, one of the potentially best physics channels has already headed into. Edit: But the olympiads are fine. It's about the only place where actual math problems come up. It's the jee's that are subpar.

[@PapaFlammy69]

ye, that is why I mostly tefuse to cover indian things :/

[@aradhya_purohit]

@@PapaFlammy69 papa, how do you 'tefuse'?

[@AlphaCurveMath]

@@PapaFlammy69 yeah if only to keep memeing as hard as you do right now, you've got to keep away from those who can't take a joke.

[@anmoldeepsingh9281]

Holy shitplus jee is overrated

[@aradhya_purohit]

@@AlphaCurveMath seconded.

[@RC32Smiths01]

Always enjoy these kinds of videos man! Awesome work

[@seriouslysupersonic]

What's up with all that violence? Why do the terms on both sides all have to "die"? Can't they just hapilly retire to the infinitesimal garden where all the epsilons and deltas hang out?

[@PapaFlammy69]

nah, they all ded

[@rudeus6621]

me who qualified rmo in 2018 : 👁👄👁

[@Prodbybah]

bhai please say how and which books and which important chapters?

[@kumaracademy7555]

7:17 I always heard that Germans are really honest..Today I saw....xD

[@IshanBanerjee]

There is a reason why I love RMO so much

[@gamingsquad7149]

Ayo ishan saar here! 💀

[@aniketeuler6443]

Now Jens is making Indians feel good 😂

[@blaiseragon8142]

Love your content so far !

[@PapaFlammy69]

thx :)

[@keshavb3128]

In Papa Flammy's perspective, he trapped Andrew in his basement. In Andrew's perspective, he trapped Papa Flammy in his attic. You've been trapped in Andrew's attic, Papa Flammy.

[@veralgupta8182]

Oh yes my mentor is here with the question I failed to answer

[@ishansingh2391]

Try geometry problems of mathematic olympiads as RMO is 1st stage of selection process in Olympiad. obviously some of these problems may be easy but in our Olympiads even after writing your solution one may not get a whole 17/17 he may get 13 or 14 marks as the solution requires elegance not rigor to be precise.

[@She_898]

It's really sad here in India every high school kid is just trying to crack exams whether it is JEE or RMO or anything like that but no one is really enjoying what they are doing. I wish that students had some amusement about learning maths and physics rather than just clearing this exams

[@mathsman5219]

If you really love Mathematics ❤️ You will never want to check yourself through any exams (as they can't .)

[@gamingstars8956]

In most of inequalities just equate all terms to get min value a/1+a=b/1+b=c/1+c to get a=b=c=1/2

[@lauthomas7179]

abc attains its maximum only when a=b=c. So 3a/(a+1) =1 3a - a =1 a. =1/2 Max of abc = 1/8 Or abc=

[@mayankchauhan2814]

This question is used as a AM class illustration in my math class

[@timburdack7366]

Love this! ❤

[@BenjusJamentus]

Is this closer to the kind of mathematics that is done in university or that we do at the "Gymnasium" (some kind of preparation school for university in Switzerland/Germany/Austria I think as well)? I'm really interested in studying maths but I'm not sure what to expect... Greetings from Switzerland!

[@PapaFlammy69]

something in between I would say^^

[@BenjusJamentus]

@@PapaFlammy69 OK, thank you!

[@paulm5441]

After reaching to 2abc + ab + ac + bc = 1 you can just plug it into the obvious (a+b+c)^2 >=0 . No need for AM, GM etc.

[@doodelay]

dividing by 4 and introducing the AM/GM inequality haha keep showing us the way papa flam (PS, most of my mathematical applications comes from watching u work in real time)

[@MathElite]

Love pro gamer moves!!!!

[@ramanunnikrishnan7354]

finally RMO problems, never got to give it

[@kumaracademy7555]

❤️❤️🔥🔥Always love your content

[@30indrayudhdas28]

Can we use rearrangement inequality? I think so🤔

[@sisyphus645]

The meme before the math is like the prayer before the food

[@AryanSingh-ro4fi]

So satisfying

[@californium-2526]

That NaCl meme, I know it...! A zillion reposts and copies of this (low-quality) meme has been seen, mainly in r/chemistrymemes.

[@lgooch]

I tried to apply Cauchy Schwarz or AM-GM too early, I refused to multiply all of that shit out.

[@faenzarfaenzar2636]

Amazing video ! But I got one question : Where did you get that Blackboard !!!!!???????? I want one !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@Regimeducamp]

papa what's the name of your intro's song

[@hsiochen7216]

I believe given two functions, f(a, b, c) and g(a, b, c), if both f and g remain the same after reordering their argument a, b, and c. g(a, b, c) will have a local maximum or minimum at the point of a=b=c, under the restriction f(a, b, c) = constant. A lot of these types of problem comes from function with symmetric argument.

[@nandansantosh5497]

Jokes on you man this was taught to us as joke by our teacher because we had a lot of time left on our hours

[@farhannr28]

Bruh i literally just did that question on a test lmao. Lucky that i found the correct answer in last second

[@zh84]

Now solve this simple problem: a/(b + c) = b/(c + a) = c/(a + b), a, b, c all positive integers.

[@baerlauchstal]

It's not *that* crazy-hard to extremise the product of p/(1-p), q/(1-q) and (1-p-q)/(p+q) over the triangular domain p>=0, q>=0, p+q

[@Jop_pop]

Alternative solution with lagrange multipliers: Substitute x=(a+1)/a, y=(b+1)/b, and z=(c+1)/c. We can define g(x,y,z)=1/x+1/y+1/z=1 (given) and f(x,y,z)=x+y+z. Very straightforward method of Lagrange multipliers then gives x=y=z=3. This can be verified to be the min for f. Thus xyz+x+y+z >= xyz+9. Next, notice that since 1/x+1/y+1/z=1, multiplying by xyz easily shows xyz=xy+yz+xz. So xyz+x+y+z >= xy+yz+xz + 9, thus xyz-(xy+yz+xz)+x+y+z-1=8. And so (x-1)(y-1)(z-1)>=8. Hence 1/(abc)>=8 and abc

[@drabart6121]

Em, could someone explain why there are 3 lines in the 'equals' sign and he just ignores it? From what I've learnt that means congruence and not equality. (Maybe they use 3 lines in India or sth)

[@PapaFlammy69]

identically equal

[@gurkiratsingh7tha993]

Hello bro, please give me some tips to solve riemann hypothesis

[@chrisglosser7318]

So... why didn’t you just use Lagrange multipliers to show that the constrained abc have a maximum when they are equal(?)

[@adityaekbote8498]

@Chris Glosser what are Lagrange multipliers??

[@chrisglosser7318]

@@adityaekbote8498 They are a way implementing constraints in variational calculus

[@adityaekbote8498]

@@chrisglosser7318 ohh thank you

[@yoyokojo651]

Kinda tedious tbh, gotta argue compactness of constraint ect. Plus it’s kinda a lot of Algebra in this case

[@leopoldschlemmer3387]

Can’t you just state that if a>=b,c then 1 >= 3*a/(1+a), as a/(1+a) is decreasing when a is increasing. If you solve you get a

[@anuragsamanta9016]

Take c/1+c to RHS .then apply AM, GM in between a/a+1 and b/b+1...After getting three such inequality in ab bc and ac multiply all three equation. You will get the desired result

[@rvure]

Do more RMO probs my man

[@qq-rf6hv]

this is a really good solution! but I think there is way to make it easier, by using the fact that the product of numbers whose sum is constant is maximal when these numbers are equal

[@karthikboyareddygari568]

I was not aware of this fact, though I'm sure it's easy enough to show as a matter of optimization. In this case, all you would need to do is find what the common value is that satisfies the constraint, right?

[@qq-rf6hv]

Yes

[@mayankchandrakar8333]

Next video idea: Napoleon Bonaparte's triangle theorem :)

[@harshpandey6725]

Love from India🇮🇳 Really well solved!! Loved the pro gamer moves❤

[@DhirajKumar-rx8hi]

Just Apply two times AM GM inequality. First for the given 3 terms a/(a+1). Product of abc/(a+1)(b+1)(c+1)

[@karthikboyareddygari568]

This just showed up in my recommended, and I gave it a shot. I ended up doing what you did up until you divided by 4 because I was not aware of that relationship between the arithmetic and geometric means. Not having that knowledge, I treated it as an optimization problem now that the constraint wasn't so hairy and used Lagrange multipliers to conclude that a = b = c = 1/2 was a point of interest. Seeing as a, b, c \in (0,\infty), there weren't really any other points to check. Using the value of 1/2, we find that abc = 1/8, which is now either an absolute max or min. By using a = b = 1/3 and c = 1 as a random point that satisfies the constraint, we obtain abc = 1/9 < 1/8; therefore, abc \leq 1/8 for all positive real a, b, c subject to the constraint.

[@turtlelink6845]

Can also be done using Jensen :) (Should be possible once you sub x = a/(1+a) etc.)

[@paulm5441]

what is a Jensen? :)

[@udbhav5079]

RMO simps for AM > GM > HM...

[@PapaFlammy69]

ye lol

[@dijkstra4678]

I was just thinking that the problem seemed pretty symmetric and then it hit me... what if we assume a=b=c? that would mean that a/(1+a) = 1/3 which means a = 1/2 which would then mean that abc = 1/8 proving abc

[@viinisaari]

6:25 It's aRithmetic for the noun and arithMetic for the adjective.

[@cr1216]

Here is a way with no complex calculation at all. Let a=tan^2(x), b=tan^2(y), c=tan^2(z) then we get sin^2(x)+sin^2(y)+sin^2(z)=1, notice that cos^2(z)+sin^2(z)=1 by definition so this means cos^2(z)=sin^2(x)+sin^2(y) >= 2sin(x)sin(y) (last step by AM-GM) . Similarly we get cos^2(y) >= 2sin(x)sin(z) and cos^2(x)>=2sin(y)sin(z). Multiply the three together we get cos^2(x)cos^2(y)cos^2(z)>=8sin^2(x)sin^2(y)sin^2(z) and therefore 1>=8tan^2(x)tan^2(y)tan^2(z) which implies 1>=8abc.

[@physicslover1950]

Omg that concept of arithmetic mean and geometric mean was great. I think I need to revise everything 😌😌😌😌

[@nicholasthesilly]

Look up harmonic means. And their relation to the other two kinds :)

[@adityasawant3813]

Please discuss INMO 2021 P6.

[@aadityathukral5667]

How you doing flammy?

[@tvbbian]

If you let a=(tanA)^2, b=(tanB)^2, c=(tanC)^2, 0

[@jmof0464]

people that do those problems in india always endup in tech support

[@mohamedababou8777]

It is better if this is the question at world Mathematics conferences and Olympiad: "Those who believe that Numbers have no end are required to prove infinity realistically and concretely."

[@screamman2723]

how old is he tho

[@einsteingonzalez4336]

0:04 "Let's do some Sesame Street action right there!" Too long so far, right?

[@rohitkhatri3892]

7:18 I felt that.

[@JonathanMandrake]

My Analysis Prof did this as one of the problems

[@YoutubeModeratorsSuckMyBalls]

Noice, 👍. I enjoyed it

[@neilgerace355]

0:10 Three of these things belong together Three of these things are kinda the same But one of these things is not like the others Now it's time to play our game It's time to play our game!

[@User-gt1lu]

More videos like that

[@aptilious2774]

Lmao if geometry problems weren’t so damn hard in greek olympiads I would actually qualify to even selection team anyway I’m focusing on calc to become engi XD

[@avtarsingh-tz1up]

Bring more

[@MrRyanroberson1]

notice: if we substitute x,y,z = 1/a,1/b,1/c, the problem becomes the following: 1/(1+x) + 1/(1+y) + 1/(1+z) = 1 -> xyz >= 8. therefore let us multiply all by (1+x)(1+y)(1+z) to get 3 = 1+x+y+z+xy+yz+xz+xyz. so on, so forth

[@MrRyanroberson1]

why am i like this - i made such an elementary mistake. fixing time: we get (1+x)(1+y) + (1+x)(1+z) ... and it doesn't get all that much simpler, so this substitution doesn't help much besides the initially simpler representation

[@PapaFlammy69]

happens to the best, Ryan

[@Alliedmaths]

a = b = c can be easily solved by doing because it has similarity

[@jackhandma1011]

Now prove AM-GM without calculus and induction (idk how to).

[@jimboli9400]

at 1:06 my mind immediately went AM GM inequality where?!

[@1987Videolover]

When i see the question.. I know it will be using AM GM..

[@erikawimmer7908]

Me: doing math. Papa: uploads Video Me: watches math Video abd doing math. ULTIMATE POWER

[@shabushabu1453]

Wait, what? How is (1/4)^4 = (1/2)^2

[@markgh2491]

I think it’s pronounced arithmetic, not arithmetic.

[@PapaFlammy69]

yeye

[@onkarsingh8358]

Luv u grom India👈

[@SK-qc2hb]

Great video, but just one thing -- if you perchance show the solution to math problems in JEE, and mention how nice it was, you might face problems, like Jack Fraser did. He didn't say anything against Indians, but it went straight downhill for him for months after his Quora answer.. Nice video!!

[@saman6176]

Is not your mbti personality ENTP?

[@chaitanyavarma1747]

6:29 Lmao. Also first guy to comment is an Indian XD

[@Godakuri]

Good vid

[@larakalish881]

Ich mag der Inhalt ihnen Videos! Gehen sie weiter! Support from the US! Unterstützung aus Amerika!

[@vidhanp482]

Micheal Penn cameo

[@duggydo]

Since a,b,c are all reals, I just solved with compass and straightedge. Ezpz weeny squeezy!

[@PinkeySuavo]

idk why/how i ended up here at 02:42 am but quite noice btw i was "going to sleep" 30 minutes ago after watching vsauce video lol

[@quickyummy8120]

I found another amazing Olympiad video kzbin.info/www/bejne/hovRpqd6jZVlqas

[@tobiasgorgen7592]

6:28 its the arithmetic meme

[@scienceshed5466]

7:32 papa spitting the truth 😂 and i can feel that because i am one of child rhat hsa to give those fucking stupid exams that are just 😑

[@PapaFlammy69]

3:

[@butter5014]

Dear Papa Flammy (capital p because now you are deified just like Santa Claus), please help me out with the following problem from one of the Bulgarian mathematical olympiads: A hotel has 26 rooms. Each room has a number from 1 to 26. Different rooms have different numbers. Two guests are placed in rooms x and y. If the product xy is equal to the sum of the numbers of all rooms except x and y, find |x-y|. You are allowed only a pen and some paper. Thanks in advance!

[@paulm5441]

You have x, y between 1...26 and xy = 1+2+3+...+26 - x- y xy = 351 - x- y xy +x+y = 351 x(y+1) + y = 351 x(y+1) + y+1 = 351 + 1 (x+1)(y+1) = 352. 352 = 11*2^5 and the only product of two numbers x=1 and y+1 equalling 352 and having x+1 and y+1 in 2...27 is 22*16. So x+1 = 22 and y+1 = 16 or x+1 = 16 and y+1 = 22, giving us x= 15 and y = 21or vice-versa. It doesn't matter since |x-y| is the same, namely 21-15 = 6

[@butter5014]

@@paulm5441 Thanks for that! I hope you enjoyed this problem.

[@jamiejoseph7823]

Lmao the gate for the Indian math tho

[@juijani4445]

papa whats the point of including mathematical olympiad in the title? isnt MO in RMO nuff? is this for algorithm??

[@PapaFlammy69]

dunno

[@zombiekiller7101]

Why does it look like that your maths teacher is sitting behind the camera with a ruler and shouting at you to STATE ALL THE FRIKIN STEPS AMD REASON

[@CrankinIt43]

lol ur noob at math

[@V-for-Vendetta01]

Please do the whole of India lmao

[@tszhanglau5747]

Can you talk about am-gm inequality in detail?

[@PapaFlammy69]

Can do! :)

[@patricius6378]

@@PapaFlammy69 and maybe motivate the geometric mean a little? :) bc I've never heard about this thing :O

[@shayangfkk7948]

Always is the indian guy doing german mathematics . Now german boi , doing indinans .