Solving IMO 2020 Q2 in 7 Minutes!! | International Mathematical Olympiad 2020 Problem 2

[@lucasm.b.4390]

I loved how you walked us through your insights, rather than just solve it. Amazing content! Keep up with the good work.

[@letsthinkcritically]

Thank you very much!!

[@mathgoldmedalist]

Just solution is not enough always.methods and ideas are important as well

[@mrlegend3983]

Man I can imagine the level of hardwork you have done to achieve this mastery . Man I hope your hardwork pays off one day I hope you get best in life . Love from India

[@mathgoldmedalist]

👌👍

[@makmesanovic8568]

Absolutely brilliant. You just made a very intimidating problem look like a piece of cake that everybody can solve. Awesome video

[@letsthinkcritically]

Thank you very much!!

[@mathgoldmedalist]

I agree with u, that's his ability

[@Triskelion345]

@@letsthinkcritically youre amazing

[@thebruhtruth8973]

truly....i am sooo grateful to you...you are really helping us the olympiad aspirants amp up their approach thank you soo soo much

[@chhabisarkar9057]

Wow i really love your channel , hats off to your work . It also helps me in learning new techniques :)

[@letsthinkcritically]

Thank you very much!!

[@debayuchakraborti1963]

woow u r here too

[@KJ-zs7pi]

@@debayuchakraborti1963 woah

[@siamahmed5466]

Really Brilliant!!! Subscription done brother.. Love from Bangladesh ❤️🇧🇩

[@lanariddle2583]

Ur videos are absolutely awesome! You made it so easy though it looked like a nightmare at the begining😆😅 Thank u so much🤩

[@mathgoldmedalist]

👍👌

[@sk99gaming99]

Am=Gm (It has a condition to be only used for if your asked abt min valve,Terms of them should be good and factors shoukd be good) I first thought of log and this am,gm did not come to me in mind

[@keongzhiyuan6364]

U hear the Chinese accent you know this video gonna be lit

[@georgefan2977]

nah I think it’s Malaysian accent, it’s like Uncle Roger

[@stayhomedomath122]

I think that it is Hong Kong accent

[@ADEEShow]

Actually I am currently not preparing for IMO but still extremely interested in Mathematics. I solved this question in 3 min yes:) without even watching the video. But still I think I should practice more and more and more so that I can tackle such problems when just looked at them, Dont think I practice too much, but I am just interested in mathematics and proving questions :) and even pray for my IMO preparation which I think I can do , I hope I can represent my country !

[@faridaparvin4535]

Which grade are you in?

[@ADEEShow]

@@faridaparvin4535 11th Grade

[@TechyMage]

you can't even copy sol in 3 min freaking liar

[@boomer1248]

u reacheed imo ?@@ADEEShow

[@keithmembe3826]

That explanation made me wonder what the question is doing on the IMO

[@ZeonLP]

Knowing the solution makes every problem seem easy. They still feature problems with seemingly easy/quicks proofs. Doesn‘t mean it‘s easy to come up with that under pressure

[@mathgoldmedalist]

There are challenging problems but some of them are easier

[@sekarganesan]

Truly thrilling and neat every step.

[@letsthinkcritically]

Thank you!!

[@varun1702]

literally a genious approach

[@nitroNM8328]

Jai Shivaji Maharaj 🎉

[@shibainugamer]

For those of you that are wondering why a+2b+3c+4d>=a+3b+3c+3d it's because in the beginning of the problem a>b>c>d (I forgot this lol)

[@XavierXavier-ts2pe]

very cool. you can also solve it by using calculus and study the max of the function (a,b,c,d) . its more calculation but you dont need magic tricks.

[@boraolmez6622]

Did you participate in any past IMOs, IMO 2020, or want to participate in IMO 2021. Also, great video. Would like to see you cover IMO 2020, P6.

[@letsthinkcritically]

Unfortunately I could not make it into the IMO Team. I was a reserve member (a few positions behind no.6) for two years. I will look into the problem and see what can be done. I might make videos on other problems first.

[@muhammadnuralamsyah5601]

@@letsthinkcritically where are you come from?

[@letsthinkcritically]

I'm from Hong Kong.

[@adamswilliam3462]

letsthinkcritically Holy moly!! Me too😆

[@adamswilliam3462]

I’ll definitely keep in touch with this channel.

[@mahirchopra4092]

Nice explanation!👍🏻 Keep up the great work

[@mathgoldmedalist]

👍👌

[@tongvi1234]

Good thinking, easy to understand.

[@letsthinkcritically]

Thank you!!

[@pushpanjaykumar3985]

Plz can u also solve inmo problems i.e, (Indian National Mathematical Olympiad). Your explations and logical thinking are superb and it will be of great help. Thanks.

[@letsthinkcritically]

Sure! Will see what can be done.

[@naveengantyala940]

Wise person..in mathematics.brooo...

[@letsthinkcritically]

Thank you for your compliment 😄

[@erzsebetdr.ludvig3687]

Very nice math problem . Congratulation for your solution.

[@austinconner2479]

More complicated than needed. After AM-GM just observe that the left hand side is convex, so the max is achieved at one of the 4 vertices of the simplex in which a b c and d lie. Check all 4 satisfy inequality

[@mathgoldmedalist]

👌👌

[@blank7771]

Hi I'm new to this concept of convexity can you please direct me to what you mean exactly or what I would need to study to understand how an expression can be evaluated as a simplex

[@subratabhattacharjee7651]

With 1 problem 7 minutes... OMG😇😇!! It is better to leave that problem or else leave the exam..

[@narayanidhar7899]

I'm a imo silver medallist watching this

[@whitegalactico7152]

Which country?

[@piyalsana1252]

Which country

[@cracknblast8247]

I don't see you in the IMO records. Trolling?

[@editg121]

lol SOF IMO is a joke.

[@cracknblast8247]

@@editg121 Agreed

[@swayamprakashkar9664]

Can we use trigo bashing here? Although nice method... Keep going bro 😊

[@letsthinkcritically]

would you mind elaborate?

[@weggquiz]

absolutely brilliant

[@andriivedmid]

Absolutely brilliant

[@dontich]

Couldn’t you just say because the variables are symmetrical, the left side is maximized when you maximize d. This is when all 4 are = .25 That gives 2.5 * (.25)^(.25)^4 Or 2.5/4 which is < 1

[@XavierXavier-ts2pe]

no that’s not correct. the max is on the edges. a=1 and others 0

[@sasharichter]

Richard Feynman used to say "Know how to solve every problem that has been solved." So it appears that one must "know how to solve every [IMO] problem that has been solved" before in order to solve these problems.

[@mathgoldmedalist]

👍👌

[@talyamansour]

Mind=blown How can one practice to become able to solve imo questions in 7 mins

[@mathgoldmedalist]

👍👌

[@mrmathcambodia2451]

i try to know and learn more about this exam .

[@sokpisey8516]

Your solution is amazing 🤩 Thank

[@letsthinkcritically]

Thank you very much!!

[@suspendedsuplexchannel1000]

Sir please share your module on KZbin, we need to be taught by you, your explanations are perfect and please reply

[@letsthinkcritically]

What do you mean by module?

[@suspendedsuplexchannel1000]

@@letsthinkcritically module means notes ( that you have got), will you share on KZbin please

[@letsthinkcritically]

Will see what can be done 👍

[@camviethung4779]

beautiful answers. Subscribed for you

[@letsthinkcritically]

Thank you very much!!

[@huseyinyigitemekci-fm8vf]

That was amazing.

[@ratnasingh8402]

What I did was: a,b,c,d ∈R...given a>=b>=c>=d>0....given stat.@ And, as a+b+c+d=1 We get the max value of 'a' To be 1/4 So, stat. @ can be written as; a,b,c,d ∈[1/4,0) So, the max value for a,b,c and d would be a=b=c=d=1/4 *If the statement (a+2b+3c+4d)a^a.b^b.c^c.d^d

[@iandavidlorenzogarcia9392]

This is wrong. You would need to probe that f(a,b,c,d)≤f(1/4,1/4,1/4,1/4) and also I don't even think that is true

[@mcwulf25]

I confess I didn't get the last step. The LHS had a 4d in it and the RHS had a 3d in it. Could the LHS be bigger than the RHS?

[@HomeOfGames]

Well in case of coefficients of a^2R HS-LHS is b-d and since b>=d RHS is always greater or equal to LHS The similar thing can be done for other terms.

[@mcwulf25]

@@HomeOfGames Thanks I will take another look.

[@oof-inator282]

Does my solution work? (a+2b+3c+4d) is less than or equal to (a+b+c+d+a+b+c+d+2d)=(2+2d). Since 2d is at most 2*0.25, (a+2b+3c+4d) is less than or equal to 2+0.5=2.5. Now (a^a * b^b * c^c * d^d) is less than or equal to (abcd)^d since a smaller exponent would increase a number between 0 and 1. Also note that, (a^a * b^b * c^c * d^d) is less than or equal to (a)^(a+b+c+d)=a. Now, this is exactly equal when a=d or d=0.25. So now we can consider (abcd)^d with d=0.25 because that is when it is largest as per the previous inequality. By the AM-GM inequality, (abcd)^0.25 is less than or equal to (a+b+c+d)/4=0.25. Finally 2.5*0.25=0.625 so it's less than one.

[@soupayandasgupta8185]

Beautiful solution

[@zobkey]

I proved aªb^b*c^c*d^d

[@gemmajoy2046]

That was elegant!!!! It would be really great if u solve Indian mos too!!

[@letsthinkcritically]

Thank you!! Will see what can be done

[@piyalsana1252]

Brilliant man....👝💥💥

[@letsthinkcritically]

Thank you!!

[@excited3134]

3:30 how we got b1 to a1×....×bn to an. My question is why b-s have powers?

[@taopaille-paille4992]

please solve problem 6. thanks (good video, I am happy to give a like^). keep the great work

[@boraolmez6622]

Yes P6 plz...

[@letsthinkcritically]

I will look into the problem and see what can be done. I might make videos on other problems first.

[@taopaille-paille4992]

@@letsthinkcritically makes sense thanks a lot for the work

[@lucaseikipereirafujii3238]

Very good video, From Braszil

[@letsthinkcritically]

Thank you!!

[@ajaypanchal4153]

You are genius....👍🙏

[@letsthinkcritically]

That’s very kind of you 😄

[@subhajitroy5758]

In the last part of the solution how can you say the respective values in the right hand side is greater than equal to the left hand side.The coefficients are different for each if the variables.Can you explain in details?

[@Georgii1212]

a >= b >= c >= d > 0 to be used

[@subhajitroy5758]

@@Georgii1212 so where is the explanation man?give the explanation

[@Georgii1212]

@@subhajitroy5758 First we need to prove that a + 2b + 3c + 4d

[@CanadaMath]

At 6:58, you say that 4d < 3d. But how is that possible? I thought 4d > 3d. Since d is positive.

[@Singh_Gunjeet]

Notice 3b instead of 2b on RHS. Since b>=d, the inequality holds

[@chandranisahanone]

Too complex process but Genius. Easy way to do this by simple Assume method. The maximum value of a,b,c,d is ( a+b+c+d=1)=>[4a]=1=>a=1/4=b=c=d. Therefore, (a+2b+3c+4d)=a+2a+3a+4a=10a=>10×1/4=5/2. Now, max(a^a×b^b×c^c×d^d)=max(a^4a)=(1/4)^(4×1/4)=1/4. Therefore:- Maximum value of The given value(a+2b+3c+4d)×(a^a×b^b×c^c×d^d)=5/2×1/4=5/8. So by assuming yhe value of the given algebric expression we can easily say that , it is less than 1(as 5/80.

[@adityasinha146]

You are only considering the possibilities when a=b=c=d But there are possibilities too due to inequality you can't just prove any statement by showing it through some particular assumptions other than using mathmatical induction.

[@chandranisahanone]

@@adityasinha146 but their max is sum of that particular a,b,c,d expression! And that's the max value. Either it is a=b=c=d or a>=b>=c>=d.

[@adityasinha146]

@@chandranisahanone YEP you are right the function is continuous so its greatest value in domain will give its maxima what an approach bro amazing.

[@chandranisahanone]

@@adityasinha146 it's just a shortcut! There is no soln. Like this , so don't try this ! First u have to explain it briefly because sometimes the functions are discontinuous and for some reason peoples doesn't care at all and apply the shortcut tricks which approx them to wrong solutions! ❤

[@robyrogo1843]

how can a+2b+3c+4d

[@letsthinkcritically]

a >= b >= d So 2a+b >= 2d+d = 3d

[@multipontushd4626]

Why do you use "whole cubed"?! 5:41 :) Aha 1^3=1?

[@adamswilliam3462]

Fantastic 👌🏼🎉

[@letsthinkcritically]

Thank you!!

[@JoaovitorsvGamer]

a=b=c=d=(1/4) is a form to solve it??

[@walnutenterprises9159]

Sir could it be solved by log method and trivial proof method?

[@letsthinkcritically]

Taking logarithms might be a viable way, but I personally find AMGM more straightforward.

[@walnutenterprises9159]

@@letsthinkcritically sir can i show my proof on any platform? Gmail, whatsapp any

[@mathgoldmedalist]

Thanks for ur videos I've uploaded all imo problems of 2021 and 2020 and I will upload the past years soon I've also uploaded national olympiads of USA,UK,Switzerland,Belgium,Luxembourg,Netherlanda,Indonesia,....

[@aniruddhaupadhyay3967]

Power of being an Asian😂😂.. I can relate😂

[@piyush4830]

I think i have to try this with Tcheffby's inequality...

[@inqalaabzindabaad]

the max value of the left side is 5/8 which can be proved using method of lagranges multiplier. I was wondering why cant this method be used when one can get answer in one minute. Looking for expert comment.

[@yuseifudo6075]

IMO only require highschool level math

[@inqalaabzindabaad]

@@yuseifudo6075 the interested candidates knowledge is not limited to high school maths. They will explore mathematics beyond high school. Its all about curosity.

[@pradipbandyopadhyay9689]

So so elegant!!!

[@mathgoldmedalist]

👍👌

[@yingyang_.]

In 4.37, why we can substitute (a^a)(b^b)(c^c)(d^d) with a^2+b^2+c^2+d^2?

[@ogreeni]

Because we proved that a^2 + … + d^2 >= a^a … d^d using AM-GM.

[@djvalentedochp]

good video

[@letsthinkcritically]

Thank you very much!!

[@dimosthenisvallis3555]

God damn brilliant

[@letsthinkcritically]

Thanks a lot!!

[@umadutta3910]

Bro U just skipped the prove that if the terms, and I emphasize the word "terms", of this form : a^2(a+3b+3c+3d) are less than the terms of this form : a^2(a+2b+3c+4d) then the terms of form abc of RHS won't increase the overall value of RHS so that it will end up being greater than the LHS. I mean i didn't exactly get this to be very obvious so i believe this part might be crucial to the proof. If a follow up video can be made about this, that will be very very satisfying

[@anonimx3512]

Very good

[@letsthinkcritically]

Thank you!!

[@gmaonlinetutorial360]

How can I get the IMO past question

[@dkr1998]

nice

[@koreanlego]

I tried to solve this with log, 1:39 Because log(a+2b+3c+4d)

[@Deathranger999]

You need to prove that log(10a) + 4alog(a) < 0, not 1.

[@CUsGaming1121]

Wowww

[@mathgoldmedalist]

👌👌

[@littlefermat]

IMO 2020: P1) ... P2) Do you know the weighted AM-GM inequality? P3) .... 😂😂

[@mathgoldmedalist]

👍👌

[@tushroy81]

7:07 How could he say that RHS are greater though it is not mentioned that d

[@michaelma5480]

It’s given in the question that a>=b>=c>=d>0

[@theevilmathematician]

Did 2021 IMO happen yet? I'm excited to do the problems for this year!

[@chhabisarkar9057]

Not yet

[@ericzhu6620]

it'll be in July 14-24, just wait for a few months :)

[@theevilmathematician]

@@ericzhu6620 thank you for letting me know!

[@RandomGuy-yf4wf]

it did

[@shbit]

Can you explain the 4:34 inequality

[@supbro125]

Did you complete imo?

[@mathgoldmedalist]

👍👌

[@ИбадатЖұмабек]

weighted am-gm prove ??????

[@letsthinkcritically]

This is just another way to write AMGM. AMGM can be proved by induction on the number of terms doing the means.

[@ИбадатЖұмабек]

@@letsthinkcritically I want to see it

[@letsthinkcritically]

Maybe in the coming videos!

[@namikagic7804]

I think its Jensen' s inequality on f(x)=x*ln(x)

[@djvalentedochp]

you can see here en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means

[@saidiso4107]

Best

[@mathgoldmedalist]

👌👌

[@naveengantyala940]

From India..

[@mathgoldmedalist]

👌👌

[@pandawagurning]

How many problems that given for the contestants in this competition?

[@dickson3725]

6 question with 4.5×2 hour

[@pandawagurning]

@@dickson3725 ohh.....thanks🙏are they hard?

[@ravinderbhandari1974]

@@pandawagurning no, very easy.

[@NoNameAtAll2]

so it's a "do you know the trick?" type of question? reminds me of coffin problems...

[@alainrogez8485]

I am struggling a little bit because you used some stuff coming from nowhere.

[@mathgoldmedalist]

👍👌

[@Zarunias]

a+b+c+d=1 b+c+d

[@letsthinkcritically]

For the right side of the product, actually x^x is not strictly increasing. So even d < 1/4, it doesnt mean d^d < (1/4)^(1/4)

[@Zarunias]

There it is. Thank you :)

[@pittyconor2489]

x^x is decreasing (0, 1/e)

[@Rookie1706]

This test is easy.

[@王剛-m7n]

I can solve this problem by an ugly way using the 1st and 2nd differential of d(then of c, then of b), under the condition a,b are constants,and I can prove the 2nd differential of d is positive so the maximum value has to appears at the boundary of d: when d=(1-(a+b))/2 or d=0+, and when d=0+ the value is bigger than when d=(1-(a+b))/2, so the maximum value appears only when d=0+, so this problem decrease to find the values of a,b,c that max the value (a+2b+3c)aᵃbᵇcᶜ (d=0+,dᵈ=1)，similarly we can get only when c=0+, the value can be max,……，at last we get a=1-0,b=c=d=0+, that maximize this value, it =1×0+⁰⁺×0+⁰⁺×0+⁰⁺=1. here 0+⁰⁺=1 means the conclusion: when x→0+，xˣ→1（x=b,c,d）.

[@XavierXavier-ts2pe]

yup i did the same. ugly but no tricks

[@MRPESMOBILE]

i made it 1k like

[@kenichimori8533]

Solving k + k + k = 3 proof.

[@TechToppers]

I think there is a small fault... The problem states that a,b,c,d are Real numbers, not exactly positive... So the comparison around 7:00 which you did, doesn't seem right... Even if it is, you must show some more inequalities that hold true which would support your argument... What do you think...?

[@letsthinkcritically]

From the problem it says > 0, so I suppose they are positive?

[@TechToppers]

@@letsthinkcritically Oh yeah! My bad...

[@gustavocosta9574]

*Meme Alert* I'm brazilian and in my country the questions are harder than IMO. Nice video.

[@letsthinkcritically]

Thank you! Indeed, Brazil has always been one of the countries that performed very well in the IMO.

[@giovanicampos4120]

Ok, isso não quer dizer nada. Lembre-se que os problemas de olimpíadas como a IMO são de heurísticos para cima. E daí se os problemas nas olimpíadas de matemática do Brasil são mais difíceis? Quais as consequências reais disso?

[@gustavocosta9574]

@@giovanicampos4120 foi só uma piada cara, baixa a bola.

[@gyanprakashraj4062]

PROPERTIES OF TRIANGLE😂😂😂DUSRA COMBINATORICS...

[@andirijal9033]

international matematic covid 19

[@mathgoldmedalist]

👌👌

[@benYaakov]

At 4:28 Put a = 1 , b = 2 , c = 3 and d = 4 in the inequation . Clearly RHS is greater than LHS . So your solution is wrong

[@TechyMage]

Bro a+b+c+d=1, while selecting a value for the variable u should consider it.

[@benYaakov]

@@TechyMage nope

[@Deathranger999]

@@benYaakov Lol how can you just say no? It’s literally given in the problem that a + b + c + d = 1, so we work under that assumption the entire time.

[@TechyMage]

@@Deathranger999 he is an absolute genius, he can do whatever he want. Don't talk like that with him

[@Deathranger999]

@@TechyMage Right of course, how dare I.

[@mauricemoss1952]

As a sigma male, I applaud you. While all the betas are struggling, you read out a solution within seven minutes. Keep it up.

[@roboto12345]

I hate that problem

[@mathgoldmedalist]

👌👌