The Tangent Trick for Olympiad Inequalities

[@HarshSharma-th8yt]

I am going to give the Indian National Mathematical Olympiad exam on 7th March, 2021. And this trick will really help me alot to solve the inequality questions. Thank you sir..........

[@ProfOmarMath]

I hope it goes well!!

[@chandrisingh1056]

How did it go bro?

[@chandrisingh1056]

Can ya tell me date of prmo this year?

[@HarshSharma-th8yt]

@@chandrisingh1056 not that good as i have assumed it would be........... And registerstion for previous year was scheduled in November due to corona epidemic. But i think this year, MTAI will release registration dates soon. Stay updated and connected with the official website of MTAI Association.

[@ProfOmarMath]

@@chandrisingh1056 Hi Chandri. I appreciate your comments. I just wanna let you know that I prefer keeping gendered language away from the channel. Terms like "bro" might make people feel excluded.

[@mohammedsalouani7672]

I m happy too for ur back !

[@ProfOmarMath]

Thanks Mohammed!

[@akshaypanwar2393]

Once again a great video from one of my favorite mentors 🙏🙏 Welcome back sir .

[@ProfOmarMath]

Thanks Akshay! Any requests or curiosities?

[@akshaypanwar2393]

@@ProfOmarMath Not yet sir, I will tell you if there will be any query.

[@armanrasouli2779]

@@ProfOmarMath I am not able to solve this system of differential equations so I really appreciate it if you take a look at it: yy'=1 and y'y''=1

[@debayuchakraborti1963]

SUCH A NICE TRICK!! THANK YOU SIR!!!

[@ProfOmarMath]

It really is. Thanks Debayu!

[@yoyokojo651]

whatching the new Prof Omar vid instead of studying my course work in abstract algebra >>> Welcome back!

[@ProfOmarMath]

😂 Thanks! What topic are you on in that class?

[@yoyokojo651]

@@ProfOmarMath We previously looked at fields of fractions (extensions of ID's) and then moved on to Polynomial rings and factorisations in them and the last thing we talked about was kriteria for the ireducibilty of polynomials (Eisenstein). My Prof said that after the holidays he plans to move to algebraic Field extensions eventually leading to Galois theory. Although this course has by far been the most challenging/time consuming for me (I really do struggle at times), and i intially had doubts about taking it in my first year at college, It has been really rewarding so far and really ties in well with some of the other more basic classes im taking such as linear algebra. Algebra has shown me some really strong tools that i have been able to Apply in a lot of other areas which makes the heavy theory worth while. :)

[@ProfOmarMath]

That's nice material. It's a jump in abstraction but it's great exposure to nice general perspectives

[@3gp]

Your videos are really impressive.

[@ProfOmarMath]

Lee thanks so much! I'm excited you're here.

[@random-td7tf]

That’s really cool ! Please make a video on (n-1)EV !

[@haitambentayebi7804]

there is the evan chen video i think it's pretty instructive

[@mcqueen424]

Glad to have you back!!

[@ProfOmarMath]

Thanks!!

[@chhabisarkar9057]

That's such a great method !

[@ProfOmarMath]

It's really nice!

[@raghualluri4245]

Great to have you back! I have been preparing for the 2021 Putnam in February eventhough it is unofficial. It will still gauge my abilities and see how I do. Is 30 a above decent score for the Putnam?

[@ProfOmarMath]

Hi. Thanks for being here? I'm preparing some Putnam stuff so if you have questions, please share. The score really depends on what your perspective is and the distribution varies wildly.

[@raghualluri4245]

@@ProfOmarMath well, is getting top 500 considered really good? What about top 200?

[@ProfOmarMath]

@@raghualluri4245 Unfortunately I don't think there's an absolute answer of any kind. I guess I'll ask a more direct question: why do you want to perform well on it? That'll give me a better idea of how to approach the question.

[@raghualluri4245]

@@ProfOmarMath Because I love problem solving and I want to showcase as well as enhance my abilities.

[@ProfOmarMath]

@@raghualluri4245 Ok great then I would actually focus less on the goal of obtaining a certain score and much more on enhancing the skill because that will give you more joy and more clout at the end of the day. The Putnam has some patterns I've noticed too... I'll be posting a video about this soon!

[@avyakthaachar2.718]

Amazing method! Thank you so much 🙏

[@jesusalej1]

Glad to see you Profesor, go on with math.!

[@ProfOmarMath]

Thanks Jesus, excited to see you!

[@Szynkaa]

very cool trick, gotta test it on my own

[@armanrasouli2779]

WELCOME BACK SIR.

[@ProfOmarMath]

Thanks Arman!

[@王剛-m7n]

Very clear and inspiring and useful and broad

[@ProfOmarMath]

Thanks!

[@rounaksinha5309]

Hii ProfOmar. Love from India. Sir, please make a video on proving the various properties of floor function (Greatest Integer function).

[@ProfOmarMath]

Hi EduHub, thanks! What properties in particular are you interested in the most?

[@rounaksinha5309]

Thanks sir drive.google.com/file/d/1PYFtVppY5N_p81c7QjOIHKklmnipIFtL/view?usp=sharing I am interested in the property (x) and (xi) . I have shared a document in which these properties are mentioned.

[@rounaksinha5309]

And sir thank u for showing ur interest . sorry for poor english

[@ProfOmarMath]

It says the document requires access. Would you be willing to share the ideas here?

[@rounaksinha5309]

@@ProfOmarMathSure sir. One is Floor(x)+floor(x+1/n)+floor(x +2/n)+ .........+floor(x+n-1/n) =floor(nx ) And another is floor(x+y)=floor(x)+ floor(y+x-floor(×))

[@krstev29]

Very assume, but I kinda fail to understand why it works, and how did you get x=1/4 3:52

[@bobmccheese1670]

How did you decide on the point x=1/4 to take derivative on ?

[@chamsderreche5750]

Because he noticed that when a=b=c=d=1/4 there's an inequality case, so he "guessed" that the tangent on 1/4 will be below the curve

[@blank7771]

@@chamsderreche5750 does the tangent trick only work below the curve ? what is the significance of the placing of the tangent

[@chamsderreche5750]

@@blank7771 if the tangent is above the curve the inequality is flipped

[@assilboukhennoufa]

Where can i find the first question

[@thedoublehelix5661]

Welcome back!!!

[@ProfOmarMath]

Thanks!

[@anjanamajumdar2981]

Solution of 2nd Problem. x,y,z are defined as in the video. Let f(x)=(1+x)^2/(2x^2+(1-x)^2). df/dx at x=1/3 is 4. Claim-(1+x)^2/(2x^2+(1-x)^2)=0.This is what we need to prove. Factor it to get, (3x-1)^2(4x+1)>=0, which is true for 0

[@Stelios2711]

Aha, I see. You used the tangent trick to "guess" the inequality that one might need to prove to "kill" the problem. Nice one for those who are not familiar with convexity (which immediately implies that the tangent line will be "below" the curve, but requires double differentiation).

[@ProfOmarMath]

Ya that's exactly it, without knowing about convexity it's tricky

[@midnightposting]

Global convexity (in the interval [0,1] in this case) will imply that the tangent line is below. But 6x^3-x^2 is not convex in all the interval so you have to do the independent proof.

[@Stelios2711]

@@midnightposting in this particular example, it is true that you cannot use convexity. However, I was talking in general. Finally, even if we don't have convexity in a whole interval, we still have chances of proving an inequality in the whole interval by finding a single inflection point and using the (n-1)-equal-value theorem. brilliant.org/discussions/thread/inflection-point-inequality-theorem/

[@blank7771]

I understand convexity but what is the significance of the tangent placement, why do we need it below the curve

[@gmncnr]

Really helpful

[@destruidor3003]

How do you replace 4 variables by one?

[@psrakhilesh2102]

very impressive, great video

[@ProfOmarMath]

Thanks!

[@joseluishablutzelaceijas928]

Isn't there a typo in the USA-Problem? I think, the denominator of the second summand should be 2*b^2+(c+a)^2.

[@ProfOmarMath]

Jose that's right thanks!

[@joseluishablutzelaceijas928]

@@ProfOmarMath Thank you very much for the shown technique, I think it can be quite useful. Somehow, it did not belong to the first approaches I would have thought of (until now).

[@ProfOmarMath]

Definitely! It's a very interesting approach that is seldom shared

[@elnura5132]

I have learned a lot I appreciate it.

[@ProfOmarMath]

I’m happy you did!

[@DragonKidPlaysMC]

Can you prove this using the newton-girard identities?

[@ProfOmarMath]

Hi Drag. That's a great idea, I'll do it.

[@krstev29]

My only question is, what a do we fix, since as i see, it's not always the arithmetic mean

[@physfaraz]

Hi Prof! Are you on AoPS?

[@ProfOmarMath]

Hi Faraz. Surprisingly I am not! Are you?

[@nawusayipsunam1643]

Very useful.

[@ProfOmarMath]

Definitely! Thanks for watching Nawus

[@kerenelbaz2607]

the tangent at 1/3 is 4x+8/6 which makes the sum of x y and z be 8!!

[@ProfOmarMath]

Isn't it excellent!

[@yoav613]

it is!! keep up with your great videos

[@nadiabouhlel5827]

lets assume without lost of generality that a= 1/8 using chebychev another time we get (a^3+b^3+c^3+d^3)>1/4(a^2+b^2+c^2+d^2)(a+b+c+d) and using chebychev (a^2+b^2+c^2+d^2)>=1/4(a+b+c+d)(a+b+c+d) then (a^3+b^3+c^3+d^3)>=1/16 then 2(a^3+b^3+c^3+d^3)>= 1/8

[@prienietisAugis]

Nice solution!

[@djvalentedochp]

I have a lot to learn

[@ProfOmarMath]

Me too. Lifelong learning is the best!

[@adityaraajsingh4322]

I did not het how y=5x-1/8

[@yashvardhan2093]

First view and comment

[@destruidor3003]

It is not clear why you replaced the original variables by X. You need to be more didactic…

[@nadiabouhlel5827]

we can just do it with chebychev s enequality

[@ProfOmarMath]

Great idea Nadia. Please share!

[@random-td7tf]

These problems are from mathematical Excalibur lol

[@mihaipuiu6231]

Maybe you are a genius, maybe...but : 1....I don't need your video on , b/c this screen is deja small,...and I need your proof ..... for me,...so as student, I wish you to presint me... a clear write.. 2. Because,...I suppose, your proof is not for Gauss , ..or Ramanujan, but for me,... so...please do your dem. with careful letters and digits, b/c the goal is for me,as begginer to understand your demonstration. Your cariocas is very ugly and confuse, ...so please try to use a pen. If you don't respect my desire, I'll try to avoid your dem, because my time is important for me. Do you understand? My desire it's a gisheft for both.Thanks. PS: sorry for my english gramarly mistakes.

[@piiscongruentto1modk]

Tf