Solving the Legendary IMO Problem 6 in 8 minutes | International Mathematical Olympiad 1988

#IMO #IMO1988 #MathOlympiad

Here is the solution to the Legendary Problem 6 of IMO 1988!!

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I share Maths problems and Maths topics from well-known contests, exams and also from viewers around the world. Apart from sharing solutions to these problems, I also share my intuitions and first thoughts when I tried to solve these problems.

[GothicKin]

The critical part is A1 is not 0 because B^2 - k can't vanish. Blink and you miss it, still a great job laying out the proof!

[ignaciobenjamingarridoboba]

This solve is brilliant, always you assume there is a minimum (a, b) when the equation is not a perfect square, you'll always find out a smaller number than the minimum

[buxeessingh]

Remember: even Terry Tao did not find a complete proof to this question.

[kephalopod]

I saw somewhere that this problem is a particular case of the nice generalization (much harder to prove):
Let a, b positive integers. Prove that
if (ab)^(n-1) + 1 | a^n + b^n, then
(a^n + b^n) / ((ab)^(n-1) + 1) is a perfect n-th power.

[pbj]

5:47 I would like to add a few more steps here so that the jump may become clearer for those who didn't get it. A1B+1>0
=> A1B > -1
As A1 was proved to be an integer and we know B is an integer as well (natural to be more specific), A1*B must also be integral. Hence A1B can be 0 at minimum as that is the next integer after 0
=> A1B >= 0
A1 was shown not to be 0 and B is natural so it can't be 0. Hence since both A1 and B are non-zero, their product must also be non-zero and therefore it can only be >0
=> A1*B > 0
Now since B is positive by virtue of being natural, A1 must also be positive. QED

[ratulee]

It can be solved with elementary number theory without Vieta jumping. If you modify the rest of the theorem a little bit.
The size of b is between ak-a and ak.
Set b = ak - r (0<=r<a) and solve it.

[PranavGarg_]

This was the second video I watched from this channel and it was a good understandable solution. Just subscribed.

[danielontheedge]

It's quite hard to find examples of a and b that satisfy this... One example is (1, 1)

[ipbmxpq]

Thanks, author. Please make more content like that(I'm the Russian olympiad participant)

[littlefermat]

The problem that you'll see in every NT book for math Olympiad.

[kayson]

I remember seeing this problem in one of my math sessions disguised as a harmless question

And the whole class was struggling to solve it

[ren]

struggled with the contradiction a bit in the end. the trick is in order for this not to contradict, B^2 must equal k. nice trick!

[Luarhackererreape]

Excelente bro! me gusta que pongas los subtítulos en español! Ganaste un suscriptor :)

[mohamedazizghorbel]

Thank you for such a nice work, all my Support ❤️

[webtoon]

Thank you very much. this idea is very helpful for my problem that ask me to prove for positve x, y if (x²+y²+6)/xy integer then it must be perfect cubes

[ary]

This channel will get 1 million by December 2021

[MrCarlosmario]

Exelente razonamiento. Muchas Gracias.

[ankitkumar-pwpu]

Sir I don't understand any thing what should I do to understand this solution I mean any basic available

[theevilmathematician]

Very interesting number theory problem.

[jasonleelawlight]

This method is very interesting! I didn’t expect this setup could lead to a solution. I need to watch again to better evaluate what role “being a perfect square” plays in solving this problem.