The Best International Math Olympiad (IMO) Algebra Problem I've Ever Solved

[@ronbannon]

Graphing the given equation will give you a line m+n-33=0 and a point (-33, -33). This is a very nice problem!

[@tsur.livney]

Brilliant! I solved it very differently: First i set a=33 for convenience and so the equation became: A^3 = m^3+n^3+3amn. I substituted a = m+n+delta on both sides, so i got: (m+n+delta)^3 = m^3 + n^3 + 3mn(m+n+delta). Opening the brackets and subtracting both sides by the lhs and rhs of the original equation, all terms left had delta in them so delta=0 derives the set of m+n=33 solutions. Dividing by delta we're left with a quadratic equation whose determinant is -3(m-n)^2, which eliminates all solutions where m is not equal to n. Now substituting m=n in our original equation, we are left with: A^3=2n^3+3an. Now using the same trick: setting this time a=delta-n, simplifying we get: Delta^2*(delta-3n) = 0. So either delta is 0 which implies a=-n=-m which gives the last solution, or a=2n=2m which means m+n=a which is already covered in previous solutions. Overall, for a general non zero value of a: Either m=n=-a - one solution, or m+n=a, which implies all solutions (i, a-i) where min(a,0)

[@adityavsx]

you have a talent to explain complex sums so easily…gives me a lot of motiv:)

[@MrGLA-zs8xt]

I agree with you

[@iqtrainer]

Great work as always professor🎉

[@MrGLA-zs8xt]

What a brilliant and nice video professor

[@alexkaralekas4060]

Or just use from the start one of euler's identity a^3+b^3+c^3-3abc= {(a+b+c)[(a-b)^2+(b-c)^2+(c-a)^2]}/2 For a=m b=n c=-33

[@Min-cv7nt]

Isn't that vieta's formula?

[@alexkaralekas4060]

@Min-cv7nt i learned it as a factoring method for a^3+b^3+c^3 my book said it was from euler but I couldn't find it on the internet

[@ronbannon]

That is a most excellent idea! Thanks for pointing this out.

[@mathnerd5647]

This is such a nice video

[@Min-cv7nt]

You are the best professor

[@domedebali632]

What a brilliant video

[@redroach401]

Can you do ploynomial divison after you find the factor m+n-33 or no?

[@ronbannon]

Yes, that's what I did.

[@mathnerd5647]

I also did this too