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Brilliant problem clarification

Hello all, I have a little problem. The problem is deal with the current Brilliant problem about number theory.

Here's the link: https://brilliant.org/community-problem/mind-your-3s-and-2s/?group=69VNhEAzD53X

My question is Should m and n always different, or are they can be same? Thanks

Note by Leonardo Chandra
3 years, 9 months ago

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The integers m and n can be the same. The answer the website thinks is correct is actually wrong, so if you got the right answer and the website said that you were incorrect, that is why. Hopefully, the staff will fix this soon.

Jon Haussmann - 3 years, 9 months ago

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Thanks for your reply. I've just got the answer, but it's wrong. Jon, I wonder how can you solve this problem: https://brilliant.org/mathematics-problem/let-our-powers-combine/?group=vAjLywc9ZnIz,

I have already used all the try: http://tinypic.com/r/2dhcvtx/5

But, cannot get the correct answer. Maybe you can try to send me your solution via my e-mail? My e-mail: leonardochandra@hotmail.com. Thanks

Leonardo Chandra - 3 years, 9 months ago

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I don't really have a solution, but I can give you some idea of how I approached the problem.

Let \(y_i = x_i^6\), so \(x_i = y_i^{1/6}\) and \[y_1 + y_2 + \dots + y_n = n.\] Also, let \(f(y) = y^{5/6} - y^{1/3}\), and let \[S = \sum_{i = 1}^n f(y_i).\] Then the problem is to find all \(n\) for which \(S\) is always nonnegative.

It took a lot of work and persistence to even find an \(n\) for which \(S\) could be negative. The results I was getting suggested looking at an example where one of the \(y_i\) was relatively "large," and the remaining \(y_i\) were "small." Once I found an example that worked, the rest was just refining the example to get the answer.

Beyond that, there's not much more I can say, except that the answer is much bigger than 2. I can only recommend being tenacious, and keeping things simple. That's what I did.

Jon Haussmann - 3 years, 9 months ago

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