APMO 1989

Discussion on Asian Pacific Mathematical Olympiad (APMO)
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nafistiham
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APMO 1989

Unread post by nafistiham » Mon Apr 09, 2012 9:16 pm

How to crack this one ?

Prove that the equation
\[6(6a^2 + 3b^2 + c^2) = 5n^2\]
has no solutions in integers except $a = b = c = n = 0$.
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
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nayel
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Re: APMO 1989

Unread post by nayel » Mon Apr 09, 2012 9:43 pm

The first thing that came to my mind after seeing $3,6$ and the squares is
infinite descent/extreme principle/consider minimum solution - whatever you want to call it.
It should work I believe.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

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samiul_samin
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Re: APMO 1989

Unread post by samiul_samin » Mon Feb 04, 2019 6:28 pm

nafistiham wrote:
Mon Apr 09, 2012 9:16 pm
How to crack this one ?

Prove that the equation
\[6(6a^2 + 3b^2 + c^2) = 5n^2\]
has no solutions in integers except $a = b = c = n = 0$.
Solved here.

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