## APMO 1989

Discussion on Asian Pacific Mathematical Olympiad (APMO)
nafistiham
Posts: 829
Joined: Mon Oct 17, 2011 3:56 pm
Location: 24.758613,90.400161
Contact:

### APMO 1989

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.

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Re: APMO 1989

The first thing that came to my mind after seeing $3,6$ and the squares is
It should work I believe.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

samiul_samin
Posts: 999
Joined: Sat Dec 09, 2017 1:32 pm

### Re: APMO 1989

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.