Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4

[mfb]

What an innocently looking equation.
If we allow negative integers, a=4, b=-1, c=11 is a solution.

Do some tricks with divisibility?
Solve for a?
Brute force with the computer?It won't help. There are solutions, but the smallest solution has 80 digits.

What happens if we replace 4 by other integers?
a/(b+c)+b/(a+c)+c/(a+b)=178?
There are integer solutions, but the smallest one has nearly 400 million digits.

A great example how simple looking problems can have very complicated solutions.

Paper: An unusual cubic representation problem (PDF)
Here is a discussion

 

[fresh_42]

A great example how simple looking problems can have very complicated solutions.

I thought Andrew had already demonstrated it.