They need to all be positive. FTA (on the solution x = 4, y = -1, y = 11) :

> This solution is not easy to see by hand, but it’s also not hard to discover with some patience without all the machinery we are reviewing here. It’s the positive solutions that are the lair of dragons.

The question asks for positive solutions, which turns out to be a bit harder!

Sorry...I was only going off of the title of the thread.

~$ calc

calc 2.12.4.1

> a=154476802108746166441951315019919837485664325669565431700026634898253202035277999

> b=36875131794129999827197811565225474825492979968971970996283137471637224634055579

> c=4373612677928697257861252602371390152816537558161613618621437993378423467772036

> a/(b+c) + b/(a+c) + c/(a+b) 4

> a/(b+c) ~3.74500615923925922050

> b/(a+c) ~0.23213745990937924275

> c/(a+b) ~0.02285638085136153676