ELMO SL 2013 N1

Find all ordered triples of non-negative integers $(a,b,c)$ such that $a^2+2b+c$ , $b^2+2c+a$ , and $c^2+2a+b$ are all perfect squares.

Proposed by Matthew Babbitt

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.