JBMO 2005 Problem 1

Find all positive integers $x,y$ satisfying the equation $9(x^2+y^2+1) + 2(3xy+2) = 2005 .$

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.