RO-NMO 2010 Problem 82

How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$ ?

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.