UA-UMO 2005 Problem 85

Are there integers $a,b,c,d,x,y,z,t$ such that each of the numbers:

$|ay-bx|,|az-cx|,|at-dx|,|bz-cy|,|bt-dy|,|ct-dz|$

equals either $1$ or $2005$ ?

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.