Find two similar triangles with sides (A,B,C) and (A,B,D), such that D-C=20141.

It's clear that in the first triangle, C must be the smallest side, so
that C≤A≤B; the second triangle must have A≤B≤D, and C/A=A/B=D/B.

If we call C/A=R, then A=CR, B=CR² and D=CR³, so D-C=C(R³-1)=20141.

We
can write C(R³-1)=C(R-1)(R²+R+1)=11x1831, so now we should try to
equate terms... and that's as far as I got when I realized I would also
have to prove that R was an integer.