This problem is from an Olympiad handout:
Show that there exists infinitely many integer triplets $(x,y,z)$ such that $ x^3+y^3+z^3-2xyz=1$.
I tried to plug $x=y$ and use tangent lines to find solutions inductively, but the method didn't work well. (It gave $(1,1,1) \rightarrow (13,13,-23)$ ,but after that there was only rational roots)
Also I plugged the equation in the cubic formula and tried to delete the cubic root, but it also failed.
Help me please :I