All non-trivial variants of 3-LDT are equivalent

The popular 3-SUM conjecture states that there is no strongly subquadratic time algorithm for checking if a given set of integers contains three distinct elements that sum up to zero. A closely related problem is to check if a given set of integers contains distinct x 1, x 2, x 3 such that x 1+x 2=2x 3. This can be reduced to 3-SUM in almost-linear time, but surprisingly a reverse reduction establishing 3-SUM hardness was not known.