Le Monde puzzle [52|solution]

I have now received the first issue of Le Monde magazine, including the solution to puzzle #52 I solved just in time by simulated annealing! The trick is in using the following theorem:

Iter(1,x,y) is divisible by 10x-1 if and only if y is divisible by 10x-1.

Then the value to be found is divisible by 3, but not by 9, by 19, by 29, by 11, by 3 and 7, and by 31. It is thus a multiple of 3x19x29x11x7x31=3,945,711 and only this number satisfies the constraint on Iter(6,1,y).

The first puzzle of the year is as follows:

Select J>999 different numbers i_j between 1 and 2001 (1≤j≤J) such that (a) if a and b are selected then a+b is selected (assuming a+b<2002) and (b) the residual

is maximised.