Pacific Asian 2007

When a lamp in a $5 \times 5$ grid is turned on (or off), its neighbors located in the same row or column also change their state. Initially, all lamps are off. A monkey arrives and plays with the switches. In the end, only one lamp is on. Determine its (their) possible position(s).
P'tit invariant: the parity of the number of lamps turned off on the 1st, 3rd, and 5th columns that are not on the 3rd row. The same applies to the rows. The only candidates are then the lamps in positions $(2,2),(2,4),(4,2),(4,4)$ and $(3,3)$. By symmetry, it suffices to find a combination leaving only $(2,2)$ lit, and one with $(3,3)$. Have fun with the constructions...