Problem
COM-B1-M06-P013 Two Thousand Twenty-Five Lamps
#13
★★★☆☆ Level 3 of 5
There are \(2025\) lamps switched off. In one move, exactly \(100\) lamps may be toggled. Can all lamps be switched on?
The parity of the number of switched-on lamps is preserved.
If \(h\) of the chosen lamps are on, the number of on lamps changes by \(100-2h\), an even number. Initially \(0\) lamps are on, even; finally \(2025\) would be on, odd. Impossible.
Large numbers, simple invariant.