There are $n$ lamps and $k$ switches in a room. Initially, each lamp is either turned on or turned off. Each lamp is connected by a wire with $2020$ switches. Switching a switch changes the state of a lamp, that is connected to it, to the opposite state. It is known that one can switch the switches so that all lamps will be turned on. Prove, that it is possible to achieve the same result by switching the switches no more than $ \left \lfloor \dfrac{k}{2} \right \rfloor$ times. Proposed by T. Zimanov