OMM 2011 Problem 1

$25$ lightbulbs are distributed in the following way: the first $24$ are placed on a circumference, placing a bulb at each vertex of a regular $24$ -gon, and the remaining bulb is placed on the center of said circumference. At any time, the following operations may be applied:

• Take two vertices on the circumference with an odd amount of vertices between them, and change the state of the bulbs on those vertices and the center bulb.
• Take three vertices on the circumference that form an equilateral triangle, change the state of the bulbs on those vertices and the center bulb.

Prove from any starting configuration of on and off lightbulbs, it is always possible to reach a configuration where all the bulbs are on.