KöMaL 2012 K326

An interior designer is planning the illumination of a large lecture hall. He is using LEDs (light emitting diodes) arranged in concentric circles. The LEDs are uniformly spaced along each circle. The radius of each circle is the double of the previous circle. If lines are drawn from each lead of a circle through the centre, only every other line will have a LED on the next circle inside. $a$) Show that the separation of two consecutive LEDs on the same circle, as measured along the circle, is constant (that is, independent of the circle selected). $b$) Determine this distance (measured along the circle) if the radius of the largest circle is 20 metres, the number of circles is 8, and the 4th smallest circle contains 112 LEDs. $c$) What is the total number of LEDs used?