It All Came From Pencils 3

Geometry Level 3

We can make "pencilogons" by aligning multiple, identical pencils end-of-tip to start-of-tip together without leaving any gaps, as shown above, so that the enclosed area forms a regular polygon (the example above left is an 8-pencilogon).

Hazri wants to make an \(n\)-pencilogon using \(n\) identical pencils with pencil tips of angle \(7^\circ.\) After he aligns \(n-18\) pencils, he finds out the gap between the two ends is too small to fit in another pencil.

So, in order to complete the pencilogon, he has to sharpen all the \(n\) pencils so that the angle of all the pencil tips becomes \((7-m)^\circ\).

Find the value of \(m+n\).

(Assume the pencils have a rectangular body and have their tips resembling isosceles triangles)


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