If each exterior angle of a regular polygon is $1/4$ of its interior angle, the polygon has:

If each exterior angle of a regular polygon is $1/4$ of its interior angle, the polygon has:
  1. $8$ sides
  2. $10$ sides
  3. $12$ sides
  4. $15$ sides

Solution

Let exterior $= x$, interior $= 4x$. Then $x + 4x = 180 \Rightarrow x = 36^{\circ}$. So $n = \dfrac{360}{36} = 10$.

Asked in: IMO

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