A polygon has $20$ diagonals. How many sides does it have?

A polygon has $20$ diagonals. How many sides does it have?
  1. $6$
  2. $7$
  3. $8$
  4. $10$

Solution

Solve $\dfrac{n(n-3)}{2} = 20 \Rightarrow n(n-3) = 40 \Rightarrow n^{2}-3n-40 = 0 \Rightarrow (n-8)(n+5) = 0$. Hence $n = 8$.

Asked in: IMO

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