How many natural numbers lie strictly between $9^{2}$ and $10^{2}$?

How many natural numbers lie strictly between $9^{2}$ and $10^{2}$?
  1. $17$
  2. $18$
  3. $19$
  4. $20$

Solution

Between $n^{2}$ and $(n+1)^{2}$ there are $2n$ natural numbers. For $n = 9$: $2 \times 9 = 18$.

Asked in: IMO

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