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Question 26

The number of integer solutions $$(x, y)$$ of the inequality $$x^2 + y^2 \leq 10$$ is

Solution

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For each integer $$x$$ with $$x^2 \leq 10$$ (i.e., $$|x| \leq 3$$), count $$y$$ with $$y^2 \leq 10 - x^2$$:

  • $$x = 0$$: $$y^2 \leq 10$$, $$y \in [-3, 3]$$ → 7 values.
  • $$x = \pm 1$$: $$y^2 \leq 9$$, 7 each → 14.
  • $$x = \pm 2$$: $$y^2 \leq 6$$, $$y \in [-2, 2]$$ → 5 each → 10.
  • $$x = \pm 3$$: $$y^2 \leq 1$$, $$y \in [-1, 1]$$ → 3 each → 6.

Total = $$7 + 14 + 10 + 6 = \mathbf{37}$$.

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