IMO 1978 LL FRA18

Given a natural number n, prove that the number M(n) of

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IMO 1978 LL FRA18

Origin: FRA

Problem

Given a natural number n, prove that the number M(n) of points with integer coordinates inside the circle (O(0, 0), \sqrtn) satisfies \pin −5\sqrtn + 1 < M(n) < \pin + 4\sqrtn + 1.