IMO Shortlist 2016 problem A5


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Dodao/la: arhiva
3. listopada 2019.
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Consider fractions \frac{a}{b} where a and b are positive integers.

(a) Prove that for every positive integer n, there exists such a fraction \frac{a}{b} such that \sqrt{n} \le \frac{a}{b} \le \sqrt{n+1} and b \le \sqrt{n}+1.

(b) Show that there are infinitely many positive integers n such that no such fraction \frac{a}{b} satisfies \sqrt{n} \le \frac{a}{b} \le \sqrt{n+1} and b \le \sqrt{n}.

Izvor: https://www.imo-official.org/problems/IMO2016SL.pdf