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1/2 ( N / a +a ) >= \sqrt{N}

Source: Netherlands - Dutch NMO 1968 p2

January 31, 2023
algebrainequalities

Problem Statement

It holds: N,a>0N,a > 0. Prove that 12(Na+a)≥N\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}, and if N≥1N \ge 1 and a=[N]a = [\sqrt{N}]. Prove that if a≠N:12(Na+a)a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right) is a better approximation for N\sqrt{N} than aa.
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