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Smallest prime larger than n and larger prime greater than n

Source: Romanian District Olympiad 2006, Grade 9, Problem 4

March 11, 2006
inequalitiesalgebra proposedalgebra

Problem Statement

For each positive integer n≥2n\geq 2 we denote with p(n)p(n) the largest prime number less than or equal to nn, and with q(n)q(n) the smallest prime number larger than nn. Prove that ∑k=2n1p(k)q(k)<12. \sum^n_{k=2} \frac 1{p(k)q(k)} < \frac 12.
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