Competition problem — Number theory

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Number theory Difficulty 5.6 AIME, harder Prove it

(10 Given positive integers $a_{1}, a_{2}, \cdots, a_{n}$ are all composite, and pairwise coprime, prove:
$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}<\frac{1}{2}$

Solution

10 Let $p_{k}$ be the smallest prime divisor of $a_{k}$ , since $a_{k}$ is not a prime, then $a_{k} \geqslant p_{k}^{2}$ . Therefore,
$\begin{aligned} \sum_{k=1}^{n} \frac{1}{a_{k}} & \leqslant \sum_{k=1}^{n} \frac{1}{p_{k}^{2}} \\ & \leqslant \frac{1}{4}+\sum_{k=2}^{n} \frac{1}{(2 k-1)^{2}} \\ & \leqslant \frac{1}{4}+\sum_{k=2}^{n} \frac{1}{(2 k-1)^{2}-1} \\ & =\frac{1}{2}-\frac{1}{4 n}<\frac{1}{2} . \end{aligned}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.