2004 IMO 第 1 题

A1 (KOR) ${ }^{\mathrm{IMO} 4}$ Let $n \geq 3$ be an integer and $t_{1}, t_{2}, \ldots, t_{n}$ positive real numbers such that $$n^{2}+1>\left(t_{1}+t_{2}+\cdots+t_{n}\right)\left(\frac{1}{t_{1}}+\frac{1}{t_{2}}+\cdots+\frac{1}{t_{n}}\right) .$$ Show that $t_{i}, t_{j}, t_{k}$ are the side lengths of a triangle for all $i, j, k$ with $1 \leq i<j<k \leq n$.

A1 (KOR) ${ }^{\mathrm{IMO} 4}$ 设 $n \geq 3$ 为整数，$t_{1}, t_{2}, \ldots, t_{n}$ 为正实数，使得 $$n^{2}+1>\left(t_{1}+t_{2}+\cdots+t_{n}\right)\left(\frac{1}{t_{1}}+\frac{1}{t_{2}}+\cdots+\frac{1}{t_{n}}\right) 。$$ 证明 $t_{i}, t_{j}, t_{k}$ 是所有 $i, j, k$ 的三角形边长，其中 $1 \leq i<j<k \leq n$。