Question 182
Let $n \ge 3$ be an integer. Let $t_1, t_2, \dots, t_n$ be positive real numbers such that
$$
n^2 + 1 \gt (t_1+t_2+\dots+t_n)\left(\frac{1}{t_1}+\frac{1}{t_2}+\dots+\frac{1}{t_n}\right).
$$
Show that $t_i, t_j, t_k$ are the lengths of the sides of a triangle for all $i, j, k$ with $1 \le i \lt j \lt k \le n$.
$$
n^2 + 1 \gt (t_1+t_2+\dots+t_n)\left(\frac{1}{t_1}+\frac{1}{t_2}+\dots+\frac{1}{t_n}\right).
$$
Show that $t_i, t_j, t_k$ are the lengths of the sides of a triangle for all $i, j, k$ with $1 \le i \lt j \lt k \le n$.