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