This problem is inspired by an IMO (don't know which year) and was the inspiration for the previous problem.
Anyway. Here goes:
Suppose $t_1, t_2, \dots, t_n$ are $n \ge 3$ positive real numbers such that
$$ (t_1 + t_2 + \dots + t_n)\left(\frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n}\right) \lt n^2 + 1$$
Show that:
Problem 1) For any distinct $i, j, k$ the numbers $t_i, t_j, t_k$ form the sides of some triangle. [This is the original IMO problem]
Problem 2) If $n \ge 13$, show that the triangles are all acute.
Problem 3) If $n \le 12$, the triangles are not necessarily acute.
Anyway. Here goes:
Suppose $t_1, t_2, \dots, t_n$ are $n \ge 3$ positive real numbers such that
$$ (t_1 + t_2 + \dots + t_n)\left(\frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n}\right) \lt n^2 + 1$$
Show that:
Problem 1) For any distinct $i, j, k$ the numbers $t_i, t_j, t_k$ form the sides of some triangle. [This is the original IMO problem]
Problem 2) If $n \ge 13$, show that the triangles are all acute.
Problem 3) If $n \le 12$, the triangles are not necessarily acute.
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