Triangle Side Lengths

Let `n > 2` be an integer, and let ` t_1,t_2,...,t_n` be positive real numbers such that

`(t_1+t_2+...+t_n)(1/t_1 + 1/t_2 + ... + 1/t_n) < n^2+1`

Prove that for all i,j,k such that `1<=i<j<k<=n`, the triple of numbers `t_i,t_j,t_k` are the side lengths of a triangle.

Difficulty level (1 very easy - 10 very hard): 8

Topics:
Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality
Sources:
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