Problem 4

Let

n\ge 3

be an integer and

t_1,t_2,\dots,t_n

positive real numbers such that

n^2+1 > (t_1+t_2+\cdots+t_n)\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 lengths of the sides of a triangle for all

i,j,k

with

1\le i<j<k\le n

Step 2 of 4: Bound the non-triple pairs by AM–GM

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\frac{t_r}{t_s}+\frac{t_s}{t_r}\ge2\text{ for the }\binom{n}{2}-3\text{ pairs outside }\{1,2,3\}\implies \left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)<7

Detailed analysis

For each of the $\binom{n}{2}-3$ pairs $(r,s)$ not contained in $\{1,2,3\}$ , AM–GM gives $t_r/t_s+t_s/t_r\ge 2$ , so those pairs contribute at least $2\bigl(\binom{n}{2}-3\bigr)=n(n-1)-6$ . Subtracting this and the diagonal $n$ from $n^2+1$ leaves $(a/b+b/a)+(b/c+c/b)+(c/a+a/c)<7$ for the three pairs inside $\{1,2,3\}$ .