Disclaimer: I haven't actually tried submitting this.

First, you should know about the triangle inequality.

Clearly, testing all triples is too slow, so we need a faster way. First, can you solve the problem for a single range in $$$O(n\log{n})$$$?

Back to the original problem:

This should lead to an $$$O(n\log{n}\log{(\max{A_i})})$$$ solution.

Lemma: If you have the sorted sequence of $$$a[l,r]$$$. If there is an answer then it will be contiguous three elements in the sorted sequence.

Start with retrieving elements from $$$[l,r]$$$ in decreasing order. If $$$1^{st}, 2^{nd}, 3^{rd}$$$ elements doesn't satisfy triangle inequality then we know $$$3^{rd}$$$ element is $$$<1^{st}/2$$$. So we just have to retrieve $$$O(lg(A_{MAX}))$$$ elements per query and we will find the answer among them.