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2023 India Regional Mathematical Olympiad, 5
Tags: algebra , ravi s substitution , perimeter , geometry
The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\ \[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\] for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$.