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Weird sum obtains min value [Cyprus IMO TST 2018]

Source: Cyprus IMO TST 2018, Problem 3

July 1, 2018
inequalities

Problem Statement

Find all triples (α,β,γ)(\alpha, \beta, \gamma) of positive real numbers for which the expression K=α+3γα+2β+γ+4βα+β+2γ−8γα+β+3γK = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma} obtains its minimum value.
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