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If-x-y-z-gt-1-show-that-1-x-2-1-y-z-2-1-y-2-1-z-x-2-1-z-2-1-x-y-2-2-




Question Number 108270 by Muhsang S L last updated on 15/Aug/20
If x, y, z > −1, show that ((1 + x^2 )/(1 + y + z^2 )) + ((1 + y^2 )/(1 + z + x^2 )) + ((1 + z^2 )/(1 + x + y^2 )) ≥ 2
$$\mathrm{If}\:{x},\:{y},\:{z}\:>\:−\mathrm{1},\:\mathrm{show}\:\mathrm{that}\:\frac{\mathrm{1}\:+\:{x}^{\mathrm{2}} }{\mathrm{1}\:+\:{y}\:+\:{z}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}\:+\:{y}^{\mathrm{2}} }{\mathrm{1}\:+\:{z}\:+\:{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}\:+\:{z}^{\mathrm{2}} }{\mathrm{1}\:+\:{x}\:+\:{y}^{\mathrm{2}} }\:\geqslant\:\mathrm{2} \\ $$
Commented by udaythool last updated on 13/Sep/20
Do you know the proof?  It′s very interesting problem.
$$\mathrm{Do}\:\mathrm{you}\:\mathrm{know}\:\mathrm{the}\:\mathrm{proof}? \\ $$$$\mathrm{It}'\mathrm{s}\:\mathrm{very}\:\mathrm{interesting}\:\mathrm{problem}. \\ $$

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