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[Solved] Real numbers question

 
Arithemetic Basics
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June 14, 2021 9:19 am
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Real numbers a, b, c satisfy  a+b+c=0,    abc=2,   show that at least one of a,b, c is not less than 2.

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June 14, 2021 9:20 am
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Staff 
(@armor)
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Obviously at least one of a, b, c is positive. Without loss of generality, let \small a>0,  then   \small b+c=-a,    \small bc=\frac{2}{a},   that is b, c are the two roots of the quadratic equation \small x^2+ax+\frac{2}{a}=0.   Consider the discriminant    \small \Delta \geq 0\Rightarrow a^2-\frac{8}{a}\geq 0\Rightarrow a^3\geq 8\Rightarrow a\geq 2.

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