If $$ a,b,c \in R^+ $$ , and $$ a+b+c=1 $$ , prove $$ 1/a+1/b+1/c\geq 9 $$

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If $$ a,b,c \in R^+ $$ , and $$ a+b+c=1 $$ , prove $$ 1/a+1/b+1/c\geq  9 $$

Uniteasy Admin · Accepted Answer

$$ \because a+b+c =1 $$
$$ \dfrac{1}{a} +\dfrac{1}{b}+\dfrac{1}{c} $$
$$ =\dfrac{a+b+c}{a}+\dfrac{a+b+c}{b}+\dfrac{a+b+c}{c} $$
$$ =3+\dfrac{b}{a}+\dfrac{a}{b}+\dfrac{a}{c}+\dfrac{c}{a}+\dfrac{b}{c}+\dfrac{c}{b} $$
$$ \geq 3+2+2+2=9 $$
Only if $$ a=b=c $$ , two sides of the inequality are equal.

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