Simple Inequality 3

Algebra Level 3

a,ba,b and cc are three positive real numbers such that a+b+c=na+b+c=n.

What is the smallest possible value of nn such that

(ab+bc+ca)(a2b+3c+b2c+3a+c2a+3b)20(ab+bc+ca)\left(\dfrac{a}{2b+3c}+\dfrac{b}{2c+3a}+\dfrac{c}{2a+3b}\right)\ge 20

for all possible ordered triplets (a,b,c)?(a,b,c)?

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