The positive real numbers a, b, c satisfy: aligned & (a)/(2 b+1)+(2 b)/(3 c+1)+(3 c)/(a+1)=1 & (1)/(a+1)+(1)/(2 b+1)+(1)/(3 c+1)=2 aligned What is the value of (1)/(a)+(1)/(b)+(1)/(c) ?

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Correct answer: 12

(3 c)/(a+1)+(a)/(2 b+1)+(2 b)/(3 c+1)=1 ......(i) (1)/(a+1)+(1)/(2 b+1)+(1)/(3 c+1)=2 ......(ii) Add (i) and (ii) (3 c+1)/(a+1)+(a+1)/(2 b+1)+(2 b+1)/(3 c+1)=3 aligned & A.M. ≥ G.M. & (((3 c+1)/(a+1)+(a+1)/(2 b+1)+(2 b+1)/(3 c+1))/(3)) ≥(1)^ (1)/(3) & (3 c+1)/(a+1)=1 and (2 b+1)/(3 c+1)=1 & (a+1)/(2 b+1)=1 & (a+1)=(2 b+1)=(3(+1))=k & (1)/(k)+(1)/(k)+(1)/(k)=2 (3)/(k)=2 k=(3)/(2) & a=(1)/(2), b=(1)/(4), c=(1)/(6) & (1)/(a)+(1)/(b)+(1)/(c)=2+4+6=12 aligned

The positive real numbers a, b, c satisfy: aligned & (a)/(2 b+1)+(2...

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