The Vitals of Mathematics

Algebra Level 5

a+b+c=λa+b+c=\lambda for positive real variables a,b,ca,b,c .
Let S1=cyca2λaS_1=\sum \limits_{cyc} \frac{a^2}{\lambda-a} and S2=cyc1λaS_2=\sum \limits_{cyc} \frac{1}{\lambda-a}.
We define a function f(t)f(t) such that f(S2)=S1f(S_2)=S_1. If f(t)=9 f'(t) = 9 for all values of t t in the domain, determine the value of λ \lambda .

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